Graph Quadratic Functions in Standard Formlhsborn.weebly.com/uploads/1/6/5/4/16542638/unit_3_quadratic_fun… · 3. Evaluate the function for two values of x. x 5 1: y 5 0 x 5 3:

56 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.

3.1 Graph Quadratic Functions in Standard FormGoal p Use intervals of increase and decrease to

understand average rates of change of quadratic functions.

GeorgiaPerformanceStandard(s)

MM2A3b, MM2A3c

VOCABULARY

Quadratic function

Parabola

Vertex

Axis of symmetry

Minimum and maximum value

Extrema

Your Notes

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ga2_ntg_03.indd 56ga2_ntg_03.indd 56 4/19/07 3:39:20 PM4/19/07 3:39:20 PM

Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 57

Your Notes

Graph y 5 2x2 1 4x 2 3.

1. Because a 0, the parabola opens .

2. Find the vertex. First, calculate

x

y1

1the x-coordinate.

x 5 2b

}2a 5 5

Then find the y-coordinate.y 5 5

The vertex is ( , ). Plot this point.

3. Draw the axis of symmetry x 5 .

4. Identify the y-intercept c, which is . Plot the point (0, ). Then reflect this point in the axis of symmetry to plot another point (4, ).

5. Evaluate the function for x 5 1.y 5 5

Plot the point (1, ) and its reflection (3, ) in the axis of symmetry.

6. Draw a parabola through the plotted points.

Example 2 Graph a function of the form y 5 ax2 1 bx 1 c

Graph y 5 22x2 1 2. Compare the graph with the graph of y 5 x2. Identify the domain and range.

1. Make a table of values for y 5 22x2 1 2.

x 22 21 0 1 2

y

2. Plot the points from the table. x

y

2

2

y 5 x2

3. Draw a smooth throughthe points.

4. Compare the graphs. Both graphs have the same . However, the graph of

y 5 22x2 1 2 opens and is than the graph of y 5 x2. Also, its vertex is units higher.

5. Identify the domain and range. The domain is and the range is .

Example 1 Graph a function of the form y 5 ax2 1 c

ga2nt-03.indd 37 4/16/07 9:06:06 AM



Your Notes

1. Graph y 51}2x2 1 2. Identify the domain and range.

x

y

1

1

2. Graph the function. Label the

x

y

1

1

vertex and axis of symmetry.

y 5 x2 2 4x 1 2

Checkpoint Complete the following exercises.

Tell whether the function y 5 23x2 1 12x 2 6 has a minimum value or a maximum value. Then find the minimum or maximum value.

Solution

Because a 0, the function has a value. To find it, calculate the coordinates of the vertex.

x 5 2b

}2a 5 5

y 5 5

The maximum value is y 5 .

Example 3 Find the minimum or maximum value

3. Tell whether the function y 5 2x2 2 6x 1 6 has a minimum or maximum value. Then find the minimum or maximum value.

Checkpoint Complete the following exercise.

ga2nt-03.indd 38 4/18/07 3:24:29 PM



Your Notes

4. In Example 4, what is the maximum daily revenue?


Homework

Revenue A bike shop sells about 18 bikes each day when they charge $42 per bike. For each $1 decrease in price, they sell about 3 more bikes each day. How can the bike shop maximize daily revenue?

Solution1. Define the variables. Let x represent the price

reduction and R(x) represent the daily revenue.

2. Write a verbal model. Then write and simplify a quadratic function.

Revenue (dollars) 5

Price(dollars/bike) p

Numberof bikes

R(x) 5 ( 2 x) p ( 1 3x)

R(x) 5 1 2 2

R(x) 5 1 1

3. Find the coordinates (x, R(x)) of the vertex.

x 5 2b

}2a 5 2 5 Find x-coordinate.

R( ) 5 23( )2 1 108( ) 1 756

Evaluate R( ).

The vertex is ( , ). The shop should reduce the price by to maximize daily revenue.

Example 4 Find the maximum value of a quadratic function

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Copy and complete the table of values for the function.

1. y 5 3x2 2. y 5 22x2

x 22 21 0 1 2

y ? ? ? ? ?

x 22 21 0 1 2

y ? ? ? ? ?

3. y 5 − 3 }

2 x2 4. y 5

1 } 5 x2

x 24 22 0 2 4

y ? ? ? ? ?

x 210 25 0 5 10

y ? ? ? ? ?

For the following functions (a) tell whether the graph opens up or opens down, (b) fi nd the vertex, and (c) fi nd the axis of symmetry.

5. y 5 x2 1 1 6. y 5 2x2 2 4

7. y 5 x2 2 2x 8. y 5 x2 1 2x 1 1

9. y 5 3x2 2 6x 1 4 10. y 5 22x2 1 5x 2 3

LESSON

3.1 Practice

Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON

3.1 Practice continued

Match the equation with its graph.

11. y 5 2x2 1 8x 2 2 12. y 5 x2 2 2 13. y 5 2 1 } 5 x2 1 2x 2 1

A.

x

y

1

2

B.

x

y

2

2

C.

x

y

2

2

Graph the function. Label the vertex and axis of symmetry.

14. y 5 x2 1 2 15. y 5 2x2 2 4x 16. y 5 2x2 1 2x 1 3

x

y

1

1

x

y

1

1

x

y

1

1

17. y 5 4x2 1 8x 1 1 18. y 5 3x2 2 9x 2 2 19. y 5 23x2 1 12x 1 7

x

y

1

1

2

2

x

y

x

y

3

3




20. y 5 1 }

2 x2 2 2x 2 2 21. y 5 2

3 } 2 x2 1 3 22. y 5

1 }

4 x2 2

1 }

2 x

1

1

x

y

x

y

1

1

x

y

1

23. Wrist Watch Brand X wrist watches at a department store are selling for $50 at a rate of 45 per month. The marketing department determined that for every $2 decrease in price, 3 more watches would be sold per month. Write a quadratic equation in standard form that models the revenue R from watch sales. How can the store maximize monthly revenue?

