1

The Graph of a Quadratic Function (Day 11–1)

Quadratic equations are written as _______________________ ; this equation has a degree

of _________.

The graph of this equation is called a _____________________; it is _____________________.

Parabolas are functions because they ____________________________________.

Exercise 1: Consider the quadratic equation 822 xxy .

a) Graph the equation.

b) Draw in the axis of symmetry. What is its equation?

c) The formula for the axis of symmetry is a

bx

2

. Use it to see if you get the same answer

as in part b.

d) Plot a point on the turning point and label it A. What is this coordinate?

e) How does this coordinate relate to the equation of the axis of symmetry? How can the

axis of symmetry be used to find the turning point?

f) What are the zeroes/roots? Determine them algebraically to see if you get the same.

2

Exercise 2: Graph 462 xxy .

a) Why is the vertex (turning point) a maximum point?

b) Determine the zeroes of the quadratic equation to the nearest tenth.

Quadratic Equations – 2 Types

CBxAxy 2

When “A” is positive, the parabola opens: _______

Where the curve reaches a ___________________

When “A” is negative, the parabola opens: _______

Where the curve reaches a ___________________

How to Graph Parabolas:

1. Find the axis of symmetry by using the formula: ________________________

2. Make a table of values with the axis of symmetry x-value in the middle (y-values

should repeat on both sides).

3. Graph the points.

3

Classwork 11–1

1. Consider the quadratic equation 342 2 xxy .

a) Algebraically determine the axis of symmetry and the turning point.

b) Graph 342 2 xxy .

c) Describe what the axis of symmetry looks like and what it goes through.

d) Is the graph a smiley-face, or frowned-face parabola? Why?

e) Is the vertex (turning point) a minimum or a maximum point? Explain.

f) Solve for the zeroes of 342 2 xxy to the nearest tenth. Describe what these zeroes

of the parabola represent on the graph.

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2. Write a quadratic equation that has zeroes of 5x and 8x .

3. A student correctly graphed the parabola shown below. Using this graph, answer the

questions below.

a) Find the zeroes of the graph.

b) Write the equation that represents this graph.

4. Which of the following functions would look like a frowny-face parabola (upside down U)?

(1) 52 xy (3) 2452 xxy

(2) 83 xy (4) 1832 xxy

5. Mr. Jones asked his class “Is the sum of 7.4 and 6 rational or irrational? Patrick answered

that sum would be irrational.

State whether Patrick is correct or incorrect. Justify your reasoning.

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More w/ Graphing Quadratic Functions (Day 11–2)

Exercise 1: Graph xxxf 4)( 2

a) What are the roots of )(xf ?

b) What are the domain and range of )(xf ?

c) Is )(xf a function? Why or why not?

d) Determine )5(f . e) Determine x if 3)( xf .

f) Use the roots to go back and determine the equation of )(xf .

How to Graph Parabolas:

3. Find the axis of symmetry by using the formula: ________________________

4. Make a table of values with the axis of symmetry x-value in the middle (y-values

should repeat on both sides).

3. Graph the points.

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Exercise 2: What are the zeroes of the parabola on the grid?

Write an equation for this graph.

Exercise 3: Graph )24(32)( 2 xxxxf This is called an interval, which means your table

should cover ONLY the x-values from –4 to 2.

Exercise 4: Write a quadratic equation that has zeroes of 3x and 1x , and whose

turning point is a maximum.

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Classwork 11–2

1. Algebraically determine the equation of the axis of symmetry for 122 xxy .

2. Algebraically determine the coordinates of the vertex for the parabola whose equation

is 442 xxy .

3. Write a quadratic equation that has zeroes of 6x and 4x , and whose turning

point is a minimum.

4. Given the graph shown, answer the following:

a) State the zeroes of the parabola.

b) Write an equation to represent this graph.

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5. Graph 10122)( 2 xxxg .

a) What are the roots of )(xg ?

b) What are the domain and range of )(xg ?

c) Is )(xg a function? Why or why not?

d) Determine )0(g . e) Determine x if 6)( xg .

6. Algebraically determine the zeroes of 10122)( 2 xxxg .

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Solving Quadratic–Linear Systems (Day 11–3)

Recall: When you have two or more equations, you have a system. In this lesson, we will

specifically be looking at a system made of a quadratic equation (degree of _____ )

and a linear equation (degree of _____ ).

The solutions to a system are (x, y) coordinates.

Determine the solution(s) to the systems graphed below.

