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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

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**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

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**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

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**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

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**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

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**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

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**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.

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**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.

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**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.

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**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.

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**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.

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**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

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**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

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**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

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**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

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**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

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**9.4 Graphing Quadratic Functions**

Solve an Equation by Graphing 2. -4x2 - 8x = -12 x y

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**9.4 Graphing Quadratic Functions**

Solve an Equation by Graphing 3. -x2 + 3x = -4 x y

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