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C H A P T E R
11 Quadratic relations What you will learn 11.1 Key features of quadratic graphs 11.2 Plotting quadratic relations of the form y ax 2 11.3 Quadratic relations of the form y x 2 c 11.4 Quadratic relations of the form y (x h)2 11.5 Quadratic relations of the form y a(x h)2 k, where a 1 (turning point form) 11.6 The Null Factor Law 11.7 Sketching quadratic relations by finding the x- and y-intercepts 11.8 Applications of quadratic relations
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Parkes radio telescope To monitor the radio waves emitted by objects in space astronomers use radio telescopes in the shape of a threedimensional parabola called a paraboloid. The giant 64-metre Parkes radio telescope in New South Wales was built in 1966. When it is linked with the seven other smaller radio telescopes in NSW together they act like a giant radio telescope 300 km across. Continual upgrades have kept the telescope at the forefront of radio astronomy; it is ten thousand times more sensitive now than when it was built. The Parkes radio telescope is used to examine a wide range of radio waves from our galaxy and other parts of the universe. Its reflecting metal surface, in the shape of a parabola, concentrates the radio waves into a receiver. In 1969, NASA used the Parkes radio telescope as a backup for the Apollo 11 moon landing as outlined in the Australian movie The Dish. Pictures of the historic moonwalk came via the Parkes telescope. The telescope has also played a part in many other space explorations like the Voyager II encounter of Neptune in 1989 and the Mars missions in 2004. The parabolic shapes used in radio telescopes are the graphs of quadratic relations.
VELS Structure Students apply the algebraic properties (closure, associative, commutative, identity, inverse and distributive) to computation with number, to rearrange formulas, rearrange and simplify algebraic expressions involving real variables. Students verify the equivalence or otherwise of algebraic expressions. Students identify and represent quadratic functions by table, rule and graph. Students recognise and explain the roles of the relevant constants in the relationships f (x) a(x b)2 c, with reference to gradient and y-axis intercept. Students solve equations of the form f (x) k, where k is a real constant using algebraic, numerical and graphical methods.
Working mathematically Students choose, use and develop mathematical models and procedures to investigate and solve problems set in a wide range of practical, theoretical and historical contexts. Students select and use technology in various combinations to assist in mathematical inquiry, to manipulate and represent data, to analyse functions and carry out symbolic manipulation.
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S
et
llshe ki
1
CH E
R
T
EA
Do now
For the linear relation y 2x 3 draw up and complete a table of values using 2 x 2.
2
If y x2 3x 2, find the value of y when: a
3
4
a
x0
b
d
x 1
e
x 3
x 2 1 x 2
c
x5
f
x
c f i l
3x(2x 5) (3x 1)(2x 2) (2x 1)(2x 1) (2x 1)2
c
2x2 6x
c f
9 x2 a2 b2
1 2
x(x 2) (x 1)(x 2) (x 2)(x 2) (x 3)2
b e h k
2x(4x 1) (x 5)(x 3) (3 x)(3 x) (x 5)2
x2 2x
b
x2 3x
Factorise using the difference of two perfect squares. x2 4 1 x2
b e
x2 16 25 x2
Factorise by taking out a common factor then using the difference of two perfect squares. a
8
c
Factorise by taking out a common factor.
a d 7
x0
Expand the following.
a 6
b
Find the value of x(x 2) when:
a d g j 5
x1
2x2 18
b
3x2 12
c
4x2 16
c f
x2 4x 21 x2 12x 36
Factorise each of the following quadratic trinomials. a d
x2 7x 12 x2 5x 6
b e
x2 x 6 x2 6x 9
Answers 1
x
x 1 2
xy
x 5 3 1 7
0
1
2 1
2 a 0 b 2 c 20 3 a 0 b 8 c 15 d 3 e 34 f 54 4 a x2 2x b 8x2 2x c 6x2 15x d x2 3x + 2 e x2 8x + 15 f 6x2 4x 2 g x2 4x 4 h 9 x2 i 4x2 1 j x2 6x 9 k x2 10x 25 l 4x2 4x 1 5 a x(x 2) b x(x 3) c 2x(x 3) 6 a (x 2)(x 2) b (x 4)(x 4) c (3 x)(3 x) d (1 x)(1 x) e (5 x)(5 x) f (a b)(a b) 7 a 2(x 3)(x 3) b 3(x 2)(x 2) c 4(x 2)(x 2) 8 a (x 3)(x 4) b (x 2)(x 3) c (x 3)(x 7) d (x 3)(x 2) e (x 3)2 f (x 6)2
398
Essential Mathematics VELS Edition Year 9
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Key features of quadratic graphs
Relations that have rules in which the highest power of x is 2, such as y x2, y 2x2 3 or y 3x2 2x 4, are called quadratics and their graphs are called parabolas. Parabolic shapes can be seen in many practical applications such as the arches of bridges, the paths of projectiles and the surface of reflectors.
Key ideas The graph of a quadratic relation is called a parabola. Its basic shape is as shown here. The basic quadratic relation is y x2.
The general equation of a quadratic is:
axis of symmetry
vertex (maximum turning point)
vertex (minimum turning point)
axis of symmetry
y ax bx c 2
A parabola is symmetrical about a line called the axis of symmetry and it has one turning point (TP), which may be a maximum or a minimum.
y Here is an example of a quadratic graph with equation y x2 4x 3, showing all the key features.
y-intercept 3 x-intercepts
O
1 (2, −1)
2
x 3 4 turning point (2, –1)
axis of symmetry: x = 2
Chapter 11 — Quadratic relations
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Example 1
y
For the following graph state the: a b c
9
equation of the axis of symmetry type of turning point (a maximum or a minimum) coordinates of the turning point (TP)
6 3 x
–2 –1 0 −3
Solution
a b c
x2 minimum turning point TP (2, 3)
1 2 3 4 5
Explanation y
9 6 3 −2 −1 0 −3
x 1
2 3 4 5
minimum turning point (2, –3) axis of symmetry: x = 2
Example 2
y 5
For this graph state the: a b c d e
equation of the axis of symmetry type of turning point coordinates of the turning point x-intercept y-intercept Solution
a b c d e
x 3 maximum turning point TP (3, 5) x-intercepts: (5, 0) and (1, 0) y-intercept: (0, 5)
Explanation y highest point: TP = (−3, 5) 5
x-intercept −6 −5 −4 −3 −2 −1 0
axis of symmetry: x = −3
Essential Mathematics VELS Edition Year 9
1
−5
−5
400
x
−6 −5 −4 −3 −2 −1 0
x 1
y-intercept
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Exercise 11A Example
1
1
For each of the following graphs state the: i ii iii a
equation of the axis of symmetry type of turning point (a maximun or a minimum) coordinates of the turning points y y b c
0
y
d
4 3 2 1
x
−2 0 −4
x
2
x
0 −1
y
e
y
1
3
g
1
−4
5
3
1
x
−1 0
x
h
y
x
−1 0
−2 0
4
y
f
2
4
–1
2
i
y
y
6 x
0
–1 0
–3
x −5
Example
2
2
2 0
1 −3
x
For each of the following graphs state: i ii iii iv v a
the equation of the axis of symmetry the coordinates of the turning point (TP) the type of turning point (maximun or minimum) the x-intercepts the y-intercept b y y
c
y 3
3
−2
d
2
−1
x
0
1
−4
x
−1 0 1 2 3 4 −1
0
x
y
e
y
y
f 7
4
6 3
−2 −1 0
x 1
2
−1 0 1 2 3 4
x
−4 −2 −1 0 2
x
Chapter 11 — Quadratic relations
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y
The graph shows the height of a golf ball, y metres, as a function of time t seconds. a b
i ii i ii iii
12 10
At what times is the ball at a height of 9 m? Why are there two different times? At what time is the ball at its greatest height? What is the greatest height the ball reaches? After how many seconds does it hit the ground?