1. Solve the following system of equations graphically.

1

342

yx

xxy

To check on your graphing calculator (find intersection):

1) Go to (Calculate) and pick (intersection)

Enter Enter Enter

#5 2nd Trace

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2) Move cursor to wanted intersection point and hit

2. Solve the following system of equations graphically and check.

42

442

xy

xxy

3. If 152)( xxh and xxxw 6)( 2 , which ordered pair below makes )()( xwxh ?

(1) (5, 5) (3) (6, 0)

(2) (0, 0) (4) (3, –9)

4. If 82)( 2 xxxf and 14

1)( xxg , for which values of x is )()( xgxf ?

(1) –1.75 and –1.438 (3) –1.75 and 4

(2) –1.438 and 0 (4) 4 and 0

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Classwork 11–3

For #1 – 2, solve the quadratic-linear systems graphically.

1. 4

662

xy

xxy

2. 22

242 2

xy

xxy

12

3. Using your calculator, find the solution(s) to the following system:

xy

xxy

6312

442

4. Using your calculator, find the solution(s) to the following system:

2

842 2

y

xxy

5. The area of the rectangular playground at Hillview is 150 sq. meters. The length is 25

meters longer than the width. Find the dimensions of the playground.

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Solving Quadratic–Linear Systems Algebraically (Day 11–4)

Exercise 1: Solve the following system algebraically.

xy

xxy

2

22

Procedure:

1. Make sure both equations are in y form (if necessary).

2. Substitute the linear equation in for the ‘y’ of the quadratic equation, to have only one

variable left to solve in the equation.

3. Get NEW quadratic equation into standard form (______________________________).

4. Solve the quadratic equation using one of the three methods.

(How many answers should you get? _______ )

5. Must find other variable (y) by substituting your ‘x’ answers into one of the equation

and solve for y.

6. Check solutions.

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2. Find the solutions of the following systems algebraically: 1

342

yx

xxy

3. Find the solutions of the following systems algebraically: 2

1372

yx

xxy

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Classwork 11–4

1. Solve the following system algebraically: 3

122

xy

xxy

2. Solve the following system algebraically: 5

5 2

xy

xy

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3. The Math Honor Society orders t-shirts to wear for pi day. The cost of the t-shirts can be

modeled by the equation 254.21.0 2 xxC , where C is the amount it costs to buy x

shirts. How many t-shirts can they purchase for $430?

4. Alexandra purchases two doughnuts and three cookies at a doughnut shop and is

charged $3.30. Briana purchases five doughnuts and two cookies at the same shop for

$4.95. All the doughnuts have the same price and all the cookies have the same price.

Find the cost of one doughnut and find the cost of one cookie.

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Exploring Factor/Zeroes of Higher Degree Equations (Day 11–5)

1. The zeroes of the function 633)( 2 xxxf are:

(1) –2 and –1 (3) –2 and 1

(2) –1 and 2 (4) 1 and 2

2. A polynomial function has zeroes of –5, 2, and 9. What would be the factors of this

polynomial function?

3. A polynomial function contains the factors x, x – 3, and x + 6. Which graph(s) below could

represent the graph of this function? Explain your answer.

(1) (2) (3)

4. Algebraically determine the zeroes of the polynomial function 3613)( 24 xxxh .

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5. Given the following three functions, which one has the smallest minimum? Justify.

442 xxy 242)( 2 xxxf

6. A polynomial function is graphed to the right.

a) State the domain:

b) State the range:

c) Find )0(f .

What is the name for this point?

d) For what interval(s) is this function decreasing?

e) For what interval(s) is this function increasing?

f) What are the zeroes of this function?

g) State the factors that would represent this polynomial function.

h) Which equation(s) represent the graph? Explain your reasoning.

)65)(1(

)2)(3(

)124)(2(

2

2

2

xxxyIII

xxxyII

xxxyI

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Classwork 11–5

1. Which one has the highest maximum? Explain.

382 2 xxy 310)( 2 xxxf

2. A polynomial is graphed to the right.

a) State the domain:

b) State the range:

c) Find )0(f .

d) For what interval(s) is this function decreasing?

e) What are the zeroes of this function?

f) State the factors that these zeroes come from.

g) Which equation(s) represent(s) the graph? (1) )214)(1( 2 xxxy

Explain your reasoning. (2) )32)(7( 2 xxxy

(3) )76)(3( 2 xxxy

y

x

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3. The equation of the axis of symmetry of the graph of 732 2 xxy is:

(1) 4

3x (3)

4

3y

(2) 2

3x (4)

2

3y

4. A polynomial contains the factors x + 6, x – 1, and x + 4. Which graph(s) below could

represent the graph of this function? Explain your answer.

