WARM UP
4
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
3
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
2
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
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
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
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
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 Equation
1
2
The graph of y = x2 – 8 is shown a the right. Use the graph to estimate
1
2
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
1 2
and x = 4 in x – 8 = 0, you can check that
2
-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 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.
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.
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
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)
x-intercept
(2, 0)
x-intercept
y = x2 – x - 2
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)
x-intercept
(2, 0)
x-intercept
y = x2 – x - 2
9.4 Graphing Quadratic Functions
Checkpoint
Solve an Equation by Graphing
1. Use a graph to estimate the solutions of x2 + 2 = 6
2. Check your solutions
algebraically.
x
y
9.4 Graphing Quadratic Functions
Solve an Equation by Graphing
1.
x2 + 2x = 3
x
y
y = x2 – x - 2
x-intercept
9.4 Graphing Quadratic Functions
Solve an Equation by Graphing
2.
x
y
-4x2 - 8x = -12
9.4 Graphing Quadratic Functions
Solve an Equation by Graphing
3.
x
y
-x2 + 3x = -4