## SUMMER MATH PACKET. Plane Geometry COURSE 223

SUMMER MATH PACKET Plane Geometry COURSE 223 MATH SUMMER PACKET INSTRUCTIONS Attached you will find a packet of exciting math problems for your enjo...
SUMMER MATH PACKET Plane Geometry COURSE 223

The OPRFHS Math Department

1

Summer Packet For Students Entering Plane Geometry Name:

_____________________________________________________________

Welcome to Geometry! This packet contains the topics that you have learned in your previous courses that are most important to geometry. Please read the information, do the problems, and be prepared to turn this in when school begins again. Enjoy your summer!

I.

Using your TI-Nspire Graphing Calculator • • • •

II.

Be able to find the intersection of graphs You will need to find the roots(zeros) of equations on your calculator Be able to use nsolve to solve linear equations Be able to use the calculator to solve systems of equations

Cartesian Coordinate System • •

• • • •

Know the quadrants, axes, and graphs Different forms of the equation of a line: standard, point-slope o Point Slope Form ( y - y1) = m (x - x1) o Standard Form Ax + By = C o Slope – intercept form y =mx+b

change y Δy = Slope (incredibly important in geometry); think change x Δx Slope of parallel and perpendicular lines Pmeans parallel and ⊥ means perpendicular

EXAMPLE – How to write the equation of a line, parallel lines, etc. Ex.Write the equation of a line that passes through (3, -5) with a slope of 7 Point Slope: ( y − (−5)) = 7(x − 3) Write the equation of a line that is parallel to the above line and passing through the point (-2, 6). (remember parallel lines have the same slope) Point Slope:

( y − 6) = 7(x − (−2))

Write the equation of a line that is perpendicular to the above line and passing through the point

( 52 ,0 ) remember perpendicular lines have slopes that are

opposite reciprocals of each other) Point-Slope: y − 0 =− 17 ( x − 52 )

2

III.

Simplifying Algebraic Expressions • •

Remember Order of Operations: Parenthesis, Exponents, Multiplication/Division and Addition/Subtraction When simplifying rational expressions, remember to simplify the numerator and denominator separately before checking if something cancels. 14

Rules of Exponents: (x )(x ) = x , 3

5

8

x = x3 , (x5 )6 = x30 11 x

Examples: Simplify the following – a) 1− (2 − 5) 2 + 5 ÷10 • 42

b)

4(9 − 2 • 3) − 32 42 − 32

1− (−3) 2 + 5 ÷10 • 42

4(3) − 32 42 − 32

1− 9 + 5 ÷10 •16

12 − 9 16 − 9

1− 9 +

1

3 7

•16

2 1− 9 + 8 = 0 IV.

Solving Equations •

Be able to solve a single-step equation: Ex.

Solve:

5x = 70 5 70 x= 5 5 x = 14

Be able to solve a two-step equation:

Ex. Solve:

Divide both sides by 5 to isolate the variable

3 − 9x = 21 3 − 9x = 21 −3 −3

− 9x = 18 −9 18 x= −9 −9 x = −2 3

Be able to solve multi-step equations using order of operations, distributive property and inverse operations. ex. Solve:

5x − 2(x − 5) = 7x − 2 5x − 2x +10 = 7x − 2 3x +10 = 7x − 2 12 = 4x x=3

Be able to solve proportions:

2

=

x +1

10 x

2x = 10x +10 cross multiply −8x = 10 5 x=− fraction reduced 4 V.

Solving Systems of Equations • Methods to Solving Systems: Graphing, Substitution, Elimination Use the graph at the right to determine whether each system has no solution, one solution, or infinitely many solutions.

♣ y = 2x + 2

a. ♦

♥4x − 2 y = 10

Since the graphs of y = 2x + 2 and 4x − 2 y = 10 are parallel, there are no solutions.

♣ y = 2x − 5

b. ♦

♥4x − 2 y = 10

Since the graphs of y = 2x − 5 and 4x − 2 y = 10 coincide, there are infinitely many solutions.

♣ y = −3x − 5

c. ♦

♥ y = 2x + 2

Since the graphs of y = −3x − 5 and y = 2x + 2 intersect, there is one solution.

4

Use substitution to solve the system of equations.