(1) (2) (3)

5. Algebraically determine the zeroes of the polynomial function 1617)( 24 xxxh .

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Quadratic Application Word Problems (Day 11–6)

1. Let 806416)( 2 ttth represents the height of an object above the ground after t

seconds.

a) Determine the number of seconds it takes for the object to achieve its maximum

height. Justify your answer.

b) State the time interval, in seconds, during which the height of the object decreases.

Explain your reasoning.

2. After t seconds, a ball tossed in the air from the ground level reaches a height of h feet

given by the function 216144)( ttth .

a) What is the height of the ball after 3 seconds?

b) What is the maximum height the ball will reach?

c) After how many seconds will the ball hit the ground before rebound?

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3. A toy rocket is launched from the ground straight upward. The height of the rocket

above the ground, in feet, is given by the equation ttth 6416)( 2 , where t is time in

seconds. Determine the domain for this function in the given context. Explain your

reasoning.

4. A rocket carrying fireworks is launched from a hill 80 feet above a lake. The rocket will fall

into the lake after exploding at its maximum height. The rocket’s height above the

surface of the lake is given by the function 806416)( 2 ttth .

a) What is the height of the rocket after 1.5 seconds?

b) What is the maximum height reached by the rocket?

c) After how many seconds after it is launched will the rocket hit the lake?

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Classwork 11–6

1. After t seconds, a model rocket launched in the air from ground level reaches a height h

feet, where 21296 tth . The rocket then falls back to the ground.

a) What is the height of the rocket after 7.5 second?

b) What is the maximum height reached by the rocket?

c) How long will it take for the rocket to reach 180 feet?

d) After how many seconds after it is launched will the rocket hit the ground?

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2. A man in a carnival is shot out of a cannon. The height that the man above the ground,

in feet is given by the equation ttth 328)( 2 , where t is the time in seconds. Determine

the domain for this function in the given context. Justify your answer.

3. Let 40328)( 2 tth represent the height of an object above the ground after t

seconds.

a) Determine the number of seconds it takes to achieve its maximum height. Justify your

answer.

b) State the time interval, in seconds, during which the height of the object decreases.

Explain your reasoning.

4. A rock is thrown from the top of a tall building. The distance, in feet, between the rock

and the ground t seconds after it is thrown is given by 382416)( 2 tttd . How long

after the rock is thrown is it 370 feet from the ground?

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Mixed Quadratic Review (Day 11–7)

1. A company is considering building a manufacturing plant. They determine their weekly

production cost at site A to be 23)( xxA while their production cost at site B is

38)( xxB , where x represents the number of products, in hundreds, and )(xA and )(xB

are the production costs, in hundreds of dollars.

a) Graph the production cost functions on the set of axes below and label them site A

and site B.

b) State the positive value(s) of x for which the production costs at the two sites are

equal. Explain how you determined your answer.

c) If the company plans on manufacturing 200 products per week, which site should they

use? Justify your answer.

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2. Given the two graphs below, write an equation for each.

a) b)

3. The zeroes of the function 25)2()( 2 xxf are

(2) −2 and 5 (3) −5 and 2

(3) −3 and 7 (4) −7 and 3

4. Find three consecutive positive integers such that the product of the second integer and

the third integer is 20.

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Classwork 11–7

1. A swim team member performs a dive from a 14 foot high springboard. The parabola

below shows the path of her dive.

a) How far was the diver from the springboard

when she hit the water?

b) What is the greatest height the diver reaches?

c) Find x, if 14)( xf . What point does this represent?

d) Find )2(f .

e) What is the axis of symmetry?

f) Over what interval is the graph decreasing?

2. Wilson throws a tennis ball into the air and its path can be modeled by the function

7.43.348.9)( 2 ttth , where h represents the height in meters, and t represents the time

in seconds. To the nearest hundredth, at what time(s) is the tennis ball at 30 meters in the

air?

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3. Graph the quadratic function 1282 xxy .

4. Using the given graph of )(xf , answer the following

a) What is the vertex of )(xf ?

b) What is the axis of symmetry of )(xf ?

c) What are the zeroes of )(xf ?

d) Write the equation of )(xf .

e) Find )5(f .

f) Find 7)( xf .

g) State the domain and range of )(xf .