24. Motocross The path that a motocross dirt bike rider follows during a jump is given by y 5 20.4x2 1 4x 1 10 where x is the horizontal distance (in feet) from the edge of the ramp and y is the height (in feet). What is the maximum height of the rider during the jump?

0 4 8 122 6 10 x0

2

4

6

810

12

14

16

18

20y

Horizontal distance (feet)

Heig

ht

(feet)

LESSON


Name ——————————————————————— Date ————————————



3.2 Graph Quadratic Functions in Vertex or Intercept FormGoal p Graph quadratic functions in vertex form or

intercept form.


MM2A3a, MM2A3c VOCABULARY

Vertex form

Intercept form

Graph y 51}2(x 1 1)2 2 2.

1. Identify the constants a 5 , h 5 , and

k 5 . Because a > 0, the parabola opens .

2. Plot the vertex (h, k) 5 ( , ) and draw the axis of symmetry at x 5 .

3. Evaluate the function for two values of x.x 5 1: y 5 0x 5 3: y 5 6

Plot the points (1, ) and (3, )x

y

1

1

and their reflections in the axis of symmetry.

4. Draw a parabola through the plotted points.

Example 1 Graph a quadratic function in vertex form

1. y 5 2(x 2 3)2 1 4

x

y

1

1

Checkpoint Graph the function. Label the vertex and axis of symmetry.

Your Notes

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Your Notes

Graph y 5 22(x 2 1)(x 2 5).

1. Identify the x-intercepts. Because

x

y

1

1

p 5 and q 5 , the x-interceptsoccur at the points ( , 0) and ( , 0).

2. Find the coordinates of the vertex.

x 5p 1 q}

2 52

5

y 5 5

So, the vertex is ( , ).

3. Draw a parabola through the vertex and the points where the x-intercepts occur.

Example 2 Graph a quadratic function in intercept form

2. y 5 (x 2 4)(x 1 2)x

y2

2

3. y 5 2(x 1 1)(x 2 3)

x

y

1

1

Checkpoint Graph the function. Label the vertex, axis of symmetry, and x-intercepts.

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Your Notes

Towers Two generator towers are designed with an electric cable that connects them. The ends of the cable are the same height above the ground. The cable can be modeled by

y 5(x 2 1400)2}

34001 10

where x is the horizontal distance (in feet) from the left tower and y is the corresponding height (in feet) of the cable. Find the distance between the towers.

Solution

The vertex of the parabola is ( , ). The cable's lowest point is feet from either tower. The distance between the towers is d 5 ( ) 5 feet.

Example 3 Use a quadratic model in vertex form

4. Suppose in Example 3, the cable is modeled by

y 5(x 2 1500)2}

2400 1 20.

Find the distance between the towers.


ga2nt-03.indd 42 4/16/07 9:06:11 AM



Your Notes

a. y 5 3(x 1 2)(x 2 5)

b. y 5 25(x 1 2)2 1 8

Solution

a. y 5 3(x 1 2)(x 2 5) Original function

5 3 Multiply using FOIL.

5 3 Combine like terms.

5 Distributive property

b. y 5 25(x 1 2)2 1 8 Original function

5 25( )( ) 1 8 Rewrite (x 1 2)2.

5 25( ) 1 8 Multiply using FOIL.

5 25( ) 1 8 Combine like terms.

5 1 8 Distributive property

5 Combine like terms.

Example 4 Change quadratic functions to standard form

5. y 5 4(x 2 3)2 2 10

6. y 5 23(x 2 7)(x 1 6)

Checkpoint Write the quadratic function in standard form.

Homework

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Name ——————————————————————— Date ————————————

LESSON

3.2 PracticeMatch the equation with its graph.

1. y 5 (x 2 1)2 2. y 5 (x 2 2)(x 1 4) 3. y 5 22(x 1 1)2 1 3

A.

x

y

1

1

B.

x

y

1

1

C.

x

y2

4


4. y 5 (x 2 1)2 1 1 5. y 5 (x 2 3)2 1 2 6. y 5 (x 1 1)2 2 2

x

y

1

1

x

y

1

1

x

y

1

1

7. y 5 2(x 1 1)2 1 2 8. y 5 4(x 2 2)2 2 1 9. y 5 22(x 2 3)2 2 3

x

y

1

1

x

y

1

1

x

y

211



Graph the function. Label the vertex, axis of symmetry, and x-intercepts.

10. y 5 (x 2 1)(x 2 5) 11. y 5 (x 1 2)(x 2 2) 12. y 5 (x 1 6)(x 1 2)

x

y1

1

x

y1

1

x

y1

21

13. y 5 2(x 1 3)(x 2 1) 14. y 5 2(x 1 1)(x 2 2) 15. y 5 23(x 2 1)(x 1 4)

x

y

1

1

x

y1

1

x

y

3

3

Write the quadratic function in standard form.

16. y 5 2(x 2 1)2 1 1 17. y 5 2(x 1 3)2 1 5

18. y 5 3(x 2 2)2 2 7 19. y 5 (x 2 3)(x 2 1)

20. y 5 2(x 1 1)(x 1 4) 21. y 5 23(x 2 2)(x 1 3)

LESSON


Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


Find the minimum value or the maximum value of the function.

22. y 5 (x 2 3)2 1 1 23. y 5 22(x 1 1)2 1 5

24. y 5 4(x 2 2)2 2 7 25. y 5 (x 1 3)(x 1 1)

26. y 5 2(x 2 1)(x 2 5) 27. y 5 24(x 2 3)(x 1 2)

In Exercises 28 and 29, use the following information.

Golf The fl ight of a particular golf shot can be modeled by the function y 5 20.0015x(x 2 280) where x is the horizontal distance (in yards) from the impact point and y is the height (in yards). The graph is shown below.