8 6 4 2 t
−1 O
1
2
3
4
5
h
4
The graph gives the height, h m, at time t seconds, of a rocket which is fired straight up in the air. a b c d
160 140 120
100 From what height is the rocket 80 launched? 60 What is the maximum height that 40 the rocket reaches? 20 For how long is the rocket in the −1 O air? Compare the time the rocket is going up and the time it is going down.
t 1
2
3
4
5
Enrichment Th
5
Follow the steps below to construct a parabola using a dynamic geometry package. a b
c
d e f g h i
402
y
Show the coordinate axes system by selecting P Show Axes from the Draw toolbox. D i Construct a line which is parallel to the x x-axis and passes through a point F on the y-axis near the point (0, 1). A F C B ii Construct a line segment AB on this line as shown in the diagram. Hide the line AB and then construct: i a point C on the line segment AB ii a point P on the y-axis near the point (0, 1) Construct a line which passes through the point C and is perpendicular to AB. Construct the point D which is equidistant from point P and segment AB. Hint: Use the perpendicular bisector of PC. Select Trace from the Display toolbox and click on the point D. Animate point C and observe what happens. Select Locus from the Construct toolbox and click at D and then at C. Drag point P and/or segment AB (by dragging F ). (Clear the trace points by selecting Refresh drawing from the Edit menu.) What do you notice?
Essential Mathematics VELS Edition Year 9
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Plotting quadratic relations of the form y ax 2
Altering the value of a in y ax2 introduces two transformations: dilation and reflection. The dilation works parallel to the y-axis and the reflection is in the x-axis. y The graph of the quadratic relation y x2 can be plotted from a table of values. y = x2 4 x 2 x 1 0
1
2
xy
1
4
4x
1
0
3 2 1
The graph passes through the key points (1, 1) (0, 0) and (1, 1).
−2
−1
The graph of y x2 can be obtained by reflecting the graph of y x2 in the x-axis. x 2 1 0
1
y 4 1 0 1 4
x 1
2
1
2
y 1 0 −2
2
0
−1
−1
x
−2
The graph passes through the key points (1, 1) (0, 0) and (1, 1).
−3 −4
Key ideas In general, for quadratic relations of the form y ax2: • the graph of y x2 is dilated by a factor of a parallel to the y-axis (from the x-axis) • the turning point is (0, 0) • the axis of symmetry is at x 0 • the x- and y-intercepts are at (0, 0) If a > 1 or a < 1:
• If 1 < a < 1:
the graph is narrower than y x
2
e.g. y 2x
2
y
0
the graph is wider than y x2 1 2 e.g. y 2x y
y = 2x2 y = x2
x
0
y = x2 y=
1 2
x2
x
Chapter 11 — Quadratic relations
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Example 3 1 2 x. 2 Draw up and complete a table of values for 2 x 2. Plot their graphs on the same set of axes. List the key features of the graphs such as the axis of symmetry, turning point, x-intercept and y-intercept. i Is the graph of y 2x2 wider or narrower than y x2? 1 ii Is the graph of y x2 wider or narrower than y x2? 2
Complete the following for y x2, y 2x2 and y a b c d
Solution
a
Explanation
y x2
y 2x2
y
1 2 x 2
1
2
xy
1 0
1
4
x 2 x 1 0
1
2
y
0
2
8
Substitute each x value into y 2x2. e.g. If x 2, y 2(2)2 2(4) 8
x 2 x 1 0
1 1 2
2
Substitute each x value into y
2
1 e.g. If x 2, y (2)2 2 2 Plot the coordinates for each graph and and join them with a smooth curve.
xy
8 7 6 5 4 3 2 1 −2 −1 −1 0 1
c
d
404
x4
8
2
x2 1
0
2
y
b
Substitute each x value into y x2. e.g. If x 2, y 22 4
x 2 x 1 0
y = 2x2
1 2 x. 2
y = x2 y=
1 2
x2 x
2
axis of symmetry: y-axis (x 0) turning point: minimum at (0, 0) x-intercept: (0, 0) y-intercept: (0, 0) i The graph of y 2x2 is narrower than the graph of y x2. 1 ii The graph of y x2 is wider 2 than the graph of y x2.
Essential Mathematics VELS Edition Year 9
Detemine the key features by looking at the graphs.
For each value of x, 2x2 is twice that of x2. 1 For each value of x, x2 is half that of x2. 2
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Exercise 11B Example
3
1
2
1 Complete the following for y x2, y 3x2 and y x2. 3 a Draw up and complete a table of values for 2 x 2. b Plot their graphs on the same set of axes. c List the key features of the graphs such as the axis of symmetry, turning point, x-intercept and y-intercept. d i Is the graph of y 3x2 wider or narrower than y x2? 1 ii Is the graph of y x2 wider or narrower than y x2? 3 For the equations given below: i ii iii iv a
3
Draw up and complete a table of values for 2 x 2. Plot the graphs of the equations on the same set of axes. List the key features of each graph. Determine whether the graphs of the equations are each wider or narrower than the graph of y x2. 1 1 y 4x2 y x2 b y 5x2 c d y x2 4 5
For the equations given below: i ii iii iv a
Draw up and complete a table of values for 2 x 2. Plot the graphs of the equations on the same set of axes. List the key features for each graph, such as the axis of symmetry, turning point, x-intercept and y-intercept. Describe how the graph of y x2 has been transformed to obtain each of the graphs. 1 1 y 2x2 y x2 b y 3x2 c d y x2 2 3
4
For each of the following quadratics, state whether the graph will be wider or narrower than the graph of y x2. 1 a y 6x2 b y 7x2 c y 4x2 d y x2 9 2 x e f y 0.3x2 y g y 4.8x2 h y 0.5x2 7
5
Which of the graphs in Question 4 involves a reflection in the x-axis?
6
Match each of the following parabolas with the appropriate equation from the list below. i iv
y 3x2 1 y x2 2
ii
y x2
iii
y 5x2
v
y 5x2
vi
y 2x2
Chapter 11 — Quadratic relations
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y
a
y
b
5
4 3 2 1
4 3 2 1 −2 −1 0
1
2
−2 −3
x 1
−4
2
y
e
4 3 2 1 −2 −1 0
x 1
2
y
d
−2 −1 0 −1
−2 −1 0
x
y
c
f
1
y −1
0
x 1
0.5 x 1
2
−1
0
x
−5
1
Enrichment Th
406
7
Determine the equation of a quadratic relation if it has an equation of the form y ax2 and passes through: a (1, 4) b (1, 7) c (1, 1)
8
This photo shows the parabolic cables of the Golden Gate bridge. The rule of the form y ax 2 describes the shape of the parabolic cables. If the cable at the top of the pylon has the coordinates (983,67) find a possible equation that describes this shape.
9
Investigate photos of other bridges that have parabolic arcs. Can you come up with quadratic equations that match the arcs?
Essential Mathematics VELS Edition Year 9
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E NDI
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Using technology to construct tables of values and draw graphs
X
A
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Example: For the quadratic relation y x2 use technology to: a construct a table of values for 3 x 3 b draw a graph TI 84 family
TI 89 family
Press Y . Type x2 into Y1.
Press ◆ Y . Type x2 into y1.
Press 2nd TBLSET. Make tblStart 3 and ¢ Tbl 1.
Press ◆ TBLSET. Make tblStart 3 and ¢ Tbl 1.
Press 2nd TABLE and scroll up or down to view other ordered pairs.
Press ◆ TABLE and scroll to see the values.
Press GRAPH.
Press ◆ GRAPH.
Exercise 1 For each of the following quadratic relations use technology to: i construct a table of values for 3 x 3 ii draw a graph 1 a y 2x2 b y 4x2 c y x2 2 1 2 f y 4x2 d y x e y 2x2 3
Chapter 11 — Quadratic relations
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Using technology to compare graphs of the form y ax2
E NDI
X
A
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Example: a b
1 Use technology to sketch the quadratic relations y x2, y 2x2, y 3x2, y x2 on the 2 same axes. Describe how each graph transforms the graph of y x2.