♣ y − x = −4 ♦ ♥ 6x + y = 3  Step 1: Solve the first equation for y since the coefficient of y is 1. y – x = -4 First equation y – x + x = -4 + x Add x to each side. y = -4 + x Simplify. Step 2: Find the value of x by substituting -4 + x for y in the second equation. 6x + y = 3 Second equation 6x + (-4 + x)=3 y = -4 + x 7x -4 = 3 Combine like terms. 7x – 4 + 4 = 3 + 4 Add 4 to each side. 7x = 7 Simplify 7x = 7 Divide each side by 7. 7 7 x= 1 Simplify Step 3: Substitute 1 for x in either equation to find the value of y. Choose the equation that is easier to solve. y – x = -4 First equation y - 1 = -4 x=1 y – 1 +1 = -4 + 1 Add 1 to each side. y = -3 Simplify.

Step 4:

Write your solution as (x, y)

Example: (1, -3)

5

Use elimination to solve the system of equations.

♣5x − 7 y = −2 ♦ ♥ −4x + 6 y = 4  Method 1 Eliminate x 5x – 7y = -2 -4x + 6y = 4

Multiply by 4. Multiply by 5.

Now substitute 6 for y in either equation to find the value of x. 5x – 7y = -2 5x – 7(6) = -2 5x – 42 = -2 5x – 42 + 42 = -2 +42 5x = 40 5x = 40 5 5 x=8

First equation y=6 Simplify. Add 42 to each side. Simplify. Divide each side by 5.

The solution is ( 8,6 ).

6

(+)

20x -28y = -8 -20x + 30y = 20 2y = 12 2y = 12 2 2 y=6

VI. •

Factoring Rewrite an algebraic expression as the product of it’s factors

Factor: x² + 7x + 6 The leading coefficient of x² is A. The leading coefficient of x is B. The constant is C. Step 1:

The leading coefficient of x² is placed in the upper left corner of the box.

Step 2:

The constant is placed in the lower right corner of the box.

Step 3:

Multiply the leading coefficient of x² by the constant.

Step 4:

Find two numbers that will multiply to Step 3 ( 1* 6) and add up to the leading coefficient of x.

Step 5:

As seen above, 6X and 1X fit into these guidelines

Step 6:

Find the the Greatest Common Factor (GCF) going across each row and down each column.

Step 7:

The GCF or each row are placed in parenthesis and the GCF for each column are also placed in parenthesis.

Step 8:

Your final answer will be ( x + 6) ( x + 1)

Example worked out: x2 + x + 6x + 6 (x2 + x) (+6x + 6) x(x + 1) 6(x + 1) (x + 1)(x + 6)

7

Example:

Factor:

2x² + 11x +5 A=2 B = 11 C=5

Two numbers multiply to 10 add to 11 2x2 + 10x + x + 5

(2x2 + 10x) (+ x + 5) 2x(x + 5) 1(x + 5) (x + 5)(2x + 1)

8

VII. •

Transformation of graphs Be able to recognize a quadratic in vertex form:

y = a(x − h)2 + k where (h, k) is the vertex. •

Be able to shift the graph of a quadratic function vertically and horizontally Original Function Vertical Shift Horizontal Shift

y = x2 •

y = x2 ± k

y = (x ± h) 2

Be able to recognize a stretch or shrink a quadratic function Original Function Vertical Shrink Vertical Stretch

y = x2

y = ax 2

y=

1

x2

a •

Be able to reflect a quadratic function across the x-axis Original Function Reflection over the x-axis

y = −x 2

y = x2 •

Examples: o

How does the graph of y = x 2 − 7 compare to the graph of y = x 2 ? Answer:

o

How does the graph of y = (x + 3)2 compare to the graph of y = x 2 ? Answer:

o

Shifted Down 7 units

Shifted Left 3 Units

How does the graph of y = (x −1)2 + 6 compare to the graph of y = x 2 ? Answer: Shifted right 1 unit and up 6 units

o

How does the graph of y = −4x 2 compare to the graph of y = x 2 ? Answer:

o

Reflected over the x-axis and vertical shrink by factor of 4

How does the graph of y = Answer:

1 4

(x + 3)2 − 8 compare to the graph of y = x 2 ?

Stretched by a factor of 4, shifted left 3 units and down 8 units

9

VIII. Solving Quadratics • • •

2

ax + bx + c = 0 Standard form of a quadratic equation: Be able to solve quadratics using factoring, quadratic formula and graphing Remember to set equations equal to 0 before solving 3x 2 = 2 − x 3x 2 + x − 2 = 0

Factoring:

Set equation equal to 0 Factor equation

2

(3x + 3x)( - 2 x -2 ) =0 3x(x+1)-2(x+1)=0 (3x – 2)(x + 1)=0

3x − 2 = 0 2 x= 3

and and

x = -1

Solve each equation

To solve ax 2 + bx + c = 0 and a ≠ 0

Then x = ex.