0 80 160 240 x0

5

10

15

2025

30

35y

Horizontal distance (yards)

Heig

ht

(yard

s)

28. How many yards away from the impact point does the golf ball land?

29. What is the maximum height in yards of the golf shot?



3.3 Interpret Rates of Change of Quadratic Functions


MM2A3c

Your Notes

Goal p Use intervals of increase and decrease to understand average rates of change of quadratic functions.

Graph the function y 5 x2 1 x 2 2. Identify the intervals over which the graph increases and decreases.

SolutionYou can see from the graph that as you

x

y

1

1

move from left to right the value of the function on the left side of the vertex and on the right side of the vertex. The x-coordinateof the vertex is

x 5 2b

}2a 5 2

21 25 .

The graph over the interval x >

and over the interval x < .

Example 1 Identify intervals of increase and decrease

1. y 51}2x2 1 1

x

y

1

1

Checkpoint Graph the function. Identify the intervals over which the graph increases and decreases.

ga2nt-03.indd 44 4/16/07 9:06:13 AM



Your Notes

Compare the average rates of change of y 5 2x2 and y 5 2x on 0 ≤ x ≤ 1.

SolutionThe average rate of change of y 5 2x

x

y

1

1

is the slope of the line, which is 21.The points and correspond to the endpoints of the interval for y 5 2x2. The average rate of change of y 5 2x2 on 0 ≤ x ≤ 1 is

r 5 5 5 .

The average rate of change of the quadratic function is times as great as the average rate of change of the linear function on the interval 0 ≤ x ≤ 1.

Example 3 Compare average rates of change

Calculate the average rate of change of the function y 5 2x2 2 1 on the interval 21 ≤ x ≤ 0.

SolutionFind the two points on the graph of

x

y

1

1

the function that correspond to the endpoints of the interval. The average rate of change is the slope of the line that passes through these two points.

y 5 2( )2 2 1 y 5 2( )2 2 1

5 2 1 5 2 1

5 5

The points are and .

The average rate of change is:

r 5 2 1 2

5 5

Example 2 Calculate the average rate of change

2

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Your Notes

2. Calculate the average rate of change of y 5 x2 1 3x 2 2 on the interval 1 ≤ x ≤ 2.

3. Compare the average rates of change of

y 51}4x2 1 1 and y 5 2x on 2 ≤ x ≤ 4.


Baseball The path of a baseball thrown at an angle of 408can be modeled by y 5 20.05x2 1 3.2x 1 8 where x is the horizontal distance (in feet) from the release point and y is the corresponding height (in feet). Find the interval on which the height is increasing.

SolutionThe height of the baseball will be increasing from the release point until it reaches its maximum height at the vertex. The x-coordinate of the vertex is

x 5 . So, the height will be increasing

on the interval .

Example 4 Solve a real world problem

4. 0 ≤ x ≤ 32

Checkpoint For the quadratic model in Example 4, find the average rate of change on the given interval.

Homework

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Name ——————————————————————— Date ————————————

LESSON

3.3 PracticeGraph the function. Identify the intervals over which the graph increases and decreases.

1. y 5 x2 2 3x 1 2 2. y 5 (x 1 1)2 2 2

x

y

1

1

x

y

1

1

3. y 5 2(x 2 2)2 1 4 4. y 5 (x 1 4)(x 2 1)

x

y

1

1

x

y

211

Calculate the average rate of change of the function on the given interval.

5. y 5 x2 1 2, 1 ≤ x ≤ 2 6. y 5 1 } 2 x2 2 1, 2 ≤ x ≤ 4

7. y 5 3x2 1 x 2 5, 21 ≤ x ≤ 1 8. y 5 2x2 1 5, 1 ≤ x ≤ 3



Classify the given interval as an interval of increase or decrease.

9. 23 ≤ x ≤ 0 10. 3 ≤ x ≤ 5 11. 1 ≤ x ≤ 2

x

y

1

1

x

y

2

1

x

y

1

1

Compare the average rates of change of the functions on the given interval.

12. y 5 3x and y 5 2x2 1 7 on 0 ≤ x ≤ 3

13. y 5 x 1 2 and y 5 1 } 2 x2 1 x 2 2 on 0 ≤ x ≤ 2

14. Basketball The path of a basketball after being thrown can be modeled by the function y 5 20.03x2 1 1.025x 1 4 where x is the horizontal distance (in feet) from where the ball was thrown and y is the corresponding height (in feet). Find the interval on which the height is increasing. What is the average rate of change on this interval?

LESSON


Name ——————————————————————— Date ————————————



3.4 Solve x2 1 bx 1 c 5 0 by FactoringGoal p Use factoring to solve quadratic equations.Georgia

PerformanceStandard(s)

MM2A3c, MM2A4b

Your Notes

VOCABULARY

Monomial

Binomial

Trinomial

Quadratic equation

Root of an equation

Zero of a function

SPECIAL FACTORING PATTERNS

Pattern Name

Difference of a2 2 b25 ( )( )Two Squares x2 2 4 5 (x 1 2)(x 2 2)

Perfect Square a2 1 2ab 1 b2 5 ( )2

Trinomial x2 1 6x 1 9 5 (x 1 3)2

Perfect Square a2 2 2ab 1 b2 5 ( )2

Trinomial x2 2 4x 1 4 5 (x 2 2)2

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Your Notes

Factor the expression x2 1 7x 2 8.

Solution

You want x2 1 7x 2 8 5 (x 1 m)(x 1 n) where mn 5and m 1 n 5 .

Factors of 28 (m, n) 21, 1,

Sum of factors (m 1 n)

Factors of 28 (m, n) 22, 2,

Sum of factors (m 1 n)

Notice that m 5 and n 5 . So, x2 1 7x 2 8 5 ( )( ).