TI 84 family
Press Y . Type y1 e 1, 2, 3,
TI 89 family
Press ◆ Y . 1 2 fx 2
y1 e 1, 2, 3,
1 2 fx 2
Press GRAPH.
Press ◆ GRAPH.
Press TRACE and scroll up and down to compare the graphs.
Press TRACE and scroll up and down to compare the graphs.
The graph of y 2x2 is narrower than the graph of y x2. 1 The graph of y x2 is wider than the graph of y x2. 2 Exercise 1 For each set of quadratics: i Use technology to sketch each set of quadratics on the same axes. ii Describe how each changes the graph of y x2. a y x2, y 2x2, y 3x2, y 4x2 1 1 1 b y x2, y x2, y x2, y x2 2 3 4 2 2 2 c y x , y x , y 2x , y 3 x2, y 4x2
408
Essential Mathematics VELS Edition Year 9
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Quadratic relations of the form y x 2 c y
Quadratic relations can be transformed by translating (shifting) their graphs vertically up and down the y-axis.
O
x
Key ideas In general, quadratic functions of the form y x2 c can be obtained by translating the graph of y x2 parallel to the y-axis.
c0 The graph is translated by c units up.
c0 The graph is translated by c units down.
y y 4 3 2
3 2 1
1 –2
–1 0
x 1
–2
2
Turning point Axis of symmetry y-intercept
(0, c) x0 (0, c)
–1 0 –1
x 1
2
Example 4 For y x2, y x2 1 and y x2 1: a b c d
Draw up and complete a table of values for 2 x 2. Plot their graphs on the same set of axes. List the key features of y x2 1 and y x2 1. Describe how y x2 1 and y x2 1 can be obtained from the graph of y x2.
Solution
a
Explanation
y x2 x y
2 4
1 1
0 0
1 1
2 4
y x2 1 x y
2 5
1 2
0 1
1 2
2 5
1 0 0 1
1 0
2 3
y x2 1 x y
2 3
Substitute each x value into y x2. e.g. If x 2, y (2)2 4 Substitute each x value into y x2 1. e.g. If x 2, y (2)2 1 41 5 Substitute each x value into y x2 1. e.g. If x 2, y (2)2 1 41 3 Chapter 11 — Quadratic relations
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y
b
–2
d
Plot the coordinates for each graph and join them with a smooth curve.
y = x2 + 1 5 4 3 2 1
c
Page 410
–1 –1 0
y = x2 y = x2 – 1
x 1
2
Key features Axis of symmetry (all graphs): y-axis (x 0) y x2 1 TP: minimum at (0, 1) x-intercepts: none y-intercept: (0, 1) y x2 1 TP: minimum at (0, 1) x-intercepts: (1, 0) and (1, 0) y-intercept: (0, 1) The graph of y x2 1 can be obtained by translating the graph of y x2 one unit up. The graph of y x2 1 can be obtained by translating the graph of y x2 one unit down.
Detemine the key features by looking at the graph.
c = +1
c = 1
Exercise 11C Example
4
1
For y x2, y x2 2 and y x2 2: a b c d
2
For each of these equations: i ii iii a
3
Plot the graphs of the equations on the same set of axes. List the key features of each graph. Describe how the graphs can be obtained from the graph of y x2. y x2 4 b y x2 4 c y x2 5 d y x2 5
For each set of quadratic relations: i ii iii
410
Draw up and complete a table of values for 2 x 2. Plot their graphs on the same set of axes. List the key features of y x2 2 and y x2 2. Describe how y x2 2 and y x2 2 can be obtained from the graph of y x2.
Use a graphics or CAS calculator to sketch the graphs on one set of axes. Determine whether the graphs have been translated up or down. Determine which of the graphs are a reflection of y x2 in the x-axis.
Essential Mathematics VELS Edition Year 9
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Give the coordinates of the turning point, y-intercept and the equation of the axis of symmetry. 1 y x2, y x2 5, y x2 6, y x2 7, y x2 2 1 2 y x2, y x2 3, y x2 4, y x2 3, y x2 4 y x2, y 3 x2, y 4 x2, y 3 x2, y 4 x2 y x2, y x2 5, y x2 6, y x2 7, y x2
For the each of the following quadratic relations: i ii iii a d
5
6/6/06
Determine whether the graphs have been translated up or down. Determine which of the graphs involve a reflection in the x-axis. Give the coordinates of the turning point. y x2 5 b y x2 7 c y 8 x2 2 2 y x 3.6 e y 7.2 x f y x2 0.02
Match each of the following parabolas with the appropriate equation from the list below. y x2 4, y x2 4, y 4 x2, y x2 3, y x2 3, y x2 3 y y y a b c 0
8 6 4
x
0
–2
2
0
–12
–4 –3 –2 –1
y
e
4 2 –4 –2 0 –2 –4
3
4
y
d
9 6
–9
2 –4
12
x 1 2 3 4
–4 –3 –2 –1 –3 –6
x
0 –4 –2 –2 –4
3 0 x 2 4
x
y
f
4 2 2 4
1 2 3 4
x
–4 –3–2 –1 1 2 3 4 –3 –6 –9
Enrichment Th
6
Find the equation of the quadratic relation which is of the form y x2 c and passes through: a
7
(1, 4)
b
(3, 5)
c
(2, 1)
d
(2, 1)
Find the equation of the quadratic relation which is of the form y 2x2 c and passes through: a
(1, 3)
b
(1, 3) c
(3, 15)
d
(2, 6)
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Solution
11.4
Quadratic relations of the form y (x h)2
Quadratic relations can be transformed by translating (shifting) their graphs to the left or right.
Key ideas In general, the graph of y (x h)2 can be obtained by translating the graph of y x2 left or right.
h0 Translates the graph of y x 2 h units to the right. y
y = x2
h0 Translates the graph of y x 2 h units to the left. y
y = (x – 1)2
5 4 3 2 1
y = (x + 1)2
x
–2 –1 –1 0 1
2
y = x2
4 3 2 1
–3 –2 –1 –1 0 1
3
x 2
Example 5 a b c
Plot the graph of y x2 for 2 x 2 and the graph of y (x 1)2 for 1 x 3 on the same set of axes. List the key features of y (x 1)2. Describe how y (x 1)2 can be obtained from the graph of y x2. Solution
a
Explanation
y x2 x 2 x 1 0
1
2
xy
1
4
x4 1
0
Substitute each x value into y x2. e.g. If x 2, y (2 )2 4
y (x 1)2 x 1 x 0
1
2
3
xy
0
1
4
x4 1 y
y = x2
4
y = (x – 1)2
3 2 1 –2
412
–1
0 –1
x 1
2
3
Essential Mathematics VELS Edition Year 9
Substitute each x value into y (x 1)2. e.g. If x 2, y (2 1)2 (3)2 9 Plot the coordinates for each graph and and join them with a smooth curve.
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b
Axis of symmetry: x 1 TP: minimum at (1, 0) x-intercept: (1, 0) y-intercept: (0, 1)
c
The graph of y (x 1)2 can be obtained by translating the graph of y x2 one unit to the right.
Detemine the key features by looking at the graph.
