Section 1: The Parabola

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 5.1 The Parabola 419 Version: Fall 2007 5.1 The Parabola In this section you will learn how to draw the graph of the quadratic function defined by the equation f ( x ) = a ( x h ) 2 + k. (1) You will quickly learn that the graph of the quadratic function is shaped like a "U" and is called a parabola . The form of the quadratic function in equation (1) is called vertex form , so named because the form easily reveals the vertex or “turning point” of the parabola. Each of the constants in the vertex form of the quadratic function plays a role. As you will soon see, the constant a controls the scaling (stretching or compressing of the parabola), the constant h controls a horizontal shift and placement of the axis of symmetry , and the constant k controls the vertical shift. Let’s begin by looking at the scaling of the quadratic. Scaling the Quadratic The graph of the basic quadratic function f ( x ) = x 2 shown in Figure 1 (a) is called a parabola . We say that the parabola in Figure 1 (a) “opens upward.” The point at (0 , 0) , the “turning point” of the parabola, is called the vertex of the parabola. We’ve tabulated a few points for reference in the table in Figure 1 (b) and then superimposed these points on the graph of f ( x ) = x 2 in Figure 1 (a). x y 5 5 f x f ( x ) = x 2 2 4 1 1 0 0 1 1 2 4 (a) A basic parabola. (b) Table of x -values and function values satisfying f ( x ) = x 2 . Figure 1. The graph of the basic parabola is a fundamental starting point. Now that we know the basic shape of the parabola determined by f ( x ) = x 2 , let’s see what happens when we scale the graph of f ( x ) = x 2 in the vertical direction. For Copyrighted material. See: 1
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420 Chapter 5 Quadratic Functions Version: Fall 2007 example, let’s investigate the graph of g ( x ) = 2 x 2 . The factor of 2 has a doubling effect. Note that each of the function values of g is twice the corresponding function value of f in the table in Figure 2 (b). x y 5 10 f g x f ( x ) = x 2 g ( x ) = 2 x 2 2 4 8 1 1 2 0 0 0 1 1 2 2 4 8 (a) The graphs of f and g . (b) Table of x -values and function values satisfying f ( x ) = x 2 and g ( x ) = 2 x 2 . Figure 2. A stretch by a factor of 2 in the vertical direction. When the points in the table in Figure 2 (b) are added to the coordinate system in Figure 2 (a), the resulting graph of g is stretched by a factor of two in the vertical direction. It’s as if we had put the original graph of f on a sheet of rubber graph paper, grabbed the top and bottom edges of the sheet, and then pulled each edge in the vertical direction to stretch the graph of f by a factor of two. Consequently, the graph of g ( x ) = 2 x 2 appears somewhat narrower in appearance, as seen in comparison to the graph of f ( x ) = x 2 in Figure 2 (a). Note, however, that the vertex at the origin is unaffected by this scaling.
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