Example 1 Factor trinomials of the form x2 1 bx 1 c

Factor the expression.

a. x2 2 25 5 x2 2 Difference of two squares 5 ( )( )

b. m2 2 22m 1 121 Perfect square trinomial5 m2 2 2(m)( ) 1 2

5 ( )2

Example 2 Factor with special patterns

1. x2 1 7x 1 12 2. x2 2 81

Checkpoint Factor the expression. If it cannot be factored, say so.

ga2nt-03.indd 48 4/16/07 9:06:17 AM



Your Notes

Find the roots of the equation x2 2 2x 2 15 5 0.

Solution x2 2 2x 2 15 5 0 Original equation

( )( ) 5 0 Factor.

5 0 or 5 0 Zero product property

x 5 or x 5 Solve for x.

The roots are and .

Example 3 Find the roots of an equation

Patio A rectangular patio measures 20 feet by 30 feet. By adding x feet to the width and x feet to the length, the area is doubled. Find the new dimensions of the patio.

Solution

New area 5 New width p New length

2( )( ) 5 ( 1 x) p ( 1 x)

5 Multiply using FOIL.

0 5 Write in standard form.

0 5 (x 2 )(x 1 ) Factor.

x 2 5 0 or x 1 5 0 Zero product property

x 5 or x 5

Solve for x.

Reject the negative value. The patio's length and width should each be increased by feet. The new dimensions are feet by feet.

Example 4 Use a quadratic equation as a model

ga2nt-03.indd 49 4/16/07 9:06:18 AM



Your Notes

3. Find the roots of x2 1 2x 2 3.

4. Rework Example 4 where a rectangular patio measures 16 feet by 24 feet.


Example 5 Find the zeros of a quadratic function

Find the zeros of the function y 5 x2 1 5x 2 6 by rewriting the function in intercept form.

Solutiony 5 x2 1 5x 2 6 Write original equation.

5 ( )( ) Factor.

The zeros of the function are and .

CHECK Graph y 5 x2 1 5x 2 6. The graph passes through ( , 0) and ( , 0).

5. Find the zeros of the function y 5 x2 1 3x 2 40 by rewriting the function in intercept form.


Homework

ga2nt-03.indd 50 4/16/07 9:06:18 AM



Name ——————————————————————— Date ————————————

LESSON

3.4 PracticeFactor the expression. If the expression cannot be factored, say so.

1. x2 2 x 2 2 2. x2 1 4x 1 3 3. x2 2 4x 1 3

4. x2 2 5x 1 6 5. x2 1 x 2 2 6. x2 2 5x 1 4

7. x2 1 8x 1 15 8. x2 2 6x 1 8 9. x2 1 3x 2 18

10. x2 2 4 11. x2 2 4x 1 4 12. x2 1 7x 1 12

13. x2 1 2x 1 1 14. x2 2 6x 1 9 15. x2 2 16

Solve the equation.

16. x2 2 2x 2 3 5 0 17. x2 1 3x 1 2 5 0 18. x2 2 10x 1 16 5 0

19. x2 2 2x 1 1 5 0 20. x2 1 6x 1 8 5 0 21. x2 1 4x 1 4 5 0

22. x2 1 7x 2 8 5 0 23. x2 1 10x 1 25 5 0 24. x2 2 9x 1 14 5 0

25. x2 2 49 5 0 26. x2 2 4x 5 12 27. x2 5 64



Find the zeros of the function by rewriting the function in intercept form.

28. y 5 x2 1 x 2 20 29. y 5 x2 2 9 30. f (x) 5 x2 2 7x 1 6

31. y 5 x2 1 14x 1 49 32. g(x) 5 x2 1 7x 1 10 33. y 5 x2 2 6x 2 7

34. h(x) 5 x2 1 5x 2 24 35. y 5 x2 2 81 36. y 5 x2 1 3x

Find the value of x.

37. Area of the rectangle 5 28

x 1 3

x

38. Area of the rectangle 5 32

x 2 4

x

39. Hopscotch The community playground has a hopscotch pad that is 8 feet longer than it is wide. The total area of the pad is 48 square feet. What are the dimensions of the hopscotch pad?

LESSON


Name ——————————————————————— Date ————————————



3.5 Solve ax2 1 bx 1 c by FactoringGoal p Use factoring to solve equations of the form

ax2 1 bx 1 c 5 0.


MM2A4b

Your Notes Factor 3x2 2 x 2 2.

You want 3x2 2 x 2 2 5 (kx 1 m)(lx 1 n) where k and l are factors of and m and n are factors of .Because mn 0, m and n have signs.

k, l m, n (kx 1 m)(lx 1 n) ax2 1 bx 1 c

3, 1 21, ( )( )

3, 1 2, ( )( )

3, 1 1, ( )( )

3, 1 22, ( )( )

The correct factorization is

3x2 2 x 2 2 5 ( )( ).

Example 1 Factor ax2 1 bx 1 c where c < 0

Checkpoint Factor the expression. If it cannot be factored, say so.

1. 3x2 1 7x 2 20 2. 5x2 2 13x 1 6

ga2nt-03.indd 51 4/16/07 9:06:19 AM



Your Notes

Factor the expression.

a. 16x2 2 25 5 ( )2 2 2 Difference of two squares5 ( )( )

b. 9y2 1 42y 1 49 Perfect square trinomial5 ( )2 1 2( )( ) 1 2

5 ( )2

c. 24m2 2 10m 1 24

5 22( ) Factor monomial first.

5 22( )( ) Factor.

Example 2 Factor with special patterns

Checkpoint Factor the expression.

3. 4x2 2 81 4. 36x2 2 16

a. 2x2 2 x 2 21 5 0 Original equation

( )( ) 5 0 Factor.

5 0 or 5 0 Zero product property


b. 4r2 2 18r 1 24 5 6r 2 12 Original equation

5 0 Standard form

5 0 Divide each side by .

5 0 Factor.

5 0 Zero product property

r 5 Solve for r.

Example 3 Solve quadratic equations

ga2nt-03.indd 52 4/16/07 9:06:20 AM



Your Notes

5. 2x2 1 4x 2 30 5 0

6. z2 1 13z 1 12 5 5z 2 4

Checkpoint Solve the equation.