Exercise 11D Example
5
1
a
Plot the graph of y x2 for 2 x 2 and the graph of y (x 1)2 for 3 x 1 on the same set of axes.
b
List the key features of y (x 1)2. Describe how y (x 1)2 can be obtained from the graph of y x2.
c 2
Consider the relations given below. i ii iii a c
3
For each set of quadratic relations: i ii iii a b c
4
Use a graphics or CAS calculator to sketch the graphs on one set of axes. Determine whether the graphs have been translated to the right or left. Give the coordinates of the turning point, y-intercept and the equation of the axis of symmetry. y x2, y (x 1)2, y (x 5)2, y (x 4)2, y (x 3)2 y x2, y (x 1)2, y (x 2)2, y (x 3)2 y x2, y (1 x)2, y (2 x)2, y (3 x)2, y (4 x)2
For each of the following quadratics, state whether the graph of y x2 (or y x2) should be translated to the left or to the right to produce the graph of the given quadratic relation. a d g
5
Plot the graphs of the equations on the same set of axes. List the key features of each graph. Describe how the graphs below can be obtained from the graph of y x2. y (x 2)2 b y (x 2)2 y (x 3)2 d y (x 3)2
y (x 7)2 y (x 6)2 y (6 x)2
b e h
y (x 7)2 y (x 3.5)2 y (4.5 x)2
c f i
y (x 8)2 y (5 x)2 y (1 x)2
Match each of the following parabolas with the appropriate equation from the list below. i iv
y x2 y (x 3)2
ii v
y (x 2)2 y (x 5)2
iii vi
y (x 2)2 y (x 3)2
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y
b
a 4 3 2 1 –4
–3
–2
d
9
–6
e
4 3 2 1 –1 0
–6
x
–1 0
y
–2
x 2
–3
x
0
–3
0
x
–9
y
f
y 4 3 2 1
1
y
c
y
25
x
0
1
2
3
4
0
x 5
10
Enrichment Th
6
Find the equation of each of the following quadratics if their equation is of the form y (x h)2 and their graph passes through the point: a
7
b
(1, 16)
(3, 1)
c
(1, 9)
d
(3, 9)
The photo shows the span of the Sydney Harbour Bridge. Use the points (0, 20) 1 and (100, 0) to find a possible equation of the form y (x b)2, using the 500 given axes. y
x –20
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Quadratic relations of the form y a(x h)2 k, where a 1 (turning point form) Enrichment
If a quadratic relation is expressed in the general form y a(x h)2 k the turning point can be easily be determined.
Key ideas The features below can be used to sketch the graph of a quadratic relation of the form:
y a(x h)2 k
a 1 upright graph a 1 graph reflected in x-axis
Translates the graph up or down:
k0
k0
Translates the graph left or right: h0 h0 Axis of symmetry: x h Turning point (h, k)
y
k
0
turning point (h, k) h
x
Example 6 For each of the following quadratic relations give the coordinates of the turning point of their graphs. a
y (x 3)2 1 Solution
a b c
(3, 1) (0, 1) (0, 0)
b
y x2 1
c
y 5x2
Explanation
The graph of y x2 has shifted 3 units to the right and 1 unit up. The graph of y x2 has shifted 1 unit down. The graph of y x2 has not been translated.
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Example 7 For the graph of the rule y (x 2)2 1: a b c
Describe how the graph can be obtained from that of y x2. State the equation of the axis of symmetry, the coordinates of the y-intercept and turning point. Sketch the graph of the quadratic relation.
Solution
a
b
Explanation
The graph of y (x 2)2 1 is obtained by translating the graph of y x2 2 units to the right and 1 unit down. Axis of symmetry: x 2 TP: minimum at (2, 1) y-intercept: y (0 2)2 1 y3 y-intercept is (0, 3) y
c
y = (x – 2)2 – 1
4
y = (x – 2)2 – 1 2 right
1 down
xh (h, k), h 2, k 1 Let x 0. Write the coordinates. Sketch the graph of y x2 and move key points such as (1, 1), (0, 0) and (1, 1) 2 units to the right and 1 unit down.
3
y
2
4 3
1 –2
–1 0 –1
2
x 1
2
3
4
(2, –1)
(–1, 1) 1
(1, 1)
(0, 0) 0 1 2 3 4 –1
x
(1, 1) S (1 2, 1 1) (1, 0) (0, 0) S (0 2, 0 1) (2, 1) (1, 1) S (1 2, 1 1) (3, 0)
Example 8 Write the equation that results by performing the following transformations on y x2: reflection in the x-axis and translation of 4 units to the left and 2 units down. Solution
y a(x h)2 k y (x 4)2 2
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Essential Mathematics VELS Edition Year 9
Explanation
a 1, h 4 and k 2 Note: x (4) x 4
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Exercise 11E Example
6
1
For each of the following quadratic relations give the coordinates of the turning point of their graphs. a d g j m p
2
y (x 1)2 4 y (x 1)2 1 y 3x2 y x2 1 y (x 2)2 y 2 (x 5)2
b e h k n q
y (x 3)2 5 y (x 2)2 1 y 4x2 y x2 2 y (x 3)2 y 5 (x 7)2
c f i l o r
State the equation of each of the following given it is of the form y (x h)2 k or y (x h)2 k. a
b
y
c
y
9
6
6
3
3 2 –2 –1
1 01 2 3 4
x
x
0
y
d
2
–2 –1 0 –3
4
y
e
9
–6 –5 –4 –3 –2 –1
3
y
3 –3 –2 –1 –1
01
–2 –1
x
0
0 1
x 2
x 1
2
3
4
–3
–3 –4
For each of the following quadratic relations: i ii iii a d g j
4
x 1 2 3 4 5 6
f
5
7
y 9
5
Example
y (x 3)2 6 y (x 3)2 2 y 2x2 y 1 x2 y (x 1)2 y 7 ( x 2)2
Describe how the graph can be obtained from that of y x2. State the equation of the axis of symmetry, the coordinates of the y-intercept and turning point. Sketch the graph of the quadratic relation. y (x 2)2 1 b y (x 1)2 1 c y (x 2)2 1 2 2 y (x 3) 2 e y (x 4) 3 f y (x 3)2 1 y 3 ( x 2)2 h y 1 ( x 2)2 i y 4 ( x 3)2 2 2 yx 5 k y (x 2) l y 6 x2
For each set of quadratic relations use a graphics or CAS calculator to sketch the graphs on one set of axes. a b c d
i i i i
y (x 1)2 2 y (x 3)2 1 y (x 1)2 y 9 x2
ii ii ii ii
y (x 2)2 1 y (x 1)2 5 y (x 2)2 y 1 x2
iii iii iii iii
y (x 1)2 3 y (x 2)2 1 y (x 3)2 y 3 x2
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For each of the following quadratic relations, sketch the graph and show key features such as the y-intercept and turning point. a d g j m
Example
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y x2 1 y (x 2)2 y (x 1)2 1 y (x 4)2 2 y 2 (x 2)2
b e h k n
y x2 5 y (x 3)2 y (x 2)2 1 y (x 1)2 3 y 5 (x 4)2
c f i l o
y x2 2 y (x 1)2 y (x 3)2 2 y (x 2)2 1 y 3 (x 1)2
Write the equations that result by performing the following transformations on y x2. a b c d e f g
Reflection in the x-axis and translation of 1 unit left Reflection in the x-axis and translation of 2 units to the right Translation of 2 units to the left Translation of 3 units to the right Translation of 4 units to the left and 1 unit down Translation of 2 units to the right and 4 units down Reflection in the x-axis then translation of 1 unit to the left and 3 units up
Enrichment Th
7
A bike track can be approximately modelled by combining two different quadratic equations. The first part of the bike path can be modelled by the equation y 9 (x 2)2 for 2 x 5 The second part of the bike track can be modelled by the equation y (x 7)2 4 for 5 x 10 a b
c
418
Find the turning point of the graph of each quadratic equation. Sketch each graph on the same set of axes. On your sketch of the bike path you need to show the coordinates of the start and finish of the track and where it crosses the x-axis. Investigate how to sketch both graphs for the given x values on the same graph by using a graphics or CAS calculator.
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Using technology to find turning points
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A graphics or CAS calculator can be used to find the turning point of the graph of a quadratic relation. Example: For y x2 7x 4 use technology to find the turning point correct to two decimal places. TI 84 family
TI 89 family
Press Y. Type x2 7x 4 into Y1.
Press ◆ Y. Type x2 7x 4 into y1.
Press GRAPH and check to see if the WINDOW is set to the correct ordered pairs.
Press ◆ GRAPH and check to see if the WINDOW is set to the correct ordered pairs.
Press 2nd CALC and select minimum. Press ¸.
Press F5 and select Minimum. Press ¸.
