Name __________________________________________________ Date _______________ Per _________

Unit 7: Solve Quadratics 4

5

6/7 Intro to Quadratics (Section 5.1)

8 Extended Advisory Review Simplifying Radicals (if time: Discriminant)

11

12

HW: Part 1 13 / 14

Discriminant

Solve Using Completing the Square (Section 5.4)

Graphing Quadratics: Find the a, AOS, vertex, y-intercept, and roots

Solve Using the Quadratic Formula (Section 5.6)

HW: Part 2 (maybe part 3 – TBD) 15 Word Problems (Chapter 5)

Quadratic Inequality (Sections 5.1-5.3, 5.7)

HW: Part 3 & 4 18

HW: Part 4 19

Review (Chapter 5)

Test Solve Quadratics

HW: Part 5 20 / 21

HW: Review Sheet Learning Objectives: • Know the restricted range that is determined by the y coordinate of the vertex. • Know that the quadratic can have 1 or 2 real roots and 2 imaginary roots. • Translate among the different representations of a quadratic: graph, table, algebraic • Determine a quadratic from its roots, real and complex, or a graph • Compare algebraic and graphical solutions of quadratic equations • Solve quadratic equations using factoring Essential Questions: • How can you determine the max/min of a quadratic? • Describe how to use the roots of a quadratic to get to the equation. • Does the domain of a quadratic ever change? The range? • Given one representation, explain how to get the others. • What key points are needed to sketch an accurate graph? • Why do we set the equation equal to zero before solving?

HW: Part 6 22

Intro to Quadratics (Section 5.1) Vocabulary Root – Y intercept – Vertex – Axis of Symmetry –

__________

__________

__________

__________

__________

Ex: Label all parts of the graph and give the appropriate coordinate point or values based on the Sketch the graph: table. f(x) = x2 – 6x + 8 ____ ____

____

____ ____

x 1 2 3 4 5

y 3 0 -1 0 3

Velocity Formula: Any object that is thrown, kicked, or launched into the air (ie: baseball, football, or soccer ball) is a ___________________. The general function that approximates the height in reference to time is given as:

Example: Use the table, developed using the above formula, to find the following information. Sketch a graph and label all parts and their values on the graph. time (sec) height (ft) 0 0 1 20 2 35 3 40 4 35 5 20 6 0 Max height: At what time did it reach max height? How long was the object above 35 ft: Landing Time: Domain:

Range:

Independent:

Dependent:

Ex. The flight path of a rocket is modeled by the function h(t) = -16t2 + 512t. What is the initial velocity? What is the initial height? What is the constant based on Earth’s gravity? What is the maximum height? How many seconds does it take for the rocket to land? Initial velocity:

Initial height:

Max height:

Seconds to landing:

Sketch the graph 1) Label the axes height and seconds 2) Circle the x - intercepts 3) Label the vertex 4) Draw in the axis of symmetry 5) Domain: _________ 6) Range: _________

Constant based on Earth’s gravity:

Ex. The height of a person on a water slide is modeled by the function h(t) = -.025t2 – 0.5t + 120 with the person landing in a swimming pool at the ground level. What is the maximum height of slide? How many seconds does it take to get to the bottom of the slide (land in the pool)? Maximum height: Seconds to bottom: Sketch the graph 7) Label the axes height and seconds 8) Circle the x - intercepts 9) Label the vertex 10) Draw in the axis of symmetry 11) Domain: _________ Range: _________

Describe the transformation, sketch a graph, and label all parts. Ex:

y = 2 ( x − 5)2 + 7 3

What are the roots?

2

Ex: y = ( x − 4) What are the roots?

Ex:

y = −2( x + 3) 2 − 8

What are the roots?

−5

Homework PART 1: 1) y = (x + -6 Opens: AOS: Vertex: Domain: ________________ Range: __________________ What type of roots? __________ 4)2

2) y = - (x - 5)2 + 8 Opens: AOS: Vertex: Domain: ________________ Range: __________________ What type of roots? __________

Opens: AOS: Vertex: Domain: ________________ Range: __________________ What type of roots? __________

3) y = 2(x + 3)2 + 1 Real-World Problems 4) A ball is kicked across the field at a velocity of 32 ft/sec. The height of the ball above the ground is modeled by the function h(t) = –16 t 2 +32 t. Use your calculator to graph the path of the ball. You may need to change your window settings. Sketch the graph Draw the table 12) Label the axis height and time 7) Label the columns height and time 13) Circle the x - intercepts 8) Circle the roots 14) Label the vertex 9) Star the vertex 15) Draw in the axes of symmetry 16) Domain: _________ 17) Range: __________ Maximum height:

Time at maximum height:

Time at landing:

5) The highway mileage m in miles per gallon for a compact car is approximated by m(s0 = -0.025s2 + 2.45s – 30, where s is the speed in miles per hour. What is the maximum mileage for this compact car to the nearest tenth of a mile per gallon? What speed results in this mileage? Sketch the graph 1) Label the axes speed and miles 2) Circle the x - intercepts 3) Label the vertex 4) Draw in the axis of symmetry 5) Domain: _________

6) Range: __________ Maximum mileage:

Speed needed to achieve this mileage:

Review Simplifying Radicals Ex: 32

Imaginary Numbers: Why do we need i? Ex: 48

−1 = i 2 −1 = i 2 = i Ex: − 12

Ex: −12

Ex: − −96

Ex: 1 3 −63

Ex: 5 −121

Ex: 7 2 − (3)(3)

Ex: b 2 − 4ac , a = 1, b = 3, c = 2

−(2) ± (2)2 − 4(−1)(15) Ex. 2(−1)

−(3) ± (3) 2 − 4(2)(−7) Ex: 2(2)

Homework Part 2: Simplifying Radicals 1.

−49

5. 3 −12

9.

−48

2.

60

3. −3 −144

6.

7 ⋅ −14

7.

45

8. − 24

10.

80

11. − −128

12. − 75

−(−5) ± (−5) 2 − 4(3)(−6) 13. 2(3)

16.

−(1) ± (1) 2 − 4(3)(−6) 2(3)

−(8) ± (8)2 − 4(2)(5) 14. 2(2)

17.

−(−7) ± (−7)2 − 4(1)(5) 2(1)

4. 5 −28

−(7) ± (7) 2 − 4(3)(16) 15. 2(3)

18.

−(4) ± (4) 2 − 4(.5)(15) 2(.5)

The Discriminant (section 5.6)

2

2

Ex: y = x − 6 x + 9

2

OYO: y = x − 8 x − 10

Ex: y = x + 3 x − 7

Ex: y = x − 6 x + 11

2

Solve using the Quadratic Formula (5.6) Must leave answers in fraction form!


Memorize it! 2

Ex: y = x + 3 x − 7

2

Ex: y = x − 6 x + 11

Quad Form Cont 2

Ex: y = x + 9

Ex:

y = 2x2 + 8

Ex:

y = −2 x 2 + 3 x + 6

Ex:

y = 4 x2 + 5x + 6

Homework Part 3: Determine the number of solutions AND describe the solutions. 1. 5x2 + 7x – 10 = 0 2. 2x2 + 18x + 81=0 3. 8x2 + 9x + 2=0

Identify a, b, & c. Solve each quadratic function by using the quadratic formula. 4. x2 + 18x + 81 = 0 5. x2 - 11x + 24 = 0 6. 10x2 - 5x = 30

7. x2 + 6x + 3 = 0

8. 4x2 + 5x + 12 = 6

9. 3x2 - 27 = 0

10. – 3x2 - 6x + 10 = 0

11. 4x2 + 5x – 9 = 0

12. 3x2 + 10x + 12 = 0

13. 5x2 - 28x - 12 = 0

14. 4x2 - 24x – 108 = 0

15. x2 - 16x + 64 = 0

16. x2 + 8x – 20 = 0

17. 3x2 + 30x = -3

18. x2 - 12x + 20 = 11

Solve Using Complete the Square (5.4) STEPS: 1. Add blanks

2 Ex: x + 12 x = 28

2 Ex: x + 8 x = 49

2. 3. Factor & simplify 4. Solve Ex: x 2 + 4 x + 13 = 0

Ex: x 2 − 8 x + 18 = 0

Ex: x 2 + 10 x + 26 = 0

Homework Part 4: Completing the square x2 - 4x – 8 = 0

2. x2 + 6x – 7 = 0

3. x2 - 2x – 1 = 0

Graphing Quadratics (Ch. 5) Sketch the graph, find a, AOS, vertex, y-intercept, and roots (if imaginary – find them). Ex. y = −2 x2 + 8x + 10