7. Rework Example 4 where the length is 6 inches more than the width.


Mirror The area of a mirror is 20 square inches and the length is 3 more inches than 2 times the width. Find the length of the mirror.

Solution

Area of mirror (square inches)

5 Width of mirror (inches)

pLength of mirror

(inches)

5 p ( )

0 5 Write in standard form.

0 5 ( 2 )( 1 ) Factor.

2x 2 5 0 or x 1 5 0 Zero product property


Reject the negative value. The length of the mirror is

inches or inches.

Example 4 Use a quadratic equation as a model

ga2nt-03.indd 53 4/16/07 9:06:22 AM



Your Notes

8. In Example 5, if the store were to only decrease the price by $2, what would be their weekly revenue?


Homework

Game A store sells about 80 board games per week when it charges $10 per game. For each decrease of $1, the store sells 20 more board games. How much should the store charge to maximize revenue? What is the maximum weekly revenue?

Solution1. Define the variables. Let x represent the number of $1

price decreases and R(x) represent the weekly revenue.

2. Write a verbal model. Then write and simplify a quadratic function.

Weekly sales (dollars) 5

Number of board games

soldp

Price of board game

(dollars/game)

5 p

5

3. Identify the zeros and find their average. The zeros are and . The average of the zeros is . To maximize revenue, the store should charge

.

4. Identify the maximum weekly revenue.

.

The maximum weekly revenue is .

Example 5 Solve a multi-step problem

ga2nt-03.indd 54 4/18/07 3:25:30 PM



Name ——————————————————————— Date ————————————

LESSON

3.5 PracticeFactor the expression. If the expression cannot be factored, say so.

1. 2x2 1 5x 1 2 2. 2x2 1 3x 1 1 3. 3x2 1 5x 2 2

4. 2x2 2 3x 1 1 5. 4x2 1 2x 2 2 6. 6x2 2 7x 2 3

7. 5x2 1 x 2 4 8. 9x2 2 3x 2 6 9. 4x2 1 13x 1 3

10. 6x2 1 2x 2 4 11. 4x2 2 9 12. 4x2 2 4

13. 2x2 2 10x 1 12 14. 3x2 2 9x 2 12 15. 27x2 2 3

16. 8x2 2 20x 2 12 17. 8x2 1 24x 1 18 18. 30x2 1 5x 2 10

Solve the equation.

19. 2x2 2 3x 1 1 5 0 20. 2x2 1 5x 1 3 5 0 21. 6x2 2 7x 1 2 5 0

22. 3x2 2 8x 2 3 5 0 23. 4x2 2 7x 1 3 5 0 24. 4x2 2 4x 2 15 5 0

25. 2x2 2 2x 2 12 5 0 26. 6x2 2 15x 2 9 5 0 27. 12x2 1 4x 2 8 5 0



Find the zeros of the function by rewriting the function in intercept form.

28. y 5 2x2 2 6x 1 4 29. y 5 3x2 1 6x 2 9 30. f (x) 5 5x2 1 10x 2 40

31. y 5 4x2 2 12x 1 9 32. g(x) 5 18x2 2 2 33. y 5 16x2 1 64x 1 60


34. Area of the square 5 81 35. Area of the rectangle 5 16

3x

3x

3x 1 2

x

36. Pool A pool deck of uniform width is going to be built

15 ft

20 ftx ft

x ft

x ft

x ft

around a rectangular pool that is 20 feet long and 15 feet wide. After the deck is built, a total of 414 square feet will be occupied. How wide is the deck encompassing the pool?

LESSON


Name ——————————————————————— Date ————————————



3.6 Solve Quadratic Equations by Finding Square RootsGoal p Solve quadratic equations by finding square roots.Georgia


MM2A4b

Your NotesVOCABULARY

Rationalizing the denominator

Conjugates

Form of denominator Multiply numerator and denominator by

Ï}

b Ï}

b

a 1 Ï}

b a 2 Ï}

b

a 2 Ï}

b a 1 Ï}

b

RATIONALIZING THE DENOMINATOR

ga2nt-03.indd 55 4/16/07 9:06:23 AM



Your Notes

Simplify (a) Î}7}3

and (b) 4}5 1 Ï

}

3.

a. Î}7}3 5 p 5

b. 4}5 1 Ï

}

35

4}5 1 Ï

}

3p

5

5

Example 1 Rationalize denominators of fractions

1. Ï}

5 p Ï}

10 2. Î}9

}11

Checkpoint Simplify the expression.

Find the solutions of 1}4

(y 2 6)2 5 8.

1}4

(y 2 6)2 5 8 Write original equation.

5 Multiply each side by .

5 Take square roots of each side.

y 5 Add to each side.

y 5 Simplify.

The solutions are and .

Example 2 Finding solutions of a quadratic equation

ga2nt-03.indd 56 4/16/07 9:06:24 AM



Your Notes

3. Solve the equation 2x2 2 16 5 34.

4. In Example 4, suppose that the water balloon is dropped from a height of 27 feet. How long does it take for the balloon to hit the sidewalk?


Water Balloon A water balloon is dropped from a window 59 feet above the sidewalk. How long does it take for the water balloon to hit the sidewalk?

Solution h 5 216t2 1 h0 Write height function.

5 216t2 1 Substitute for h and for h0.

5 216t2 Subtract from each side.

5 t2 Divide each side by .

5 t Take square roots of each side.

ø t Use a calculator.

Reject the negative solution, , because time must be positive. The water balloon will fall for about seconds before it hits the ground.

Example 3 Model a dropped object with a quadratic function

Homework

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Simplify the expression.

1. Ï}

98 2. Ï}

9 p 3 Ï}

27 3. Ï}

12 p Ï}

7

4. Î}

25

} 16

5. Î}

49

} 9 6. Î}

100

} 25

7. Î}

27

} 1 8. Î}

45

} 3 9. Î}

4 } 5 p Î}

3 } 5

10. 1 }

3 1 Ï}

3 11.