Scroll to the left of the turning point and press ¸.
Scroll to the left of the turning point and press ¸.
Scroll to the right of the turning point and press ¸.
Scroll to the right of the turning point and press ¸.
Scroll close to the turning point for a guess and press ¸.
The minimum turning point will be displayed.
The turning point is (3.5, 8.25).
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Exercise 1
Use technology to find the turning point of each of the following correct to two decimal places. a d g j
2
420
y x2 5x 9 y x2 6x 9 y 3x2 x 1 y 3x2 12x 12
b e h k
y x2 5x 7 y x2 4x 5 y 2x2 x 4 y 4x2 9x 10
c f i l
y x2 2x 3 y x2 7x 8 y 6x2 18x 5 y 42 36x 100x2
Use technology to determine the turning point and x-intercepts of each of the following quadratic relations correct to two decimal places. a
y 0.5x2 5x 1
d
y
x2 x 6 20 4
b
1 y x2 8x 9 4
e
y
Essential Mathematics VELS Edition Year 9
3x2 x 12 10 5
c
y 2x2 0.5x 0.1
f
y
x2 x 6 3 2
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The Null Factor Law
Any number multiplied by zero is zero, for example 3 0 0. This leads to the Null Factor Law, which states that if the product of two numbers is zero then one or both numbers must be zero. That is, if ab 0 than a 0, b 0 or both a and b equal 0. Quadratic equations of the general form ax2 bx c 0 can be solved using the Null Factor Law. This then leads to an alternative method for sketching quadratic relations.
Key ideas Null Factor Law (NFL): If a b 0 then a 0 or b 0 or both a and b zero.
Example 9 Solve each of following equations. x (x 1) 0
a
b
(x 3)(x 2) 0
Solution
a
b
c
c
3(1 x )(2x 7) 0
Explanation
x (x 1) 0 x 0 or x 1 0 x 0 or x1 (x 3)(x 2) 0 x 3 0 or x 2 0 x 3 or x2 3(1 x )(2x 7) 0 1 x 0 or 2x 7 0 x 1 or 2x 7 7 x 1 or x 2
Let each factor equal 0. Note: x is one of the factors. Solve for x. Let each factor equal 0. Solve for x. Let each factor equal 0. Since 3 is a constant it can be ignored. Solve for x.
Example 10 Solve the following equation: (x 2)2 0 Solution
Explanation
(x 2) 0 (x 2) (x 2) 0 x20 x2 2
(x 2)2 is the same as (x 2) (x 2). Since the factors are equal we only need to let one of them equal 0. Solve for x.
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Example 11 Solve each of the following equations. x2 4x 0
a
2x2 8x
b
Solution
Explanation
x2 4x 0 x (x 4) 0 x 0 or x 4 0 x 0 or x 4 2x2 8x 2x2 8x 0 2x (4 x ) 0 2x 0 or 4 x 0 x 0 or x 4
a
b
Factorise by taking out the common factor x. Let each factor equal 0. Note: x is one of the factors. Solve for x. Make the RHS equal to zero. Factorise by taking out the common factor of 2x Let each factor equal 0. Solve for x.
Example 12 Solve each of the following equations. x2 9 0
a
b
x2 25
Solution
a
b
c
c
x2 5x 6 0
Explanation
x2 9 0 (x 3)(x 3) 0 x 3 0 or x 3 0 x 3 or x 3 2 x 25 x2 25 0 (x 5)(x 5) 0 x 5 0 or x 5 0 x 5 or x 5 x2 5x 6 0 (x 3)(x 2) 0 x 3 0 or x 2 0 x 3 or x2
Factorise by using the difference of perfect squares. Let each factor equal 0. Solve for x. Make the RHS equal to zero. Factorise by using difference of perfect squares. Let each factor equal 0. Solve for x. Factorise. Let each factor equal 0. Solve for x.
Exercise 11F Example
9
1
Solve each of the following equations. a d g
422
x (x 1) 0 (x 2)(x 1) 0 2x (x 3) 0
b e h
x (x 2) 0 (x 3)(x 4) 0 5x (x 1) 0
Essential Mathematics VELS Edition Year 9
c f i
x (x 5) 0 (x 5)(x 2) 0 3x(x 2) 0
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j m p Example
10
2
11a
3
11b
4
12a
5
12b
6
12c
7
l o r
3(x 2)(x 9) 0 (3x 1)(x 5) 0 (2x 9)(3x 1) 0
(x 1)2 0 (x 7)2 0 (1 x)(1 x) 0 (4x 3)2 0 (6 5x)2 0
b e h k n
c f i l o
(x 2)2 0 (1 x )2 0 2(3 x)(3 x) 0 (2x 5)2 0 (2 5x)2 0
x2 6x 0 x2 4x 0 2x2 4x 0 x2 x 0
c f i l
x2 5x 0 x2 12x 0 3x2 21x 0 3x2 12x 0
6x2 3x 5x2 45x 5x2 0 9x x2
c f i l
x2 4x 7x2 49x x2 1 5x 2x2
x2 1 0 36 x2 0 4x2 9 0 2x2 18 0
c f i l
x2 16 0 1 x2 0 25x2 36 3x2 12 0
x2 9 x2 36 4x2 9
c f i
x 2 16 x2 4 16x2 25
x2 6x 9 0 x2 8x 16 0 x2 14x 49 0 x2 10x 11 0 x2 5x 50 0 2x2 22x 20 0 5x2 10x 40 0
c f i l o r u
x2 10x 25 0 x2 6x 9 0 x2 18x 81 0 x2 x 20 0 x2 x 12 0 3x2 18x 24 0 2x2 2x 12 0
x2 1 2x x2 8x 16 3x2 3x 126 0 x2 10x 11 x2 15x 50
c f i l o
x2 6x 9 x2 6x 9 0 x2 5x x2 x 20 x2 9x 18
(x 1)2 0 (2 x )2 0 (5 x)(5 x) 0 (2x 1)2 0 (5 3x)2 0
x2 4x 0 x2 10x 0 2x2 10x 0 6x x2 0
b e h k
4x2 2x x2 3x x2 25 5x x2
b e h k
x2 4 0 100 x2 0 16x2 25 0 3x2 27 0
b e h k
x2 4 x2 25 4x2 1
b e h
Solve the following equations. a d g j m p s
8
5(x 2)(x 7) 0 (x 2)(2x 3) 0 (4x 5)(5x 3) 0
Solve the following equations. a d g
Example
k n q
Solve the following equations. a d g j
Example
2(x 2)(x 1) 0 (2x 1)(x 3) 0 (3x 4)(2x 3) 0
Solve the following equations. a d g j
Example
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Solve each of the following equations. a d g j
Example
3:22 AM
Solve each of the following equations. a d g j m
Example
6/6/06
x2 2x 1 0 x2 4x 4 0 x2 8x 16 0 x2 12x 36 0 x2 15x 50 0 2x2 8x 6 0 4x2 8x 4 0
b e h k n q t
Solve the following equations. a d g j m
x2 12x 36 x2 7x x2 7x 12 x2 12x 35 x2 3x
b e h k n
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Enrichment 9
Th
A block of land is in the shape of a right-angled triangle. The hypotenuse is 2 metres longer than the shortest side. The third side is 1 metre longer than the shortest side. a b c d
10
A rectangular swimming pool has the dimensions shown in the diagram. a b c d
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x+3 Write an expression for the area in terms of x. 2 Given that its surface area is 20 m write a quadratic equation of the form x2 bx c 0. x+2 Use the Null Factor Law to help you find the dimensions of the pool. Investigate other pool dimension combinations (e.g. x 1 and x 4) and surface areas that allow you to use the Null Factor Law to find the exact pool dimensions.
Using technology to solve quadratic equations
X
A
Draw a diagram. Use Pythagoras’ theorem to find a relationship between the side lengths. Express the relationship you found in part b in the form x2 bx c 0. Use the Null Factor Law to help you find the length of each side.
