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Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Section 8.1
Quadratic Functions and Their Graphs
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Objectives
• Graphs of Quadratic Functions
• Min-Max Applications
• Basic Transformations of Graphs
• More About Graphing Quadratic Functions (Optional)
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
The graph of any quadratic function is a parabola.
The vertex is the lowest point on the graph of a parabola that opens upward and the highest point on the graph of a parabola that opens downward.
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
The graph is symmetric with respect to the y-axis. In this case the y-axis is the axis of symmetry for the graph.
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
Use the graph of the quadratic function to identify the vertex, axis of symmetry, and whether the parabola opens upward or downward.a. b.
Vertex (0, 2)Axis of symmetry: x = –2Open: up
Vertex (0, 4)Axis of symmetry: x = 0Open: down
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
Find the vertex for the graph of Support your answer graphically.Solutiona = 2 and b = 8
Substitute into the equation to find the y-value.
2( ) 2 8 3.f x x x
82
2(2)x
2
bx
a
2( ) 2( 2) 8( 2) 3
8 16 3
11
f x
The vertex is (2, 11), which is supported by the graph.
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
Identify the vertex, and the axis of symmetry on the graph, then graph.SolutionBegin by making a table of values.Plot the points and sketch a smooth curve.
The vertex is (0, –2)axis of symmetry x = 0
2( ) 2f x x
x f(x) = x2 – 2
3 7
2 2
1 1
0 2
1 1
2 2
3 7
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
Identify the vertex, and the axis of symmetry on the graph, then graph.SolutionBegin by making a table of values.Plot the points and sketch a smooth curve.
The vertex is (2, 0)axis of symmetry x = 2
2( ) ( 2)g x x
x g(x) = (x – 2)2
0 4
1 1
2 0
3 1
4 4
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
Identify the vertex, and the axis of symmetry on the graph, then graph.SolutionBegin by making a table of values.Plot the points and sketch a smooth curve.
The vertex is (2, 0)axis of symmetry x = 2
2( ) 2 3h x x x
x h(x) = x2 – 2x – 3
2 5
1 0
0 3
1 4
2 3
3 0
4 5
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
Find the maximum y-value of the graph of
SolutionThe graph is a parabola that opens downward because a < 0. The highest point on the graph is the vertex.a = 1 and b = 2
2( ) 2 3.f x x x
( 2)1
2 2( 1)
bx
a
2( ) ( 1) 2( 1) 3
4
f x
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
A baseball is hit into the air and its height h in feet after t seconds can be calculated by a. What is the height of the baseball when it is hit?b. Determine the maximum height of the baseball.Solutiona. The baseball is hit when t = 0.
b. The graph opens downward because a < 0. The maximum height occurs at the vertex. a = –16 and b = 64.
2( ) 16 64 2.h t t t
2( ) 16 64 2h t t t 2(0) 16(0) 64(0) 2
2
h
64 642
2 2( 16) 32
bx
a
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Example (cont)
2( ) 16 64 2 h t t t
2(2) 16(2) 64(2) 2
66
h
The maximum height is 66 feet.
2( ) 16 64 2 h t t t
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Basic Transformations of Graphs
The graph of y = ax2, a > 0.As a increases, the resulting parabola becomes narrower.When a > 0, the graph of y = ax2 never lies below thex-axis.
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
Compare the graph of g(x) = –4x2 to the graph of f(x) = x2. Then graph both functions on the same coordinate axes.SolutionBoth graphs are parabolas.The graph of g opens downward and is narrower than the graph of f.