2 }

5 2 Ï}

6 12.

1 1 Ï}

5 }

5 1 Ï}

5

LESSON

3.6 Practice

Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


Solve the equation.

13. x2 5 16 14. x2 5 144 15. x2 5 121

16. x2 2 4 5 0 17. x2 2 64 5 0 18. x2 2 8 5 0

19. 2x2 5 32 20. 3x2 5 75 21. 1 }

3 x2 5 12

22. x2 1 12 5 13 23. x2 2 1 5 6 24. 20 2 x2 5 229

25. 2x2 2 1 5 7 26. 3x2 2 9 5 0 27. 1 }

2 x2 1 8 5 17

When an object is dropped, its height h (in feet) above the ground can be modeled by h 5 216t2 1 h0 where h0 is the object’s initial height (in feet). Find the time it takes an object to hit the ground when it is dropped from a height of h0 feet.

28. h0 5 100 29. h0 5 196 30. h0 5 480

31. h0 5 600 32. h0 5 750 33. h0 5 1200

34. Multiple Choice What are all the solutions to 3x2 1 x 1 6 5 x2 1 x 1 42?

A. 22 Ï}

2 , 2 Ï}

2

B. 23, 3

C. 23 Ï}

2 , 3 Ï}

2

D. 3 Ï}

2

35. New Car From 1980 to 2000, the average cost of a new car C (in dollars) can be approximated by the model C 5 45.5t2 1 6100 where t is the number of years since 1980. During which year was the average cost of a new car $15,025?



3.7 Complete the SquareGoal p Solve quadratic equations by completing

the square.GeorgiaPerformanceStandard(s)

MM2A3a, MM2A4b

Your Notes

VOCABULARY

Completing the square

Find the value of c that makes x2 1 12x 1 c a perfect square trinomial. Then write the expression as the square of a binomial.

Solution

1. Find half the coefficient of x.2

5

2. Square the result of Step 1. 5

3. Replace c with the result of Step 2.

The trinomial x2 1 12x 1 c is a perfect square when c 5 . Then

5 ( )( ) 5 ( )2.

Example 1 Make a perfect square trinomial

COMPLETING THE SQUARE

Words To complete the square for the expression

x2 1 bx, add 1 22.

Algebra x2 1 bx 1 1 22 5 1x 1b}2 21x 1

b}2 2

5 1 22

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Your Notes Checkpoint Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial.

1. x2 2 24x 1 c 2. x2 1 10x 1 c

Solve x2 2 10x 1 13 5 0 by completing the square.

Solution

x2 2 10x 1 13 5 0 Write original equation.

5 Write left side in the form x2 1 bx.

5 Complete the square.

5 Write left side as a binomial squared.


x 5 Solve for x.

x 5 Simplify.


Example 2 Solve ax2 1 bx 1 c 5 0 when a 5 1

ga2nt-03.indd 59 4/16/07 9:06:27 AM



Your Notes

Solve 3x2 2 12x 1 27 5 0 by completing the square.

Solution

3x2 2 12x 1 27 5 0 Write original equation.

5 Divide each side by the coefficient of x2.

5 Write left side in the form x2 1 bx.

5 Complete the square.

5 Write left side as binomial squared.


x 5 Write in terms of the imaginary unit i.


Example 3 Solve ax2 1 bx 1 c 5 0 when a Þ 1

Checkpoint Solve the equation by completing the square.

3. x2 2 8x 1 7 5 0 4. 2x2 2 20x 1 24 5 0

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Your Notes

Profit A store's profit is modeled by P 5 (200 1 8x)(35 2 x). Rewrite in vertex form to find the number of units x that maximizes the profit.

Solution

P 5 Write original function.

P 5 Use FOIL.

P 5 Combine like terms.

P 5 Prepare to complete the square.

P 5

Add and subtract.

P 5 Write a perfect square trinomial as the square of a binomial.

The vertex is , so the number of units that maximizes P is .

Example 4 Find the maximum value of a quadratic equation

5. Rework Example 4 where P 5 (100 1 5x)(50 2 x).


Homework

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Solve the equation by fi nding square roots.

1. x2 1 2x 1 1 5 9 2. x2 1 6x 1 9 5 1 3. x2 2 4x 1 4 5 16

4. x2 2 10x 1 25 5 4 5. x2 2 14x 1 49 5 7 6. x2 1 20x 1 100 5 12

7. x2 2 x 1 1 }

4 5 1 8. 2x2 1 16x 1 32 5 14 9. 4x2 1 12x 1 9 5 16

Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial.

10. x2 1 4x 1 c 11. x2 2 2x 1 c

12. x2 1 18x 1 c 13. x2 1 24x 1 c

14. x2 2 14x 1 c 15. x2 2 5x 1 c

16. x2 1 x 1 c 17. x2 1 7x 1 c

LESSON

3.7 Practice

Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


Solve the equation by completing the square.

18. x2 2 2x 2 2 5 0 19. x2 1 6x 1 3 5 0

20. x2 1 8x 2 2 5 0 21. x2 1 2x 1 5 5 0

22. x2 1 10x 1 11 5 0 23. x2 2 14x 1 10 5 0

24. x2 2 x 1 1 5 0 25. x2 2 x 2 3 5 0

Write the quadratic function in vertex form. Then identify the vertex.

26. y 5 x2 1 8x 1 5 27. y 5 x2 2 12x 1 1

28. y 5 x2 1 4x 1 12 29. y 5 x2 2 10x 1 3


30. Area of rectangle 5 40 31. Area of rectangle 5 78

x 1 3

x

x 1 7

x

32. Area of triangle 5 16 33. Area of triangle 5 40

x 1 4

x

x 2 2

x



3.8 Use the Quadratic Formula and the DiscriminantGoal p Solve quadratic equations using the quadratic

formula.