A graphics or CAS calculator can be used to solve quadratic equations. Example: Use technology to solve x2 8x 20 0. TI 84 family CAS screens
TI 89 family
From the MATH Menu select 0:Solver . . . and type in the equation.
Press F2 and select 1:solve(.
Press ALPHA. Solve. Change the bounds to get a second solution or choose a different guess for x.
Type (x2 8x 20 0, x) and press ¸.
Exercise 1 Solve using technology: a x2 2x 1 0 d x2 8x 6 0 g x2 x 5 0
424
b e h
x2 2x 1 0 x2 7x 2 0 4x2 3x 5 0
Essential Mathematics VELS Edition Year 9
c f i
x2 3x 1 0 x2 x 9 0 x2 x 1 0
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Sketching quadratic relations by finding the x- and y-intercepts
An alternative method of sketching the graph of a quadratic relation requires us to find the x-intercepts and y-intercept first.
Key ideas All parabolas have one y-intercept but they may have two, one or zero x-intercepts.
Two x-intercepts
One x-intercept
Zero x-intercepts
The x-intercepts can be found using the Null Factor Law. If the graph has two x-intercepts the turning point can be found by determining the location of the axis of symmetry which lies halfway between the two x-intercepts (x1 and x2) • calculating the x value of the turning point, the midpoint of x1 and x2, that is:
•
x
x1 x2 2
• calculating the y value of the turning point by substituting the x-coordinate into the rule for the quadratic relation.
Example 13 For each of the following quadratic relations: i ii a
find the x-intercepts find the y-intercept y x( x 1) b
y (x 2)(x 3) Explanation
Solution
a
i
ii b
i
ii
x-intercepts (let y 0): x(x 1) 0 x 0 or x 1 0 x 0 or x 1 y-intercept (let x 0): y 0(0 1) y0 y (x 2)(x 3) has two x-intercepts x-intercepts (let y 0): (x 2)(x 3) 0 x 2 0 or x 3 0 x 2 or x3 y-intercept (let x 0): y (2)(3) y6
Let y 0 to find the x-intercepts. Let each factor equal 0 and solve. Solve for x. Let x 0 to find the y-intercept. There are two different factors. Let y 0 to find the x-intercepts. Let each factor equal 0 and solve. Let x 0 to find the y-intercept.
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Example 14 For each of the following quadratic relations: a b c d e f
Factorise the relation. Find the x-intercepts. Find the y-intercept. Find the axis of symmetry. Find the turning point. Sketch the graph clearly showing all the key features. Explanation
Solution
a b c d e
y x2 2x x(x 2) x-intercepts (let y 0): 0 x(x 2) x 0 or x 2 y-intercept (let x 0): y 0 02 1 Axis of symmetry: x 2 Turning point occurs when x 1. When x 1, y 12 2(1) y 1 There is a minimum turning point at (1, 1).
Take out common factor of x. Let y 0 to find the x-intercepts. Let each factor equal 0 and solve. Let x 0 to find the y-intercept. The axis of symmetry is halfway between the x-intercepts. Substitute x 1 into y x2 2x to find the y-coordinate. x 1 and y 1
y
f
The coefficient of x2 is positive therefore the basic shape is . Sketch the graph showing the key features.
3 2 1
–1
0 –1
x 1
2
3
(1, –1)
Example 15 For quadratic relation y x2 2x 8: a c e
426
Factorise the relation. Find the y-intercept. Find the turning point.
b d f
Find the x-intercepts. Find the axis of symmetry. Sketch the graph clearly showing all the key features.
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Explanation
Solution
a b c d e
f
y x 2x 8 Factorise the quadratic. (x 4)(x 2) x-intercepts (let y 0): 0 (x 4)(x 2) Let y 0 to find the x-intercepts. x 4 or x 2 Let x 0 to find the y-intercept. y-intercept (let x 0): y 8 4 2 1 Axis of symmetry: x The axis of symmetry is halfway between 2 the x-intercepts. Turning point occurs when x 1. Substitute x 1 into y x2 2x 8 to 2 When x 1, y (1) 2(1) 8 find the y-coordinate. 9 x 1 and y 9 There is a minimum turning point at (1, 9). y The coefficient of x2 is positive therefore the basic shape is Sketch the graph x showing the key features. 2
–5 –4 –3 –2 –1 0 –1
1 2 3
(–1, –9) –9
Exercise 11G Example
13
1
For each of the following quadratic relations: i a d g
Example
14
2
find the x-intercepts y x( x 4) y (x 5) (x 1) y (x 5)(x 1)
ii find the y-intercept b y x(x 2) e y (x 2)(x 6) h y (x 5)(x 1)
c f i
y x(x 3) y (x 3)(x 7) y (4 x)(3 x)
For each of the following quadratic relations: i iii v vi a d g
Factorise the relation. ii Find the x-intercepts. Find the y-intercept. iv Find the axis of symmetry. Find the turning point Sketch the graph clearly showing all the key features. y x2 3x b y x2 x c y 2x x2 e y 3x x2 f 2 2 y x 8x h y 2x x i
y x2 5x y 5x x2 y x2 x
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15
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For each of the following quadratic relations: i iii v vi a d
4
6/6/06
Factorise the relation. ii Find the x-intercepts. Find the y-intercept. iv Find the axis of symmetry. Find the turning point. Sketch each graph clearly showing all the key features. y x2 3x 2 b y x2 2x 3 c y x2 4x 3 2 2 y x 2x 1 e y x 4x 4 f y x2 2x 8
For each of the following relations sketch the graph clearly showing the x- and y-intercepts and the turning point. a d g j m
y x2 1 y x2 4x y 2x2 3x y x2 5x 4 y x2 2x 1
b e h k n
y x2 4 y x2 5x y 2x2 6x y x2 3x 4 y x2 6x 9
c f i l o
y 9 x2 y x2 3x y 4x2 8x y x2 x 12 y x2 10x 25
Enrichment Th
5
From the first golf hole Chris walks 5 metres towards the second golf hole which is 50 metres away, then hits the golf ball which follows a path described by a quadratic relation of the form y (x p)(x q) where p and q are positive integers. The ball lands 15 metres from the hole. Taking the origin as the first golf hole: a b c d e f g h
428
How far has the ball travelled horizontally? Find the values of p and q. Find the x-intercepts. Find the axis of symmetry. Find the turning point. Find the highest point reached by the golf ball. Find how far horizontally the ball has travelled when it reaches its highest point. Sketch the graph of the ball’s path for suitable values of x.
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Using technology to find x-intercepts
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A graphics or CAS calculator can be used to solve quadratic equations or find x-intercepts of graphs. Example: For y x2 4x 6 use technology to find the x-intercepts correct to two decimal places. TI 84 family GC key strokes
GC key strokes
TI 89 family
Press Y . Type x2 4x 6 into Y1.
Press ◆Y . Type x2 4x 6 into y1.
Press GRAPH.
Press ◆GRAPH.
Press 2nd CALC and select Zero. Press ¸.
Press F5 and select Zero. Press ¸.
Scroll to the left of the first x-intercept and press ¸.
Scroll to the left of the first x-intercept and press ¸.
Scroll to the right of the first xintercept and press ¸.
Scroll to the right of the first x-intercept and press ¸.
Scroll close to the first intercept for a guess and press ¸.
The first x-intercept is x 1.162. Repeat the process to find the second xintercept: x 5.162.
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The first x-intercept is x 1.162. Repeat the process to find the second x-intercept: x 5.162. Exercise 1 For each of the following quadratics use technology to find the x-intercepts correct to two decimal places. a y x2 3x 1 b y x2 5x 1 c y x2 3x 1 e y x2 7x 2 f y x2 x 9 d y x2 8x 6 h y 4x2 3x 5 i y 3x2 3x 4 g y 2x2 x 5 2 2 k y 2x x 3 l y 4x2 8x 1 j yx x1
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Applications of quadratic relations
Quadratic relations and their graphs can be used to solve practical problems in which maximum or minimum values need to be determined. You also need to ensure that the solution satisfies the physical constraints of the problem. For example, if the variables represent lengths any negative results should be rejected.
