158 CHAPTER 3 Linear and Quadratic Functions Many applications require a knowledge of quadratic functions. For example, suppose that Texas Instruments collects the data shown in Table 7, which relate the number of calculators sold to the price p (in dollars) per calculator. Since the price of a product determines the quantity that will be purchased, we treat price as the independent variable.The relationship between the number x of calculators sold and the price p per calculator is given by the linear equation x p 21,000 150 = − In Words A quadratic function is a function defined by a second-degree polynomial in one variable. Figure 12 R p p p 150 21,000 2 ( ) =− + 800,000 0 0 140 Figure 13 Path of a cannonball Price p per Calculator, (in dollars) Number of Calculators, x 60 65 70 75 80 85 90 12,000 11,250 10,500 9,750 9,000 8,250 7,500 Table 7 Then the revenue R derived from selling x calculators at the price p per calculator is equal to the unit selling price p of the calculator times the number x of units actually sold. That is, ( ) ( ) = = − = − + R xp R p p p p p 21,000 150 150 21,000 2 = − x p 21,000 150 So the revenue R is a quadratic function of the price p . Figure 12 illustrates the graph of this revenue function, whose domain is p 0 140, ≤ ≤ since both x and p must be nonnegative. A quadratic function also models the motion of a projectile. Based on Newton’s Second Law of Motion (force equals mass times acceleration, F ma = ), it can be shown that, ignoring air resistance, the path of a projectile propelled upward at an inclination to the horizontal is the graph of a quadratic function. See Figure 13 for an illustration. 1 Graph a Quadratic Function Using Transformations Figure 14 shows the graph of three functions of the form f x ax a , 0, 2 ( ) = > for a a 1, 1 2 , = = and a 3. = Figure 15 shows the graphs of f x ax2 ( ) = for a 0. < Notice that these graphs are reflections about the x -axis of the graphs in Figure 14. The graphs in Figures 14 and 15 are typical of the graphs of all quadratic functions, which are called parabolas . Figure 14 y ax a , 0 2 = > 4 21 24 4 Y2 5 x2 1 2 Y1 5 x2 Y3 5 3x2 Figure 15 y ax a , 0 2 = < 1 24 24 4 Y2 5 2 x2 1 2 Y1 5 2x2 Y3 5 23x2
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