8.5. Quadratic Function A function f is a quadratic function if f(x) ax 2 bx c, where a, b, and c are real numbers, with a 0.

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1 8.5 Quadratic Functions, Applications, and Models In the previous section we discussed linear functions, those that are defined b firstdegree polnomials. In this section we will look at quadratic functions, those defined b second-degree polnomials. Ais Quadratic Function A function f is a quadratic function if f() a b c, where a, b, and c are real numbers, with a. (, ) (, ) Verte (, ) (, ) (, ) f() = FIGURE The simplest quadratic function is defined b f. This function can be graphed b finding several ordered pairs that satisf the equation: for eample,,,,,,,,,,,,,,, 3, 9, and 3, 9. Plotting these points and drawing a smooth curve through them gives the graph shown in Figure. This graph is called a parabola. Ever quadratic function has a graph that is a parabola.

2 8.5 Quadratic Functions, Applications, and Models 35 Focus Ais Parabolic reflector FIGURE 5 Parabolas are smmetric about a line (the -ais in Figure.) Intuitivel, this means that if the graph were folded along the line of smmetr, the two sides would coincide. The line of smmetr for a parabola is called the ais of the parabola. The point where the ais intersects the parabola is the verte of the parabola. The verte is the lowest (or highest) point of a vertical parabola. Parabolas have man practical applications. For eample, the reflectors of solar ovens and flashlights are made b revolving a parabola about its ais. The focus of a parabola is a point on its ais that determines the curvature. See Figure 5. When the parabolic reflector of a solar oven is aimed at the sun, the light ras bounce off the reflector and collect at the focus, creating intense heat at that point. In contrast, when a lightbulb is place at the focus of a parabolic reflector, light ras reflect out parallel to the ais. Graphing Quadratic Functions The first eample shows how the constant a affects the graph of a function of the form g a. EXAMPLE Graph the functions defined as follows. (a) g For a given value of, the corresponding value of g will be the negative of what it was for f. (See the table of values with Figure 6(a).) Because of this, the graph of g is the same shape as that of f, but opens downward. See Figure 6(a). This is generall true; the graph of f a b c opens downward whenever a. = 3 = _ = The screen illustrates the three graphs considered in Eample and Figures 6(a) and 6(b). f() = (, ) (, ) (, ) g() = (, ) (, ) (a) (, ) f() = (, _ ) (, ) (b) g() = _ (, ) (, _ ) FIGURE 6 (b) g Choose a value of, and then find g. The coefficient will cause the resulting value of g to be smaller than for f, making the parabola wider than the graph of f. See Figure 6(b). In both parabolas of this eample, the ais is the vertical line and the verte is the origin,.

3 36 CHAPTER 8 Graphs, Functions, and Sstems of Equations and Inequalities The net few eamples show the results of horizontal and vertical shifts, called translations, of the graph of f. EXAMPLE Graph g. B comparing the tables of values for g and f shown with Figure 7, we can see that for corresponding -values, the -values of g are each less than those for f. Thus, the graph of g is the same as that of f, but translated units down. See Figure 7. The verte of this parabola (here the lowest point) is at,. The ais of the parabola is the vertical line. f() = g() = Compare with Figure 7. g() = 3 3 f() = (, ) (, ) f() = g() = FIGURE 7 EXAMPLE 3 Graph g. Comparing the tables of values shown with Figure 8 shows that the graph of g is the same as that of f, but translated units to the right. The verte is at,. As shown in Figure 8, the ais of this parabola is the vertical line. f() = g() = ( ) Compare with Figure 8. g() = ( ) f() = = (, ) (, ) f() = g() = ( ) FIGURE 8 Errors frequentl occur when horizontal shifts are involved. To determine the direction and magnitude of horizontal shifts, find the value that would cause the epression h to equal. For eample, the graph of f 5 would be shifted 5 units to the right, because 5 would cause 5 to equal. On the other hand, the graph of f would be shifted units to the left, because would cause to equal. The following general principles appl for graphing functions of the form f a h k.