Key ideas To solve application problems you may need to follow all or some of these steps. Draw a diagram. Define the variables and determine what you are trying to find. Determine any restrictions. Write an equation in terms of one variable only. Solve the quadratic equation. Draw a graph.
Example 16 The quadratic rule h t2 2t 8 gives the height, h metres, of a ball thrown vertically upwards from the top of a balcony at time t seconds. a b c d e
From what height was the ball thrown? What is the height of the ball after two seconds? When does the ball hit the ground? What is the maximum height that the ball reaches? Sketch the graph of h for suitable values of t.
Solution
a b
c
When t 0, h 8 The height is 8 m. When t 2, h (2)2 2(2) 8 6 The height is 6 m. Let h 0: 0 (t2 2t 8) 0 (t 4)(t 2) t 4 0 or t 2 0 t 4 or t 2 Since t 0 (time is positive) the ball hits the ground after 4 seconds.
Explanation
Substitute t 0. Substitute t 2.
When the ball hits the ground its height is zero. Take out 1 as a common factor. Factorise the quadratic. Let each factor equal zero. Solve for t. Answer the question in words.
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t-intercepts: t 4 or t 2 Turning point occurs at t 1. When t 1, h (1)2 2(1) 8 9 The ball reaches a maximum height of 9 m.
e
h
The y-coordinate of the turning point gives the maximum height. The turning point occurs halfway between the two t-intercepts. Find the height when t 1. Answer the question in words. Since t 0 the graph should only be drawn for 0 t 4.
(1, 9)
8 6 4 2 t
0
1
2
3
4
5
Example 17 Leonie plans to fence a rectangular paddock for her horses so that they have the maximum area of grass available for grazing on. a
i
b
c d
If the length of the paddock is x metres and its width is y metres, write a rule for its perimeter, P m, in terms of x and y. ii If Leonie plans to use 100 m of fencing show that y 50 x. i If the area of the paddock is A m2, write a rule for its area in terms of x and y. ii Show that its area can be written as A x(50 x). Sketch the graph of A for suitable values of x. i For what value of x does the maximum area occur? ii What is the largest area the fence can enclose?
Solution
a
Explanation ym
i xm
ii
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Perimeter of paddock, P 2x 2y 100 2x 2y 50 x y y 50 x
Essential Mathematics VELS Edition Year 9
Draw a diagram.
Let the perimeter equal 100. Write y in terms of x.
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b
A length width A xy Substitute y 50 x: A xy A x(50 x)
i ii
Substitute y 50 x.
A
c
Since the fencing is 100 m long 0 x 50.
625
0
d
Length x and width y
x 25
50
The maximum area occurs at x 25.
i
When x 25, A 25(50 25) 625 That is, the paddock has a maximum area of 625 m2. ii
TP (25, 625) The turning point occurs halfway between the two x-intercepts. Substitute x 25. Answer the question in words.
Exercise 11H Example
16
1
The quadratic rule h t2 4t 12 gives the height, h metres, of a toy rocket fired vertically upwards from the top of a building at time t seconds. a b c d e
2
From what height was the toy rocket fired? What is the height of the toy rocket after two seconds? When does the toy rocket hit the ground? What is the maximum height the toy rocket reaches? Sketch the graph of h for suitable values of t.
The height, h metres, of a model aeroplane at time t seconds is given by h 64 t2. a b c d e f
What is the initial height of the aeroplane? What is the height of the aeroplane after 4 seconds? When does the aeroplane land on the ground? For how long is the aeroplane in the air? What is the maximum height the aeroplane reaches? Sketch the graph of h for suitable values of t.
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An organising committee found that the profit, $P, from a concert depends on the ticket price x. The estimated profit is given by
5
b c d 6
i
If the length of the paddock is x metres and its width is y metres, write a rule for its perimeter in terms of x and y. ii If Amy plans to use 80 m of fencing show that y 40 x. i If the area of the paddock is A m2, write a rule for its area in terms of x and y. ii Show that its area can be written as A x(40 x). Sketch the graph of A for suitable values of x. i For what value of x does the maximum area occur? ii What is the largest area the fence can enclose?
One side of a rectangular farm is bounded by a straight river as indicated in the diagram. a
b
c d 7
P x2 60x 500 If the ticket price is $35 what is the profit? Sketch the graph of P for suitable values of x. Find the ticket price that will maximise the profit. Find the maximum possible profit.
Amy plans to fence a rectangular paddock so that the maximum area is enclosed. a
i
If the length of the paddock is x metres and its width is y metres, write a rule for its perimeter in x terms of x and y. ii If the farmer has 80 m of fencing with which to construct the fence as shown in the diagram, show that y 80 2x. i If the area of the paddock is A m2, write a rule for its area in terms of x and y. ii Show that its area can be written as A 2x(40 x). Sketch the graph of A for suitable values of x. i At what point does the maximum area occur? ii What is the largest area the fence can enclose?
y
A piece of wire 72 m long is bent into the shape of a rectangle. a b c d
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Lasno Inc. manufactures soft toys. The daily profit, $P, is given by P x2 100x 1600 where x is number of toys produced each day. a If they produce 40 soft toys a day what is the profit? b Sketch the graph of P for suitable values of x. c How many toys do they need to produce before they start making a profit? d Determine the maximum profit and the number of toys for which it occurs.
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a b c
Example
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If x cm is the length and y cm is the width write a rule connecting x and y. If the area of the rectangle is A cm2 write an equation that connects A and x. Sketch the graph of A for suitable values of x. Determine the maximum area enclosed by the rectangle.
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8
A rectangular playground is to be marked out so that it has a perimeter of 100 metres. a Write an expression for the area, A cm2. b Sketch the graph of A for suitable values of the length and width of the rectangle. c Determine the maximum area of the playground. d What are the dimensions of the playground that give this maximum area?
9
The sum of two numbers, x and y, is 10. a b c
Write a rule connecting x and y. If the product of the two numbers is P, write a rule connecting P and x. Find the two numbers so that their product is a maximum.
Enrichment Th
10
Two spiders are racing each other. In the following rules, d is the distance in centimetres travelled from the starting line, and t is the time they have travelled in minutes. The distance travelled by the first spider, Redback, is given by the rule d 2t 2 5t 5 The distance travelled by the second spider, Hunter, is given by the rule d 2t 2 a What distance has each travelled in the first 5 minutes? b i At what time do the Redback and Hunter pass each other? ii How far from the starting point are they at this time? c Sketch the graphs for both spiders on the same set of axes over suitable values of t. Show all the key features of both graphs.
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Mathematically
N G Quadratics
1 Pop concert The price of a ticket at a concert is based on the costs associated with staging the production and on what people are prepared to pay. Promoters know that a concert will not sell out when the ticket price exceeds a certain amount.
Revenue and cost of tickets Assume that for a concert by a particular band promoters know that: the venue caters for a maximum of 10 000 people it will be a ‘sell out’ if the cost of the tickets is $50 or less for each $10 increase above $50 in ticket price, the number of tickets sold decreases by 10% a Complete the following table. 50 60 Number of tickets sold 10 000 9000 Revenue ($) Cost of tickets ($)
b
c
E HPE ICT PL S Th
Essential Mathematics VELS Projects
d
70 8100
80
90
100
110
120
Use a graphics calculator to complete the following. i Enter the data for the cost of tickets in L1 and the revenue in L2 by first pressing STAT and selecting Edit. ii Plot the graph of the ‘Revenue’ versus the ‘Cost of tickets’ using the STAT PLOT. You will need to adjust the WINDOW to show all the data points. Discuss the shape of the graph and any important features. From your graph determine how much the promoters should charge if they want to maximise the revenue. Use STAT CALC QuadReg on the graphics calculator to find a quadratic rule that best fits the data. Use this rule to determine the revenue if the cost of each ticket was: i $65 ii $95
Revenue and number of tickets a b c
Plot the graph of the ‘Revenue’ versus the ‘Number of tickets sold’ on the graphics calculator. Discuss the shape of the graph and any important features. From your graph determine how many tickets need to be sold to maximise the revenue. Use the graphics calculator to determine a quadratic rule that best fits the data. Use this rule to determine the revenue if the number of tickets sold is: i 5000 ii 7000
Varying factors a
b
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Explore what effect each of the following changes would have on the maximum possible revenue and the number of tickets sold. i The venue is changed and the number of tickets available increases to 15 000. (All other factors are kept the same as the bullet points above.) ii For each $10 increase in ticket price the number of tickets sold decreases by 20%. (All other factors are kept the same as in the bullet points above.) Should the promoter consider factors other than maximising revenue? Discuss.
