Quadratic Functions and Inequalities

1. Graph and solve quadratic equations and inequalities

2. Write quadratic equations and functions3. Analyze graphs of quadratic functions

Quadratic functions can be used to model real-world phenomena like the motion of a falling object.

Quadratic functions can be used to model the shape of architectural structures such as the supporting cables of a suspension bridge.

Can you think of examples?

6.1 Graphing quadratic functions

1. Quadratic function – ax2 + bx + c.2. Parabola – graph of a quadratic

function3. Y – intercept is c4. Axis of symmetry and x-coordinate of

the vertices - x = (-b/2a)5. The y-coordinate is the maximum

and minimum value.

6.1 continued Opens up and has a minimum value

when a > o. Opens down and has a maximum

value when a < 0. Minimum point and minimum value

are not the same. Minimum value is the y-coordinate of the vertex. Minimum point is the ordered pair (x,y) of the vertex.

F(x) = x2 + 8x + 9

Find the y-intercept Find the axis of symmetry Find the vertex Find the maximum or minimum value

Does the graph open up or down?

9

-8/2 = -4

(-4,-7)

-7, min.

Opens up

6.1 examples

1. Graph f(x) = x2 + 3x – 1 by making a table of values. Find the vertex point first. Then use 2 number less and 2 numbers greater.

x -3 -2 -3/2 0 1

f(x) -1 -3 -13/4

-1 3

6.1 examples

2. f(x) = 2 – 4x + x2

a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex.

b. Make a table of values that includes the vertex.

c. Graph the functiona. 2, x = 2, 2

b. (0,2),(1,-1),(2,-2),(3,-1),(4,2

6.1 Classwork

1. f(x) = -x2 + 2x + 3 a. Determine whether the function

has a maximum and minimum value. b. State the maximum and minimum

value of the function. a. Maximum value

b. 4

6.1 Classwork see ex. 4 p. 289

2. A souvenir shop sells about 200 coffee mugs each month for $6 each. The shop owner estimates that for each $.50 increase in the price, he will sell about 10 fewer coffee mugs per month.

a. How much should the owner charge for each mug in order to maximize the monthly income from their sales?

b. What is the maximum monthly income the owner can expect to make from these items?

$8

$1280

6.1 Classwork

Consider the equations –x2 + 8x – 16 and x2 – 8x + 16.

• What is true about the equations?• What is true about the graphs of the

equations?

1. They have the same solution

2. They open in opposite directions

6.2 Solve quadratic equations by graphing

ax2 + bx + c = 0 The solutions are called the roots or

the zeros. The solutions are the x-intercepts.

Solutions of quadratic equations – 1 real solution or 2 real solutions or no

real solution.

6.2 examples

1. Solve x2 -3x – 4 = 0 by graphing.2. Solve x2 – 4x = -4 by graphing.3. Find two numbers whose sum is 4

and whose product is 5 or show that no such numbers exist.

4. Solve x2 – 6x + 3 = 0 by graphing. Estimate.

-1,4

2

No real solution

Between 0 and 1, between 5 and 6

6.3 solve quadratic equations by factoring

1. x2 = 6x x2 -6x = 0 x(x-6) = 0 x = 0 x – 6 = 0 x = 62. 2x2 + 7x = 15 2x2 + 7x – 15 = 0 (2x – 3)(x + 5) = 0 2x – 3 = 0 x + 5 = 0 x = 3/2 x = -5

6.3 examples

Solve by factoring1. x2 = -4x2. 3x2 = 5x + 23. x2 – 6x = -94. Write a quadratic equation with -2/3

and 6 as its roots. (see example 4 p. 303)

{0,-4}

{-1/3, 2}

{3}

3x2 – 16x – 12 = 0

6.4 Completing the square

Solve using the square root property x2 + 10x + 25 = 49 (x + 5)2 = 49 x + 5 = + x = -5 + 7 x = -12 x = 2

49

continued

2. x2 – 6x + 9 = 32 (x – 3)2 = 32 x – 3 = + x = 3 + 4 x = 8.7 x = -2.7

32

2

Complete the square

1. x2 + 8x – 20 = 0 x2 + 8x + ___ = 20 + ___ x2 + 8x + 16 = 20 + 16 (x + 4)2 = 36 x + 4 = +6 x = -4 + 6 x = -10, x = 2

continued

2. x2 + 4x + 11 = 0 x2 + 4x + _____ = -11 + ___ x2 + 4x + 4 = -11 + 4 (x + 2)2 = -7 x + 2 =

No real solution

7

continued

3. 2x2 – 5x + 3 = 0 x2 – (5/2)x + (3/2) = 0 x2 – (5/2)x + ____ = -(3/2) + ___x2 – (5/2)x + (25/16)= –(3/2)+ (25/16) (x – (5/4))2 = (1/16) x – (5/4) = +(1/4) x =+(1/4) + (5/4) x = 3/2 x = 1

6.4 Examples Solve

1. x2 + 14x + 49 = 642. x2 – 10x + 25 = 123. x2 + 4x – 12 = 0

4. x2 + 2x + 3 = 05. 3x2 – 2x + 1 = 0

{-15,1}

{5 + 2 } 3

{-6,2}

{no real solution}

{-1/3,1}

6.5 Quadratic Formula and discriminant Quadratic formula Discriminant b2 -4ac b2 – 4ac > 0 2 real roots b2 -4ac = 0 1 real root b2 – 4ac < 0 no real roots

6.5 Examples

1. Solve x2 – 8x = 33 using the quadratic formula

2. Solve x2 - 34x + 289 = 03. Solve x2 – 6x + 2 = 04. Solve x2 + 13x = 6x

6.5 Examples continued

Find the value of the discriminant and describe the nature and types of roots.

1. x2 + 6x + 9 = 02. x2 + 3x + 5 = 03. x2 + 8x – 4 =04. x2 – 11x + 10 = 0

6.6 Analyzing Graphs

Vertex form y = a(x-h)2 + k (h,k) vertex x = h axis of symmetry Translations h units to left if h is + h units to right if h is – k units up if k is + k units down if k is -

If a > 0, then the graph opens up If a < 0, then the graph opens down If abs(a) > 1, then the graph is

narrower than y = x2

If abs(a)< 1, then the graph is wider than y = x2

How to use vertex form to graph

1. Plot the vertex2. Draw the axis of symmetry3. Find and plot 2 points on one side of

the axis of symmetry4. Use symmetry to complete the graph

6.6 Examples

1. Write y = x2 + 2x + 4 in vertex form. Analyze the function.

2. Write y = -2x2 – 4x + 2 in vertex form. Analyze and graph the function.

3. Write an equation for the parabola whose vertex is at (1,2) and passes through (3,4)

6.7 Graphing and solving quadratic inequalities

1. Graph the inequality and decide if the parabola should be solid or dashed.

2. Test a point inside the parabola to see if it is a solution.

3. If it is a solution, shade inside.4. If it is not a solution, shade outside.

6.7 Examples

1. Graph y > x2 – 3x + 22. Solve x2 – 4x + 3 > 0 by graphing.3. Solve 0 < -2x2 – 6x + 1 by graphing.4. Solve x2 + x < 2 algebraically.

Chapter 6 Study Guide1. Solve by factoring2. Solve by completing the square3. Solve using quadratic formula4. Solve using graphing calculator5. Set up word problems using quadratic

equations and solve6. Solve using a method of your choice7. Give the discriminant and describe the

nature of the roots8. Write a quadratic equation with the given

roots.