1. WARM UP 4
PRODUCT OF POWERS Write the expression as a single power of the base (LESSON 8.1).
x2 • x5
(-5) • (-5)8
x2 • x4 • x6
x • x4 • x3

2. 3 WARM UP x2 • x5 (-5) • (-5)8 x2 • x4 • x6 x • x4 • x3
PRODUCT OF POWERS Write the expression as a single power of the base (LESSON 8.1).
x2 • x5
(-5) • (-5)8
x2 • x4 • x6
x • x4 • x3

3. 2 WARM UP x2 • x5 (-5) • (-5)8 x2 • x4 • x6 x • x4 • x3
PRODUCT OF POWERS Write the expression as a single power of the base (LESSON 8.1).
x2 • x5
(-5) • (-5)8
x2 • x4 • x6
x • x4 • x3

4. 1 WARM UP x2 • x5 (-5) • (-5)8 x2 • x4 • x6 x • x4 • x3
PRODUCT OF POWERS Write the expression as a single power of the base (LESSON 8.1).
x2 • x5
(-5) • (-5)8
x2 • x4 • x6
x • x4 • x3

5. WARM UP x2 • x5 (-5) • (-5)8 x2 • x4 • x6 x • x4 • x3
PRODUCT OF POWERS Write the expression as a single power of the base (LESSON 8.1).
x2 • x5
(-5) • (-5)8
x2 • x4 • x6
x • x4 • x3

6. 9.5 Solving Quadratic Equations by Graphing
GOAL
Use a graph to find or check a solution of a quadratic equation
KEY WORDS
X-intercept
Roots of a Quadratic Equation

7. 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.

8. 9.5 Solving Quadratic Equations by Graphing
EXAMPLE 1 Use a Graph to Solve an Equation
The graph of y = x2 – 8 is shown a the right. Use the graph to estimate the solutions of x2 – 8 = 0.
SOLUTION
The graph appears to intersect the x-axis
at (-4, 0) and (4, 0). By substituting x = -4
and x = 4 in x2 – 8 = 0, you can check that
-4 and 4 are solutions of the given equation.

9. 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.

10. 9.5 Solving Quadratic Equations by Graphing
ESTIMATING SOLUTIONS BY GRAPHING
The 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 = 0
STEP 2 Sketch the graph of the related quadratic function
y = 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.

11. 9.5 Solving Quadratic Equations by Graphing
EXAMPLE 2 Graph Quadratic Function with Positive a-Value
Sketch the graph of x2 - x = 2
SOLUTION
STEP 1 Write the equation in the standard form.
x2 – x = 2 ORIGINAL EQUATION
x2 – x – 2 = 0 SUBTRACT 2 FROM EACH SIDE.

12. 9.5 Solving Quadratic Equations by Graphing
EXAMPLE 2 Graph Quadratic Function with Positive a-Value
Sketch the graph of y = x2 - x = 2
SOLUTION
STEP 2 Sketch the graph of the related function y = x2 – x – 2 .
y = x2 – x - 2

13. 9.5 Solving Quadratic Equations by Graphing
EXAMPLE 2 Graph Quadratic Function with Positive a-Value
Sketch the graph of y = x2 - x = 2
SOLUTION
STEP 3 Estimate the values of the x-intercepts. The x-intercepts appear to be -1 and 2.
(-1, 0)
(2, 0)
x-intercept
x-intercept
y = x2 – x - 2

14. 9.5 Solving Quadratic Equations by Graphing
EXAMPLE 2 Graph Quadratic Function with Positive a-Value
Sketch the graph of y = x2 - x = 2
CHECK YOUR ANSWER BY SUBSTITUING
(-1)2 – (-1) – 2 = 0 ? (2) 2 – (2) – 2 = 0 ?
(-1, 0)
(2, 0)
x-intercept
x-intercept
y = x2 – x - 2

15. 9.4 Graphing Quadratic Functions
Checkpoint Solve an Equation by Graphing
Use a graph to estimate the solutions of x2 + 2 = 6
Check your solutions
algebraically. x y

16. 9.4 Graphing Quadratic Functions
Solve an Equation by Graphing
1. x2 + 2x = 3 x y
(2, 0)
(-1, 0)
x-intercept
y = x2 – x - 2
x-intercept

17. 9.4 Graphing Quadratic Functions
Solve an Equation by Graphing
2. -4x2 - 8x = -12 x y

18. 9.4 Graphing Quadratic Functions
Solve an Equation by Graphing
3. -x2 + 3x = -4 x y