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2 Cars and stopping distance Driving over the speed limit even by a small amount can significantly increase the risk of not stopping in time and causing an accident. Statistics show that the risk of injury doubles for every 5 km/h above 60 km/h. One reason for this increased risk is reaction time—the time it takes between a person perceiving a danger and reacting to it. Once the brake has been depressed, it takes additional time before the car comes to a complete stop. The table below gives formulae for a car travelling at v metres per second. The reaction distance, dr metres; t is the reaction time
dr tv
The braking distance, db metres; a is the car’s deceleration
db
The total stopping distance, ds metres
ds tv
v2 2a v2 2a
Alert drivers An alert person driving will have a reaction time of about 1.5 seconds and their deceleration will be about 10 m/s2. a Write a formula in terms of v for: i reaction distance ii braking distance iii total stopping distance Sketch a graph of each one on the same axes over an appropriate range of t values. b Complete a table like the one below. Extend the table to include 65 km/h, 80 km/h, 100 km/h and 120 km/h. Speed in km/h
Speed in m/s
Reaction distance
Braking distance
Total stopping distance
40 km/h
Essential Mathematics VELS Projects
60 km/h c
d
How much further, in total, will the car travel if the driving speed is i increased by 5 km/h above 60 km/h? ii increased by 20 km/h (from 40 km/h to 60 km/h, from 60 km/h to 80 km/h, etc.)? iii doubled (from 40 km/h to 80 km/h and from 60 km/h to 120 km/h)? Approximately how many car lengths is each answer in c? Assume that a car length is approximately 4 metres.
Distracted drivers A driver who is distracted (e.g. using a mobile phone) takes twice as long as an alert driver to react. Repeat the above procedure if their deceleration remains at 10 m/s2. Compare your results with those above.
Deceleration Investigate what effect there will be on stopping distances if the deceleration is changed from 10 m/s2 to 15 m/s2. (Use the reaction time of 1.5 seconds used above.)
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Review
Chapter summary Key features of quadratic relations: y ax 2 bx c y
y-intercept
3
x-intercepts O
x 0 1
2 3 4 turning point
axis of symmetry
Transformations of y x 2 y a(x h)2 k a 0 minimum turning point a 0 graph reflected in x-axis (maximum turning point) The graph is dilated by a factor of a parallel to the y-axis. Axis of symmetry: x h
Translates the graph up or down: k k Translates the graph left or right: h0S h0d Turning point (h, k)
Null Factor Law (NFL) If a b 0 then a 0 or b 0 or both a and b 0. e.g. If (x c)(x d) 0 then x c or x d x-intercepts and turning points Number of x-intercepts
0
1
2
Shape of graph
If the graph has x-intercepts the turning point can be found by first determining the location of the axis of symmetry, which lies halfway between the two x-intercepts. Applications To solve maximum and minimum application problems you may need to: Draw a diagram. Define the variables and determine what you are trying to minimise or maximise. Determine any restrictions. Write an equation in terms of one variable only. Draw a graph. Solve the quadratic equation.
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Multiple-choice questions
2
3
For the graph shown the equation of the axis of symmetry and the coordinates of the turning point are respectively A x 4; (1, 4) B x 1; (1, 4) C x 1; (4, 1) D x 3; (1, 4) E x 1; (0, 3)
y 3 2 1 –2 –1–1 –2 –3 –4
x
0 1
2
3
4
The quadratic equation y 3x2 has which of the following table of values for 2 x 2? A x 2 1 0 1 2 y
12 3
0
3
12
B
x y
2 1 7 4
0 3
1 4
2 7
C
x y
2 1 6 3
0 0
1 3
2 6
D
x y
2 1 12 3
0 0
1 3
2 12
E
x y
2 1 12 3
0 3
1 3
2 12
The curve shown in the diagram is the graph of A y (x 3)2 B y x2 3 C y (x 3)2 D y (x 3)2 E y (x 3)2
y 9
O
x 3
6
4
Compared to y x2, the graph of y x2 2 would be A twice as wide B moved 2 units to the left C moved 2 units to the right D moved 2 units up the y-axis E moved 2 units down the y-axis
5
Which of the following is the coordinates of the turning point of the parabola whose equation is y (x 4)2 2? A (0, 2) B (0, 18) C (4, 2) D (4, 2) E (4, 2)
6
The y-intercept of the parabola in Question 5 is at A (18, 0) B (14, 0) C (0, 2) D (0, 16)
E (0, 18)
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1
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7 The curve shown in the diagram is the graph of A y (x 2)2 1 B y (x 2)2 1 C y (x 1)2 2 D y (x 2)2 1 E y (x 1)2 2
y 5 4 3 2 1 x
–1 –1 0 1
2
3
8 If x2 2x 3 0 then x is equal to A 3 or 1 B 3 or 1 C 3 or 1
D 1 or 3
E 2 or 3
9 If 3x 12 0 then x is equal to A 2 B 2 C ;2
D ;4
E 4
4
2
10 The curve shown in the diagram is the graph of A y (x 1) ( x 3) B y (x 1) ( x 3) C y (x 1) ( x 3) D y (x 1) ( x 3) E y (x 3) ( x 3)
y
–3
0 –3
x 1
Short-answer questions 1 For each of the following quadratics: i Construct and complete a table of values for 2 x 2. ii Plot the graph iii State key features, such as the axis of symmetry, turning point and y-intercept. iv Describe how the graph can be obtained by transforming the graph of y x2. 1 a y 2x2 b y x2 2 2 d y (x 3)2 c yx 2 2 Sketch the graphs of the following on the axes below. You must show the y-intercept and the position of the turning point. b y (x 3)2 a y x2 3 2 c y (x 1) 2 d y (x 3)2 2 3 Solve each of the following for x. a (x 5)(x 6) 0 b 2(x 9)(x 2) 0 c x(x 3) 0 d x2 6x 0 f 2x2 8 0 e x2 7x 10 0
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For each of the following quadratics find: i the y-intercept ii the x-intercepts iii the equation of the axis of symmetry iv the coordinates of the turning point a y (x 2)(x 3) b y x(x 4) 2 d y x2 6x 8 c y x 4x 5
5
a
For the graph shown state the: i x-intercepts ii y-intercept iii equation of the axis of symmetry iv coordinates of the turning point b Determine the equation of this parabola.
Review
4
y 3 2 1 –1
0
x 1
2
3
–1
Extended-response questions 1
A paper aeroplane is thrown upwards from a balcony. The path the aeroplane follows is given by the equation y 64 (x 2)2 where y is the height above the ground in metres and x is the time in seconds. a How high is the balcony? b When does the aeroplane land on the ground? c How far above the ground is the highest point reached by the aeroplane? d Sketch a graph of the aeroplane’s path for 0 x 10.
2
The cost of producing n surf boards is given by C 50(32 3n n2). The surf boards are sold for $750 each. a Show that the profit, P, is given by P 50(18n 32 n2). b If 10 surf boards are produced what is the profit? c Draw a graph of P for suitable values of n. d How many surf boards do they need to produce before they start making a profit? e Find the number of surf boards which should be produced in order to make a maximum profit. f What is the maximum possible profit? MC
T E ST D&D
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T E ST