Set each factor equal to 0

x +1 = 0

−b ± b2 − 4ac 2a

Solve 5x + 8x = −3 using the quadratic formula 2

a = 5, b = 8, c = 3

5x 2 + 8x + 3 = 0 x=

−8 ± 82 − 4(5)(3) 2(5)

x=

−8 ± 4 10

Split the plus and minus apart

x = (-8 + 2)/10 x = (-8 – 2)/10 x=−

3

or

x = −1

5

10

Solving Quadratics by Graphing: Graph:

y = x 2 + x − 6 on your graphing calculator

- the solution(s) are where the graph crosses the x-axis

Solutions: x=__-3___ & x = ___2_____

IX.

Problem Solving Skills for Word Problems 1.

Read the problem carefully.

2. Draw a picture (if you can) illustrating the problem. 3. Identify unknown quantities you're being asked to find. 4. Choose a variable to represent one quantity. 5. Write down, in words, what the variable (you chose in 4) represents. 6. Represent other quantities to be found in terms of the variable (you chose in 4). 7. Write the word problem as an equation (hard step--takes practice). 8. Solve the equation for the unknown quantity. 9. Answer the question asked in the word problem. 10. Check your solution in the original word problem.

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Review problems: Complete the following problems, showing work where necessary. Feel free to do your work on separate sheets of paper and remember you will be required to turn this in. An answer key is provided for you but in geometry the work is usually more important than the answer!

I.

Using the TI-Nspire (graphing calculator)

1.

Graph the following on your calculator and sketch the graph: a)

2 y = − x+4 3

b)

y = x−4

c)

2. Find the points of intersection of the following:

2x + y = 5 x − y = 3

y = x 2 +1 y = 2x + 3

3. Find the roots (y = 0) of the following: y =x − 2x +1 2

II.

Cartesian Coordinate System

4. Which pair of lines are perpendicular?

y= a)

y = 3x - 4 y = 3x + 9

b)

3

x+8

2 3

2 y = − x +1

y=3 c)

y = − x−4 2

y=−

1 3

12

5

d)

y=

5 2

x−6

5).Write an equation in point-slope form for the line through the given point and slope. a)

(-2, 8), m = -

4

b)

3

(1,5), m = -6

6. Write an equation for the line through the given points. Use the form given only. b) ( 3,1) , ( 0, 7 ) Slope Intercept Form a) (1, 4 ) , ( 2,5 ) Point-slope form

7. Find the equation of the line perpendicular to the given line that contains the following point. Use slope intercept form.

1 y = - x - 2; (0, 7 ) 4

8. Find the equation of the line parallel to the given line that contains the following point. Use point-slope form. y = -

3 4

x + 7; (1,8 )

13

9. Graph the following lines:

10

10

8

8

6

6

4

4

2

2

-10 -8 -6 -4 -2

A.

2

6

8 10

-2

-4

-4

-6

-6

-8

-8

-10

-10

y+2=7

4x - 3y = 12

-10 -8 -6 -4 -2

-2

B.

10

8

8

6

6

4

4

2

2 2

4

6

8 10

-10 -8 -6 -4 -2

-2

-2

-4

-4

-6

-6

-8

-8

-10

-10

D.

14

2

4

6

8 10

2

4

6

8 10

y + 2x =7

10

-10 -8 -6 -4 -2

C.

4

(y - 3) = 0.4(x + 2)

III.

Simplifying Algebraic Expressions

10. Simplify the following expressions:

a)

b)

19 − (4 + 2× 32 )

8 ÷ 4× 6 42 − 52 9 − 4 +11− 42

c)

Evaluate: 2(x − 6) − 3x2 if x = -2

d)

3x + 2x 2 − x − 2(5x + 9x 2 )

IV. Solving Linear Equations and Inequalities 11. Solve the following equations:

17x = −51

a)

c)

b)

-9x + 4(2x - 3) = 5 (2x-3) + 7

15

9x − 8 = −15

1

d)

3

(45 y −18) = 15( y +1)

V.

Graph the following lines

12.

y=

2 5

e)

x−6

13.