4 8.5 Quadratic Functions, Applications, and Models 37 General Principles for Graphs of Quadratic Functions a. The graph of the quadratic function defined b f() a( h) k, a, is a parabola with verte h, k, and the vertical line h as ais.. The graph opens upward if a is positive and downward if a is negative. 3. The graph is wider than that of f if. The graph is narrower than that of f if a. f() = ( + 3) + EXAMPLE Graph f 3. The parabola opens downward (because a ), and is narrower than the graph of f, since a, and. This parabola has verte at 3,, as shown in Figure 9. To complete the graph, we plotted the ordered pairs, and,. f() = ( + 3) + Compare with Figure 9. The verte is 3,. = 3 3 FIGURE 9 When the equation of a parabola is given in the form f a b c, it is necessar to locate the verte in order to sketch an accurate graph. This can be done in two was. The first is b completing the square, as shown in Eample 5. The second is b using a formula which can be derived b completing the square. 5 The trajector of a shell fired from a cannon is a parabola. To reach the maimum range with a cannon, it is shown in calculus that the muzzle must be set at 5. If the muzzle is elevated above 5, the shell goes too high and falls too soon. If the muzzle is set below 5, the shell is rapidl pulled to Earth b gravit. EXAMPLE 5 Find the verte of the graph of f 5. To find the verte, we need to epress 5 in the form h k. This is done b completing the square. (See the previous chapter.) To simplif the notation, replace f b. 5 5 Transform so that the constant term is on the left. 5 Half of is ;. Add to both sides. Combine terms on the left and factor on the right. Add to both sides. Now write the original equation as f. As shown earlier, the verte of this parabola is,.

5 38 CHAPTER 8 Graphs, Functions, and Sstems of Equations and Inequalities Johann Kepler (57 63) established the importance of a curve called an ellipse in 69, when he discovered that the orbits of the planets around the sun were elliptical, not circular. The orbit of Halle s comet, shown here, also is elliptical. See For Further Thought at the end of this section for more on ellipses. A formula for the verte of the graph of the quadratic function a b c can be found b completing the square for the general form of the equation. In doing so, we begin b dividing b a, since the coefficient of must be. a c a a b c a b a c a a c a b a b a b b a a a b ac b a a a b a b ac a a a b ac b a a a b h a ac b a k Divide b a. c Subtract. a Add a. Combine terms on left and factor on the right. Transform so that the -term is alone on the left. Multipl b a. The final equation shows that the verte h, k can be epressed in terms of a, b, and c. However, it is not necessar to memorize the epression for k, since it can be obtained b replacing b b. Using function notation, if f, the -value a of the verte is f b. a b f() = 6 Notice the slight discrepanc when we instruct the calculator to find the verte (a minimum here). This reinforces the fact that we must understand the concepts and not totall rel on technolog! EXAMPLE 6 Use the verte formula to find the verte of the graph of the function f 6. For this function, a, b, and c 6. The -coordinate of the verte of the parabola is given b b a. The -coordinate is. f b a f f Finall, the verte is, 5.

6 8.5 Quadratic Functions, Applications, and Models 39 A general approach to graphing quadratic functions using intercepts and the verte is now given. Graphing a Quadratic Function f a b c Step : Decide whether the graph opens upward or downward. Determine whether the graph opens upward (if a ) or opens downward (if a ) to aid in the graphing process. Step : Find the -intercept. Find the -intercept b evaluating f. Step 3: Find the -intercepts. Find the -intercepts, if an, b solving f. Step : Find the verte. Find the verte either b using the formula or b completing the square. Step 5: Complete the graph. Find and plot additional points as needed, using the smmetr about the ais. f() = 6 This split screen illustrates that the points, 6, 3,, and, lie on the graph of f 6. Compare with Figure 3. EXAMPLE 7 Graph the quadratic function f 6. Because a, the parabola will open upward. Now find the -intercept. f 6 f 6 Find f. f 6 The -intercept is, 6. Now find an -intercepts. f Let f. Factor. 3 or Set each factor equal to and solve. 3 or The -intercepts are 3, and,. The verte, found in Eample 6, is, 5. Plot the points found so far, and plot an additional points as needed. The smmetr of the graph is helpful here. The graph is shown in Figure 3. = _ f() = 6 (, ) (3, ) This table provides other points on the graph of (, ) (, 6) (, ) _ ( ), 5 Y X X 6. FIGURE 3