MM2A4b, MM2A4c

Your Notes

VOCABULARY

Quadratic formula

Discriminant

THE QUADRATIC FORMULA

Let a, b, and c be real numbers such that a Þ 0. The solutions of the quadratic equation ax2 1 bx 1 c are:

x 5 6 Ï

}}

2 4}}

2

Example 1 Solve an equation with two real solutions

Solve x2 1 7x 5 6.

x2 1 7x 5 6 Original equation

x2 1 7x 5 0 Standard form

x 5 6 Ï

}}

2 4}}

2 Quadratic formula

x 5 6 Ï}}

2 4}}}

2 a 5 , b 5 ,

c 5

x 5 Simplify.

The solutions are x 5 ø and

x 5 ø .

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Your Notes Example 2 Solve an equation with one real solution

Solve 2x2 2 8x 1 8 5 0.

Solution2x2 2 8x 1 8 5 0 Original equation

x 5 6 Ï}}

2 4}}}

2

x 5 Simplify.

x 5 Simplify.

The solution is .

a 5 , b 5 ,c 5

Solve 2x2 1 2x 5 5.

Solution

2x2 1 2x 5 5 Original equation

2x2 1 2x 5 0 Standard form

x 5 6 Ï}}

2 4}}}

2

x 5 Simplify.

x 5

x 5 Simplify.


Example 3 Solve an equation with imaginary solutions

a 5 , b 5 ,c 5

Rewrite using the imaginary unit i.

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Your Notes Checkpoint Use the quadratic formula to solve the equation.

1. 2x2 1 12x 5 216 2. 4x2 2 13x 5 7x 2 25

3. 3x2 2 6x 1 6 5 0 4. x2 2 3x 1 3 5 0

USING THE DISCRIMINANT OF ax2 1 bx 1 c 5 0

When b2 2 4ac > 0, the equation has . The graph has x-intercepts.

When b2 2 4ac 5 0, the equation has . The graph has x-intercept.

When b2 2 4ac < 0, the equation has . The graph has x-intercepts.

ga2nt-03.indd 64 4/16/07 9:06:31 AM



Your Notes

Homework

Find the discriminant of the quadratic equation and give the number and type of solutions of the equation.

a. x2 1 6x 1 5 5 0

b. x2 1 6x 1 9 5 0

c. x2 1 6x 1 13 5 0

Discriminant Solution(s)

b2 2 4ac x 52b 6 Ï

}

b2 2 4ac}}

2aa.

b.

c.

Example 4 Use the discriminant

Checkpoint Find the discriminant of the quadratic equation and give the number and type of solutions of the equation. Then solve the equation.

5. x2 2 8x 1 17 5 0 6. x2 1 4x 1 3 5 0

7. 2x2 1 2x 2 1 5 0 8. x2 1 6x 1 4 5 0

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Write the equation in standard form. Identify a, b, and c.

1. 2x2 1 x 1 4 5 0 2. x2 2 2x 1 3 5 6

3. 23x2 2 2 5 x2 1 3x 4. 5x 5 2x2 2 x 1 9

Find the discriminant and use it to determine if the solution has one real, two real, or two imaginary solution(s).

5. x2 1 4x 1 1 5 0 6. x2 2 2x 1 1 5 0

7. x2 1 2x 1 5 5 0 8. 2x2 1 3x 1 1 5 0

9. 2x2 2 4x 2 6 5 0 10. x2 2 5x 2 6 5 0

11. 22x2 1 x 1 4 5 0 12. 5x2 1 7x 1 6 5 0

LESSON

3.8 Practice

Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


Use the quadratic formula to solve the equation.

13. x2 2 3x 1 2 5 0 14. x2 1 5x 1 2 5 0 15. x2 2 3x 1 1 5 0

16. 3x2 1 x 2 4 5 0 17. 2x2 2 4x 2 1 5 0 18. 2x2 2 4x 1 1 5 0

19. 3x2 1 2x 5 0 20. 22x2 2 2x 2 1 5 0 21. 5x2 2 9x 1 3 5 0

22. 2x2 1 3x 2 4 5 2 23. 2x2 2 1 5 3x 1 4 24. x2 2 4x 5 23x 1 2

25. 3x2 1 2x 5 x2 1 x 1 1 26. 2x2 2 2 5 3x2 2 5 27. 2x2 2 x 1 3 5 3x 1 7

Solve the equation using the quadratic formula. Then solve the equation by factoring to check your solution(s).

28. x2 2 x 2 2 5 0 29. x2 1 5x 1 6 5 0 30. x2 2 2x 2 8 5 0

31. x2 1 7x 1 6 5 0 32. 2x2 1 x 2 6 5 0 33. 3x2 1 2x 2 1 5 0


34. Area of rectangle 5 17.6 35. Area of parallelogram 5 40.5

x 1 2.3

x

x

2x

36. Horseshoes A contestant tosses a horseshoe from one pit to another with an initial vertical velocity of 50 feet per second. The horseshoe is released 3 feet above the ground. Use the model h 5 216t2 1 50t 1 3 where h is the height (in feet) and t is the time (in seconds) to tell how long the horseshoe was in the air.



3.9 Graph and Solve Quadratic InequalitiesGoal p Graph and solve quadratic inequalities.Georgia


MM2A4d

Your Notes

VOCABULARY

Quadratic inequality in two variables

Quadratic inequality in one variable

Bridge A suspension bridge can support a maximum weight W (in pounds) if W ≤ 3000d2 where d is the diameter of the steel cables (in inches) used to suspend the bridge. Graph the inequality.

Solution

Graph W 5 3000d2 for nonnegativevalues of d. Because the inequality symbol is ≤, make the parabola

. Test the point (5, 10,000) which is below the parabola.

10 2 3 4 5 6 7

12500

0

25000

37500

50000

62500

75000

87500

(5, 10,000)

d

W

W ≤ 3000d2

≤? 3000( )2

Because (5, 10,000) a solution, shade the region the parabola.

