WARM UP

PRODUCT OF POWERS Write the expression as a single power of the base (LESSON 8.1).

1. x2 • x5

2. (-5) • (-5)8

3. x2 • x4 • x6

4. x • x4 • x3

4

WARM UP

PRODUCT OF POWERS Write the expression as a single power of the base (LESSON 8.1).

1. x2 • x5

2. (-5) • (-5)8

3. x2 • x4 • x6

4. x • x4 • x3

3

WARM UP

PRODUCT OF POWERS Write the expression as a single power of the base (LESSON 8.1).

1. x2 • x5

2. (-5) • (-5)8

3. x2 • x4 • x6

4. x • x4 • x3

2

WARM UP

PRODUCT OF POWERS Write the expression as a single power of the base (LESSON 8.1).

1. x2 • x5

2. (-5) • (-5)8

3. x2 • x4 • x6

4. x • x4 • x3

1

PRODUCT OF POWERS Write the expression as a single power of the base (LESSON 8.1).

1. x2 • x5

2. (-5) • (-5)8

3. x2 • x4 • x6

4. x • x4 • x3

WARM UP 0

GOAL•Use a graph to find or check a solution of a quadratic equationKEY WORDS•X-intercept•Roots of a Quadratic Equation

9.5 Solving Quadratic Equations by Graphing

The x-intercepts of graph y = ax2 + bx + c are the solutions of the related equations ax2 + bx + c = 0.

Recall that an x-intercept is the x-coordinate of a point where a graph crosses the x-axis.

At this point, y = 0.

9.5 Solving Quadratic Equations by Graphing

EXAMPLE 1 Use a Graph to Solve an EquationThe graph of y = x2 – 8 is shown a the right. Use the graph to estimate

the solutions of x2 – 8 = 0.

SOLUTIONThe graph appears to intersect the x-axis at (-4, 0) and (4, 0). By substituting x = -4and x = 4 in x2 – 8 = 0, you can check that-4 and 4 are solutions of the given equation.

9.5 Solving Quadratic Equations by Graphing

Checkpoint Use a Graph to Solve and Equation.The graph o f y = 2x2 – 4x is shown. Use the graph to estimate the

solutions of 2x2 – 4x = 0. Check your solutions algebraically by substituting each one for x in the given equation.

9.5 Solving Quadratic Equations by Graphing

ESTIMATING SOLUTIONS BY GRAPHINGThe solutions of a quadratic equation in one variable x can be estimated by graphing. Use the following steps:STEP 1 Write the equation in the standard for ax2 +bx + c = 0STEP 2 Sketch the graph of the related quadratic functiony = ax2 +bx + c STEP 3 Estimate the values of the x-intercepts, if any.

The solutions, or roots, of ax2 +bx + c = 0 are the x-intercepts of the graph.

9.5 Solving Quadratic Equations by Graphing

EXAMPLE 2 Graph Quadratic Function with Positive a-ValueSketch the graph of x2 - x = 2SOLUTIONSTEP 1 Write the equation in the standard form.

x2 – x = 2 ORIGINAL EQUATION

x2 – x – 2 = 0 SUBTRACT 2 FROM EACH SIDE.

9.5 Solving Quadratic Equations by Graphing

EXAMPLE 2 Graph Quadratic Function with Positive a-ValueSketch the graph of y = x2 - x = 2SOLUTIONSTEP 2 Sketch the graph of the related function y = x2 – x – 2 .

9.5 Solving Quadratic Equations by Graphing

y = x2 – x - 2

EXAMPLE 2 Graph Quadratic Function with Positive a-ValueSketch the graph of y = x2 - x = 2SOLUTIONSTEP 3 Estimate the values of the x-intercepts. The x-intercepts appear

to be -1 and 2.

9.5 Solving Quadratic Equations by Graphing

y = x2 – x - 2

(-1, 0) (2, 0)

x-intercept x-intercept

EXAMPLE 2 Graph Quadratic Function with Positive a-ValueSketch the graph of y = x2 - x = 2CHECK YOUR ANSWER BY SUBSTITUING(-1)2 – (-1) – 2 = 0 ? (2) 2 – (2) – 2 = 0 ?

9.5 Solving Quadratic Equations by Graphing

y = x2 – x - 2

(-1, 0) (2, 0)

x-intercept x-intercept

Checkpoint Solve an Equation by Graphing

1. Use a graph to estimate the solutions of x2 + 2 = 62. Check your solutionsalgebraically.

9.4 Graphing Quadratic Functions

x

y

Solve an Equation by Graphing

1. x2 + 2x = 3

9.4 Graphing Quadratic Functions

x

y

x-intercept

x-intercept

y = x2 – x - 2

Solve an Equation by Graphing

2. -4x2 - 8x = -12

9.4 Graphing Quadratic Functions

x

y

Solve an Equation by Graphing

3. -x2 + 3x = -4

9.4 Graphing Quadratic Functions

x

y