SECTION 3.3 Quadratic Functions and Their Properties 159 Two conclusions can be drawn about the graph of f x ax .2 ( ) = • As a increases, the graph is vertically stretched (becomes “taller”), and as a gets closer to zero, the graph is vertically compressed (becomes “shorter”). • If a 0 > , the graph opens “up,” and if a 0 < , the graph opens “down.” Refer to Figure 16, where two parabolas are pictured. • The parabola on the left opens up.We describe this by saying the graph is concave up. Notice that the graph has a lowest point, where there is an absolute minimum. • The parabola on the right opens down. We describe this by saying the graph is concave down. Notice the graph has a highest point, where there is an absolute maximum. The lowest or highest point of a parabola is called the vertex. The vertical line passing through the vertex in each parabola is called the axis of symmetry (usually abbreviated to axis) of the parabola. The axis of a parabola can be used to find additional points on the parabola. The parabolas shown in Figure 16 are the graphs of a quadratic function f x ax bx c a , 0. 2 ( ) = + + ≠ Notice that the coordinate axes are not included in the figure. Depending on the numbers a, b, and c, the axes could be placed anywhere. The important fact is that the shape of the graph of a quadratic function will look like one of the parabolas in Figure 16. In the following example, techniques from Section 2.5 are used to graph a quadratic function f x ax bx c a , 0. 2 ( ) = + + ≠ The method of completing the square is used to write the function ƒ in the form f x a x h k. 2 ( ) ( ) = − + Need to Review? Completing the square is discussed in Section A.3, p. A29. Graphing a Quadratic Function Using Transformations Graph the function f x x x 2 8 5. 2 ( ) = + + Find the vertex and axis of symmetry. Solution EXAMPLE 1 Recall that to complete the square, the coefficient of x2 must equal 1. So we factor out 2 on the right-hand side. f x x x x x x x x 2 8 5 2 4 5 2 4 4 5 8 2 2 3 2 2 2 2 ( ) ( ) ( ) ( ) = + + = + + = + + + − = + − The graph of f can be obtained from the graph of y x2 = using transformations as shown in Figure 17. Now compare this graph to the graph in Figure 16(a). The graph of f x x x 2 8 5 2 ( ) = + + is a parabola that is concave up and has its vertex (lowest point) at 2, 3 . ( ) − − Its axis of symmetry is the line x 2. = − Figure 16 Graphs of a quadratic function, f x ax bx c a , 0 2 ( ) = + + ≠ (b) Concave down Vertex is highest point Axis of symmetry a , 0 (a) Concave up Axis of symmetry Vertex is lowest point a . 0 Factor out the 2 from + x x 2 8 . 2 Complete the square of + x x4 2 by adding 4. Notice that the factor of 2 requires that 8 be added and subtracted. Figure 17 Replace x by x 1 2; Shift left 2 units. x y 23 3 (21, 2) (22, 0) (23, 2) 3 23 Subtract 3; Shift down 3 units. Axis of symmetry x 5 22 Vertex x y 3 (21, 21) (22, 23) (23, 21) 3 23 Multiply by 2; Vertical stretch y 22 2 x 3 (23, 1) (22, 0) (21, 1) 23 x y (1, 1) (0, 0) (21, 1) 3 23 (c) y 5 2(x 1 2)2 (d) y 5 2(x 1 2)2 2 3 (b) y 5 (x 1 2)2 (a) y 5 x2 Now Work PROBLEM 37
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