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

Quadratic Functions and Equations 8 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 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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

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

Example Graph: Solution 1 –1 2 –2 8 (0, 0) (1, 2) (–1, 2) (2, 8) y (-1,2 ) (2,8) (-2,8) -5 -4 -3 -2 -1 1 2 3 4 5 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) Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Graph: Solution 1 –1 2 –2 –3 –12 (0, 0) (1, –3) (–1, –3) y -5 -4 -3 -2 -1 1 2 3 4 5 -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) Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

4 3 6 2 5 1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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

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

Example Graph: Solution vertex 1 –1 2 3 4 9 (0, 4) (1, 1) (–1, 9) y -5 -4 -3 -2 -1 1 2 3 4 5 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 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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

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

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

Example Graph: Solution vertex –1 –2 –3 –4 –5 -11/2 –3/2 (0, -11/2) y -5 -4 -3 -2 -1 1 2 3 4 5 -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 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley