Copyright © Cengage Learning. All rights reserved. 2 Nonlinear Functions and Models

Copyright © Cengage Learning. All rights reserved. 2.1 Quadratic Functions and Models

3 Quadratic Function A quadratic function of the variable x is a function that can be written in the form f (x) = ax 2 + bx + c or y = ax 2 + bx + c where a, b, and c are fixed numbers (with a ≠ 0). Function form Equation form

4 Quadratic Functions and Models Quick Example f (x) = 3x 2 – 2x + 1 Every quadratic function f (x) = ax 2 + bx + c (a ≠ 0) has a parabola as its graph. a = 3, b = –2, c = 1

5 Quadratic Functions and Models Following is a summary of some features of parabolas that we can use to sketch the graph of any quadratic function. Features of a Parabola The graph of f (x) = ax 2 + bx + c (a ≠ 0) is a parabola. If a > 0 the parabola opens upward (concave up) and if a < 0 it opens downward (concave down):

6 Quadratic Functions and Models Vertex, Intercepts, and Symmetry Vertex The vertex is the highest or lowest point of the parabola (see the above figure). Its x-coordinate is. Its y-coordinate is.

7 Quadratic Functions and Models x-Intercepts (if any) These occur when f (x) = 0; that is, when ax 2 + bx + c = 0. Solve this equation for x by either factoring or using the quadratic formula. The x-intercepts are If the discriminant b 2 – 4ac is positive, there are two x-intercepts. If it is zero, there is a single x-intercept (at the vertex). If it is negative, there are no x-intercepts (so the parabola doesn’t touch the x-axis at all).

8 Quadratic Functions and Models y-Intercept This occurs when x = 0, so y = a(0) 2 + b(0) + c = c. Symmetry The parabola is symmetric with respect to the vertical line through the vertex, which is the line x =

9 Quadratic Functions and Models Note that the x-intercepts can also be written as making it clear that they are located symmetrically on either side of the line x = –b/(2a). This partially justifies the claim that the whole parabola is symmetric with respect to this line.

10 Example 1 – Sketching the Graph of a Quadratic Function Sketch the graph of f (x) = x 2 + 2x – 8 by hand. Solution: Here, a = 1, b = 2, and c = –8. Because a > 0, the parabola is concave up (Figure 1). Figure 1

11 Vertex: The x coordinate of the vertex is To get its y coordinate, we substitute the value of x back into f (x) to get y = f (–1) = (–1) 2 + 2(–1) – 8 = 1 – 2 – 8 = –9. Thus, the coordinates of the vertex are (–1, –9). Example 1 – Solution cont’d

12 Example 1 – Solution x-Intercepts: To calculate the x-intercepts (if any), we solve the equation x 2 + 2x – 8 = 0. Luckily, this equation factors as (x + 4)(x – 2) = 0. Thus, the solutions are x = –4 and x = 2, so these values are the x-intercepts. y-Intercept: The y-intercept is given by c = –8. cont’d

13 Example 1 – Solution Symmetry: The graph is symmetric around the vertical line x = –1. Now we can sketch the curve as in Figure 2. (As we see in the figure, it is helpful to plot additional points by using the equation y = x 2 + 2x – 8, and to use symmetry to obtain others.) Figure 2 cont’d

14 Applications

15 Applications We know that the revenue resulting from one or more business transactions is the total payment received. Thus, if q units of some item are sold at p dollars per unit, the revenue resulting from the sale is revenue = price  quantity R = pq.

16 Example 3 – Demand and Revenue Alien Publications, Inc. predicts that the demand equation for the sale of its latest illustrated sci-fi novel Episode 93: Yoda vs. Alien is q = –2,000p + 150,000 where q is the number of books it can sell each year at a price of $p per book. What price should Alien Publications, Inc., charge to obtain the maximum annual revenue?

17 Example 3 – Solution The total revenue depends on the price, as follows: R = pq = p(–2,000p + 150,000) = –2,000p ,000p. We are after the price p that gives the maximum possible revenue. Formula for revenue. Substitute for q from demand equation. Simplify.

18 Example 3 – Solution Notice that what we have is a quadratic function of the form R(p) = ap 2 + bp + c, where a = –2,000, b = 150,000, and c = 0. Because a is negative, the graph of the function is a parabola, concave down, so its vertex is its highest point (Figure 5). cont’d Figure 5

19 Example 3 – Solution The p coordinate of the vertex is This value of p gives the highest point on the graph and thus gives the largest value of R(p). We may conclude that Alien Publications, Inc., should charge $37.50 per book to maximize its annual revenue. cont’d

20 Fitting a Quadratic Function to Data: Quadratic Regression

21 Fitting a Quadratic Function to Data: Quadratic Regression Here, we see how to use technology to obtain the quadratic regression curve associated with a set of points. The quadratic regression curve is the quadratic curve y = ax 2 + bx + c that best fits the data points in the sense that the associated sum-of-squares error is a minimum. Although there are algebraic methods for obtaining the quadratic regression curve, it is normal to use technology to do this.

22 Example 5 – Carbon Dioxide Concentration The following table shows the annual mean carbon dioxide concentration measured at Mauna Loa Observatory in Hawaii, in parts per million, every 10 years from 1960 through 2010 (t = 0 represents 1960). a. Is a linear model appropriate for these data? b. Find the quadratic model C(t) = at 2 + bt + c that best fits the data.

23 Example 5(a) – Solution To see whether a linear model is appropriate, we plot the data points and the regression line (Figure 8). From the graph, we can see that the given data suggest a curve and not a straight line: The observed points are above the regression line at the ends but below in the middle. (We would expect the data points from a linear relation to fall randomly above and below the regression line.) Figure 8

24 Example 5(b) – Solution The quadratic model that best fits the data is the quadratic regression model. As with linear regression, there are algebraic formulas to compute a, b, and c, but they are rather involved. However, we exploit the fact that these formulas are built into graphing calculators, spreadsheets, and other technology and obtain the regression curve using technology (see Figure 9): C(t) = 0.012t t Coefficients rounded to two significant digits Figure 9 cont’d

25 Example 5(b) – Solution Notice from the previous graphs that the quadratic regression model appears to give a far better fit than the linear regression model. This impression is supported by the values of SSE: For the linear regression model SSE  58, while for the quadratic regression model SSE is much smaller, approximately 2.6, indicating a much better fit. cont’d