Ex. y = 3x2 + 6 x

Opens: AOS: Vertex: y-int: Roots:

Opens: AOS: Vertex: y-int: Roots:

Ex. y = x2 + 4

Ex: y = −2 x2 + 8x − 9

Opens: AOS: Vertex: y-int: Roots:

Opens: AOS: Vertex: y-int: Roots:

Homework Part 5: Graph 1) h( x) =

1 2 x − 2x − 4 2

Opens: AOS: Vertex: Y-intercept: X-intercepts:

2) y = Opens:

2x2 +

AOS: Vertex: Y-intercept: Zeros:

3x – 3

3)

f ( x) = − x 2 + 4 x − 1

5) y = x2 + 3x – 10

Opens:

Opens:

AOS:

AOS:

Vertex:

Vertex:

Y-intercept:

Y-intercept:

Roots:

Solutions:

2 4) y = x − 3x + 2

6)

y = x2 + 2 x + 3

Opens:

Opens:

AOS:

AOS:

Vertex:

Vertex:

Y-intercept:

Y-intercept:

X-intercepts:

Roots:

Quadratic Inequality (5.7)

≤

Shading Line type




≥

Ex: y ≥ x2 − 2 x − 3 Vertex: Roots:

Ex: y < 2 x2 − 9 x + 4 Vertex: Roots:

Ex: y < x 2 − 4 Vertex: Roots:

*Ex: − x2 + y ≥ −6x + 2 Vertex: Roots:

Ex: y > −2 x2 + 8x Vertex: Roots: 2 y ≥ x C. ____

D. Match: 2 y ≤ x A. ____

B. 2 y ≤ − x ____

2 y ≥ − x ____

Homework Part 5 (cont.): Quadratic Inequalities (graph each one) 2 2 1. y ≥ x − 4 x + 3 4. y ≤ 4 − 3x − x

2.

y ≥ −( x + 1)2 + 5

3.

f ( x) ≥ 8 + x − x 2

5.

0 ≤ x2 − x

6.

0 < x2 + 2x + 5

7.

0 < x2 − 2x + 6

8.

f ( x) < x 2 + 2 x − 5

9.

0 > x2 − 4

Application of Quadratics Ex: A golf ball is hit from ground level with an initial vertical velocity of 80 ft/s. After how many seconds will the ball hit the ground?

Ex: A business offers educational tours to Patagonia, a region of South America that includes parts of Chile and Argentina. The profit P for x number of persons is P(x) = –25x2 + 1250x – 5000. The trip will be rescheduled if the profit is less $7500. How many people must have signed up if the trip is rescheduled? Ex: A football is kicked from ground level with an initial vertical velocity of 48 ft/s. How long is the ball in the air?

Ex: The monthly profit P of a small business that sells bicycle helmets can be modeled by the function P(x) = –8x2 + 600x – 4200, where x is the average selling price of a helmet. What range of selling prices will generate a monthly profit of at least $6000?

Homework PART 6: Quadratic Inequalities Use h = −16t 2 + v0 t + h0 where h = height (ft), t =time (sec), initial velocity = v0 and initial height = h0 . 1) A toy rocket is fired straight up from ground level with an initial velocity of 80 feet per second. During what time interval will it be at least 64 feet above the ground?

2) A ball is dropped from the roof of a 120 foot high building. During what time period will it be at least 56 feet high?

Real World Word Problems Continued (Part 1 – Spiral)

Use h = −16t 2 + v0t + h0 where h = height (ft), t =time (sec), initial velocity = v0 and initial height = h0 . 5) A toy rocket is fired straight up from ground level with an initial velocity of 80 feet per second. Write the equation ______________________ Maximum height: Time at maximum height: Time at landing: Domain: ___________

Range: ______________

6) A ball is dropped from the roof of a 120 foot high building. Write the equation ______________________ Maximum height: Time at maximum height: Time at landing (to nearest tenth): Domain: ___________

Range: ______________

7) The area of a triangle is given by the equation –h2 + h = –30, where h is the height in feet. What is the height of the triangle?

8) The area of a triangle is given by the equation h2 – 20 = 8h, where h is the height in millimeters. What is the height of the triangle?