Example 1 Use a quadratic inequality in real life

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Your Notes

1. Rework Example 1 given the

10 2 3 4 5 6 7

12500

0

25000

37500

50000

62500

75000

87500

d

W

new inequality W ≤ 2500d2.


Graph the system of quadratic inequalities.

y > x2 2 2 Inequality 1

y ≤ 2x2 2 3x 1 4 Inequality 2

Solution

2

2

x

y

1. Graph y > x2 2 2. The graph is the region (but not including) the parabola y 5 .

2. Graph y ≤ 2x2 2 3x 1 4. The graph is the region and including the parabola y 5 .

3. Identify the region where the two graphs overlap. This region is the graph of the system.

Example 2 Graph a system of quadratic inequalities

2. Graph the system.

x

y

1

1

y < 2x2 1 3

y ≥ 2x2 1 3x 2 2


ga2nt-03.indd 67 4/16/07 9:06:34 AM



Your Notes

Solve x2 1 3x ≤ 4.

Rewrite the inequality as x2 1 3x 2 4 ≤ 0. Then make a table of values.

x 25 24 23 22 21

x2 1 3x 2 4

x 0 1 2

x2 1 3x 2 4

Notice that x2 1 3x 2 4 ≤ 0 when the values of x are between and , inclusive.

The solution of the inequality is .

Example 3 Solve a quadratic inequality using a table

Solve 23x2 2 5x 1 3 ≤ 0.

The solution consists of the x-values for which the graphof y 5 23x2 2 5x 1 3 lies the x-axis.Find the graph’s x-intercepts by letting y 5 0 and using to solve for x.

0 5 23x2 2 5x 1 3

x

y

1

1

x 5

x 5

x ø or x ø

Sketch a parabola that opens and has and as x-intercepts. The graph lies the x-axis to the left of (and including) x 5 and to the right of (and including) x 5 .

The solution is approximately .

Example 4 Solve a quadratic inequality by graphing

ga2nt-03.indd 68 4/16/07 9:06:35 AM



Your Notes

3. Solve the quadratic inequality x2 1 2x ≤ 3 using a table.

4. Solve the quadratic inequality x

y

2112x2 1 3x ≥ 5 by graphing.


ga2nt-03.indd 69 4/16/07 9:06:36 AM



Your Notes

Homework

Solve x2 1 x ≥ 12.

First, write and solve the equation obtained by replacing ≥ with .

Write corresponding equation.

Write in standard form.

Factor.

Zero product property

The numbers are the critical x-values of the inequality x2 1 x ≥ 12. Plot on a number line, using dots. The critical x-values partition the number line into three intervals. Test an x-value in each interval to see if it satisfies the inequality.

Test x 5 : Test x 5 :

Test x 5 :

The solution is .

Example 5 Solve a quadratic inequality algebraically

5. Solve the inequality algebraically.

22x2 1 12x ≤ 16


ga2nt-03.indd 70 4/16/07 9:06:36 AM



Name ——————————————————————— Date ————————————

LESSON

3.9 PracticeDetermine whether the ordered pair is a solution of the inequality.

1. y < x2 1 4x 1 1, (0, 0) 2. y > x2 2 x 1 4, (2, 7)

3. y ≤ 2x2 2 5x 1 2, (2, 1) 4. y ≥ 2x2 1 3x 1 2, (23, 4)

5. y < 2x2 1 6x, (6, 25) 6. y > x2 2 2x 1 8, (21, 24)

Match the inequality with its graph.

7. y ≥ x2 1 2x 2 2 8. y ≤ 2x2 2 2x 1 2 9. y ≤ x2 1 4x 2 3

10. y < x2 2 2x 1 2 11. y > 2x2 1 2x 1 2 12. y > 2x2 2 4x 1 3

A.

x

y

2

2

B.

x

y

1

1

C.

x

y

1

1

D.

x

y

2

1

E.

x

y

1

2

F.

2

2

x

y



Graph the inequality.

13. y ≥ x2 1 2x 14. y < 2x2 2 5x 1 1 15. y ≤ x2 2 3

x

y

1

1

x

y

2

2

x

y

1

1

16. y < x2 2 4x 17. y > 2x2 1 2x 1 1 18. y ≤ 2x2 2 x 1 1

x

y1

1

x

y

1

1

x

y

1

1

19. y ≥ x2 1 5x 2 2 20. y > 2x2 2 3x 1 3 21. y ≤ x2 1 6x 2 1

x

y2

2

x

y

1

1

x

y

1

1

LESSON


Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


Match the system of inequalities with its graph.

22. y > x2 2 2 23. y > 2x2 2 2 24. y < x2 2 2

y < x2 1 2 y < x2 1 2 y > 2x2 1 2

A.

x

y

1

1

B.

x

y

1

1

C.

x

y

1

2

Graph the system of inequalities.

25. y > x2 26. y ≤ 2x2 1 2x 1 1 27. y < x2 1 1

y < 2x2 1 3 y > x2 2 1 y ≥ 2x2 2 2

x

y

1

1

x

y

1

1

x

y

1

1

In Exercises 28 and 29, use the following information.

Construction A paint can is dropped from the top of a 500 foot tall building being con-structed. The height of the paint can can be modeled by h 5 216t2 1 500 where h is the height (in feet) and t is the time (in seconds). A platform is 100 feet from the ground.

28. Write an inequality that shows when the paint can is above the platform.

29. For what values of t is the paint can above the platform?



Words to ReviewGive an example of the vocabulary word.

Quadratic function

Vertex

Minimum and maximum values

Intercept form

Binomial

Quadratic Equation

Parabola

Axis of symmetry

Vertex form

Monomial

Trinomial

Root of an equation

ga2nt-03.indd 71 4/18/07 4:10:23 PM



Zero of a function

Conjugates

Quadratic formula

Quadratic inequality in two variables

Rationalizing the denominator

Completing the square

Discriminant

Quadratic inequality in one variable

ga2nt-03.indd 72 4/16/07 9:06:38 AM


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