7 CHAPTER 8 Graphs, Functions, and Sstems of Equations and Inequalities A Model for Optimization As we have seen, the verte of a vertical parabola is either the highest or the lowest point of the parabola. The -value of the verte gives the maimum or minimum value of, while the -value tells where that maimum or minimum occurs. Often a model can be constructed so that can be optimized. Galileo Galilei (56 6) died in the ear Newton was born; his work was important in Newton s development of calculus. The idea of function is implicit in Galileo s analsis of the parabolic path of a projectile, where height and range are functions (in our terms) of the angle of elevation and the initial velocit. According to legend, Galileo dropped objects of different weights from the tower of Pisa to disprove the Aristotelian view that heavier objects fall faster than lighter objects. He developed a formula for freel falling objects that is described b d 6t, where d is the distance in feet that a given object falls (discounting air resistance) in a given time t, in seconds, regardless of weight. Problem Solving In some practical problems we want to know the largest or smallest value of some quantit. When that quantit can be epressed using a quadratic function f a b c, as in the net eample, the verte can be used to find the desired value. EXAMPLE 8 A farmer has feet of fencing. He wants to put a fence around three sides of a rectangular plot of land, with the side of a barn forming the fourth side. Find the maimum area he can enclose. What dimensions give this area? FIGURE 3 A sonic boom is a loud eplosive sound caused b the shock wave that accompanies an aircraft traveling at supersonic speed. The sonic boom shock wave has the shape of a cone, and it intersects the ground in one branch of a curve known as a hperbola. Everone located along the hperbolic curve on the ground hears the sound at the same time. See For Further Thought at the end of this section for more on hperbolas. α Figure 3 shows the plot. Let represent its width. Then, since there are feet of fencing, length length length. Sum of the three fenced sides is feet. Combine terms. Subtract. The area is modeled b the product of the length and width, or A. To make the area (and thus ) as large as possible, first find the verte of the graph of the function A. A Standard form Here we have a and b. The -coordinate of the verte is b 3. a

8 8.5 Quadratic Functions, Applications, and Models A() = 6 The verte is 3, 8, supporting the analtic result in Eample 8. The verte is a maimum point (since a ), so the maimum area that the farmer can enclose is A square feet. The farmer can enclose a maimum area of 8 square feet, when the width of the plot is 3 feet and the length is 3 6 feet. As seen in Eample 8, be careful when interpreting the meanings of the coordinates of the verte in problems involving maimum or minimum values. The first coordinate,, gives the value for which the function value is a maimum or a minimum. Read the problem carefull to determine whether ou are asked to find the value of the independent variable, the dependent variable (that is, the function value), or both. FOR FURTHER THOUGHT The circle, introduced in the first section of this chapter, the parabola, the ellipse, and the hperbola are known as conic sections. As seen in the accompaning figure, each of these geometric shapes can be obtained b intersecting a plane and an infinite cone (made up of two nappes). Circle Ellipse Parabola Hperbola For Group Discussion. The terms ellipse, parabola, and hperbola are similar to the terms ellipsis, parable, and hperbole. What do these latter three terms mean? You might want to do some investigation as to the similarities between the mathematical terminolog and these language-related terms.. Name some places in the world around ou where conic sections are encountered. 3. The accompaning figure shows how an ellipse can be drawn using tacks and string. Have a class member volunteer to go to the board and using string and chalk, modif the method to draw a circle. Then have two class members work together to draw an ellipse. (Hint: Press hard!) The Greek geometer Apollonius (c. 5 B.C.) was also an astronomer, and his classic work Conic Sections thoroughl investigated these figures. Apollonius is responsible for the names ellipse, parabola, and hperbola. The margin notes in this section show some was that these figures appear in the world around us.