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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. Quadratic Functions and Equations8
Quadratic Functions and Equations
8.1 Quadratic Equations
8.2 The Quadratic Formula
8.3 Applications Involving Quadratic Equations
8.4 Studying Solutions of Quadratic Equations
8.5 Equations Reducible to Quadratic
8.6 Quadratic Functions and Their Graphs
8.7 More About Graphing Quadratic Functions
8.8 Problem Solving and Quadratic Functions
8.9 Polynomial and Rational Inequalities
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3. Quadratic Functions and Their Graphs
8.6
Quadratic Functions and Their Graphs
The Graph of f (x) = ax2
The Graph of f (x) = a(x – h)2
The Graph of f (x) = a(x – h)2 + k
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4. The Graph of f (x) = ax2
All quadratic functions have graphs similar to y = x2. Such curves are called parabolas. They are U-shaped and symmetric with respect to a vertical line known as the parabola’s axis of symmetry. For the graph of f (x) = x2, the y-axis is the axis of symmetry. The point (0, 0) is known as the vertex of this parabola.
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5. Example Graph: Solution 1 –1 2 –2 8 (0, 0) (1, 2) (–1, 2) (2, 8)y
(-1,2 )
(2,8)
(-2,8) 4 3 6 2 5 1
(1,2)
(0, 0) -1 -2 7 8
Solution x y
(x, y) 1 –1 2 –2 8
(0, 0)
(1, 2)
(–1, 2)
(2, 8)
(–2, 8)
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6. Example Graph: Solution 1 –1 2 –2 –3 –12 (0, 0) (1, –3) (–1, –3)y -3 2 -2 3 -1 1 6 5 4 -4 -5 x y
(x, y) 1 –1 2 –2 –3
–12
(0, 0)
(1, –3)
(–1, –3)
(2, –12)
(–2, –12)
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7. 4 3 6 2 5 1
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8. Graphing f (x) = ax2
The graph of f (x) = ax2 is a parabola with x = 0 as
its axis of symmetry. Its vertex is the origin.
For a > 0, the parabola opens upward. For a < 0,
the parabola opens downward.
If |a| is greater than 1, the parabola is narrower than y = x2.
If |a| is between 0 and 1, the parabola is wider than y = x2.
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9. The Graph of f (x) = a(x – h)2
We could next consider graphs of f (x) = ax2 + bx + c, where b and c are not both 0. It turns out to be convenient to first graph f (x) = a(x – h)2, where h is some constant. This allows us to observe similarities to the graphs drawn in previous slides.
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10. Example Graph: Solution vertex 1 –1 2 3 4 9 (0, 4) (1, 1) (–1, 9)y 4 3 6 2 5 1 -1 -2 7 8
Solution x y
(x, y) 1 –1 2 3 4 9
(0, 4)
(1, 1)
(–1, 9)
(2, 0)
(3, 1)
(4, 4)
vertex
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11. Graphing f (x) = a(x – h)2
The graph of f (x) = a(x – h)2 has the same shape
as the graph of y = ax2.
If h is positive, the graph of y = ax2 is shifted h
units to the right.
If h is negative, the graph of y = ax2 is shifted |h|
units to the left.
The vertex is (h, 0) and the axis of symmetry is
x = h.
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12. The Graph of f (x) = a(x – h)2 + k
Given a graph of f (x) = a(x – h)2, what happens if we add a constant k? Suppose that we add 2. This increases f (x) by 2, so the curve is moved up. If k is negative, the curve is moved down. The axis of symmetry for the parabola remains x = h, but the vertex will be at (h, k), or equivalently (h, f (h)).
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

13. Graphing f (x) = a(x – h)2 + k
The graph of f (x) = a(x – h)2 + k has the same shape
as the graph of y = a(x – h)2.
If k is positive, the graph of y = a(x – h)2 is
shifted k units up.
If k is negative, the graph of y = a(x – h)2 is
shifted |k| units down.
The vertex is (h, k), and the axis of symmetry is x = h.
For a > 0, k is the minimum function value. For a < 0, k is the maximum function value.
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14. Example Graph: Solution vertex –1 –2 –3 –4 –5 -11/2 –3/2 (0, -11/2)y -3 2 -2 3 -1 1 -4 -5 -6 -7 -8 x y
(x, y) –1 –2 –3 –4 –5
-11/2
–3/2
(0, -11/2)
(–1, –3)
(–2, –3/2)
(–3, –1)
(–4, –3/2)
(–5, –3)
vertex
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