1

5 3 3 x− = x+ 2 8 4 8

y + 3 = −4(x −1)

10

10

8

8

6

6

4

4

2

2 -10 -8 -6 -4 -2

2

4

6

-10 -8 -6 -4 -2

8 10

2 -2

-2

-4

-4

-6

-6

-8

-8

-10

-10

16

4

6

8 10

VI.

Solving Systems of Linear Equations

14. Graph the following systems and state the solution.

10

10

8

8

6

6

4

4

2

2

-10 -8 -6 -4 -2

A.

2

8 10

-10 -8 -6 -4 -2

2

-4

-4

-6

-6

-8

-8

-10

-10

1 y =− x+4 2

y = 2x - 4

y = 3x

6

-2

B.

y=

1

10

8

8

6

6

4

4

2

2 2

4

6

8 10

-10 -8 -6 -4 -2

2

-2

-2

-4

-4

-6

-6

-8

-8

-10

-10

y – 1 = 2 (x + 4)

D.

17

6x - 2y = 2

4

6

8 10

6

8 10

y − 0 = 2(x − 6)

x+3

2

10

-10 -8 -6 -4 -2

C.

4

-2

9x - 3y = 1

4

15.

Solve using Substitution or Elimination Method

a).

b).

y = −x + 5

4x − 5 y = 11 6x + 7 y = 31

2x + 4y = 2 ♦x = y + 7 

d).

-2x + 5y = 26 ♦3x - 2y = 5 

Substitution Only!

17.

Elimination Only!

c).

16.

y = 2x +11

-3x + 2y = 16  ♥y = -x + 3 

3x + 2y = 17  ♥3x - 2y = -5 

18

VII.

Transformations of Graphs 18.

Describe the transformations (compared to y = x 2 ) in the graph of y = −(x + 4)2 −11

19.

Describe the transformations (compared to y = x 2 ) in the graph of y =

1

(x − 9)2

3

VIII. Polynomials 20.

21.

Multiply the following expressions:(write your answer in descending order)

a).

(a + 3)(a - 7)

b).

y(3y2 - 5y - 10)

c).

(2a + 3)(2a - 3)

d).

(3m - 1)2

Factor the following polynomials, if the polynomials does not factor write PRIME

a).

y2 - 10y + 25

b).

x2 + 12x +36

c).

b2 - 100

d).

c2 + 9

19

e).

x2 - 7x + 6

f).

8x5 − 32x3

22. Solve each quadratic equation.

a)

x 2 - 144 = 0

b) x 2 + 8 = 72

c)

2x 2 - 10 = -4

d) 4x 2 - 8 = 50

23. Solve by factoring:

a)

(8x - 3)( 4x + 1) = 0

b) x 2 + 14x + 13 = 0

20

c

x 2 + 7x = 8

Part 1: Graphing calculator 1a)

1b)

1c)

2.

21

3.

4.

d

5.

a)

6.

a)

7.

y = 4x + 7

8.

y −8 =

9. 10.

(see graphs at the end of the answer key) a) -3 b) undefined c) -28 d) -6x2 – 8x

11.

a) x = -3

4 b) y − 8 = − (x + 2) y − 5 = −6(x −1) 3 y − 4 = 1(x −1) or y − 5 = 1(x − 2) b) y = -2x + 7 4

3

(x −1)

b) x = -7/9

c) x = -4/11

12. 13.

22

d) no solution

e) x = -4

14)

a) (3.2, 2.4)

15)

a) (-2, 7)

b) (10, 8)

c) (9, 27)

b) (4, 1)

d) No solution

c) (5, -2)

d) (7, 8)

16. (-2, 5) 17) (2, 5.5) 18)Flipped over the x-axis, shifted left 4 units and down 11 units 19) Stretched by 1/3, shifted right 9 units b)

c)

a 2 − 4a − 21 4a 2 − 9

a) c) e)

(y – 5)(y-5) (b – 10)(b + 10) (x – 6)(x – 1)

b) (x + 6)(x + 6) d) Does not factor f) 8x3 (x – 2)(x + 2) b) x = -8 & x = 8 b) x = -13 & x = -1

20) a) 21)

22) 23)

a) x = 12 & x = -12 a) x = 3/8 & x = -1/4

d)

3y 3 − 5y 2 −10 y 9m2 − 6m +1

23

c) c)

x = 3.808 & x -3.808 x = -8 & x = 1