Name Period Date Introduction to Quadratic Functions Activity 2 Imagine yourself standing on the roof of the 1450-foot-high Willis Tower (formerly called the Sears Tower) in Chicago. When you release and drop a baseball from the roof of the tower, the ball s height above the ground, H (in feet), can be described as a function of the time, t (in seconds), since it was dropped. This height function is defined by: H(t) = 16t 2 + 1450 1. Sketch a diagram illustrating the Willis Tower and the path of the baseball as it falls to the ground. 2. a. Complete the following table. TIME, t(sec) H(t) = 16t 2 + 1450 0 1 2 3 4 5 6 7 8 9 10 b. How far does the baseball fall during the first second? c. How far does it fall during the interval from 1 to 3 seconds? 3. Using the height function H(t) = 16t 2 + 1450, determine the average rate of change of H with respect to t over the given interval. Remember: average rate of change = change in output change in input a. 0 t 1 b. 1 t 3 1
c. Based on the results of parts a and b, do you believe that H(t) = 16t 2 + 1450 is a linear function? Explain. 4. a. What is the value of H when the baseball strikes the ground? Use the table in Problem 2a to estimate the time when the ball is at ground level. b. What is the practical domain of the height function? c. Determine the practical range of the height function. d. On the following grid, plot the points in Problem 2a that satisfy part b (practical domain) and sketch a curve representing the height function. e. Is the graph of the height function in part d the actual path of the object (see Problem 1)? Explain. 2
Some interesting properties of the function defined by H(t) = 16t 2 + 1450 arise when you ignore the falling object context. Replace H with y and t with x and consider the general function defined by y = 16x 2 + 1450 5. a. The graph of the function looks like: b. Describe the important features of the graph of y = 16x 2 + 1450. Discuss the shape, symmetry, and intercepts. Quadratic Functions The graph of the function defined by y = 16x 2 + 1450 is a parabola. The graph of a parabola is a shaped figure that opens upward,, or downward,,. Parabolas are graphs of a special category of functions called quadratic functions. Definition Any function defined by an equation of the form y = ax 2 + bx + c or f(x) = ax 2 + bx + c where a, b, and c represent real numbers and a 0, is called quadratic function. The output variable y is defined by an expression having three terms: the quadratic term, ax 2, the linear term, bx, and the constant term, c. The numerical factors of the quadratic and linear terms, a and b, are called the coefficients of the terms. H(t) = 16t 2 + 1450 defines a quadratic function. The quadratic term is 16t 2. The linear term is 0t, although it is not written as part of the expression defining H(t). The constant term is 1450. The numbers 16 and 0 are the coefficients of the quadratic and linear terms, respectively. Therefore, a = 16, b = 0, and c = 1450. 3
6. For each of the following quadratic functions, identify the value of a, b, and c. Quadratic Function a b c y = 3x 2 y = 2x 2 + 3 y = x 2 + 2x 1 y = x 2 + 4x The Constant Term c: A Closer Look Consider once again the height function H(t) = 16t 2 + 1450 from the beginning of the activity. 7. a. What is the vertical intercept of the graph? Explain how you obtained the results. b. What is the practical meaning of the vertical intercept in this situation? c. Predict what the graph of H(t) = 16t 2 + 1450 would look like if the constant term 1450 were changed to 800. That is, the baseball is dropped from a height of 800 feet rather than 1450 feet. Verify your prediction by graphing H(t) = 16t 2 + 800. What does the constant term tell you about the graph of the parabola? The constant term c of a quadratic function f(x) = ax 2 + bx + c always indicates the vertical intercept of the parabola. The vertical intercept of any quadratic function is (0, c) since f(0) = a(0) 2 + b(0) + c = c. 4
8. Graph the parabolas defined by the following quadratic equations. Note the similarities and differences among the graphs, especially the vertical intercepts. Use desmos.com on your mobile device or download the app. a. f1(x) = 1.5x 2 b. f2(x) = 1.5x 2 + 2 c. f3(x) = 1.5x 2 + 4 d. f4(x) = 1.5x 2 4 5
The Effects of the Coefficient a on the Graph of y = ax 2 + bx + c 9. a. Graph the quadratic function defined by f1(x) = 16x 2 + 1450 on the same screen as f2(x) = 16x 2 + 1450. Click on the wrench and change the y-axis to 2000 y 2000 and use your touchscreen to make adjustments. b. What effect does the sign of the coefficient of x 2 appear to have on the graph of the parabola? c. Graph the functions in the f3(x) = 16x 2 + 100, f4(x) = 6x 2 + 100, f5(x) = 40x 2 + 100 in the same window. What effect does the magnitude of the coefficients of x 2 (namely, 16 = 16, 6 = 6, 40 = 40 appear to have on the graph of that particular parabola? The results from Problems 9 regarding the effects of the coefficient a can be summarized as follows. The graph of a quadratic function defined by f(x) = ax 2 + bx + c is called a parabola. If a > 0, the parabola opens upward. If a < 0, the parabola opens downward. The magnitude of a affects the width of the parabola. The larger the absolute value of a, the narrower the parabola. 10. a. Is the graph of h(x) = 0.3x 2 wider or narrower than the graph of f(x) = x 2? b. How do the output values of h and the output values of f compare for the same input value? c. Is the graph of g(x) = 3x 2 wider or narrower than the graph of f(x) = x 2? d. How do the output values of g and f compare for the same input value? e. Describe the effect of the magnitude of the coefficient a on the width of the graph of the parabola. f. Describe the effect of the magnitude of the coefficient of a on the output value. 6
The Effects of the Coefficient b on the Turning Point Assume for the time being that you are back on the roof of the 1450-foot Willis Tower. Instead of merely releasing the ball, suppose you throw it down with an initial velocity of 40 feet per second. Then the function describing its height above ground as a function of time is modeled by H down (t) = 16t 2 40t + 1450 If you tossed the ball straight up with an initial velocity of 40 feet per second, then the function describing its height above ground as a function of time is modeled by H up (t) = 16t 2 + 40t + 1450 11. Predict what features of the graphs of H down and H up have in common with H(t) = 16t 2 + 1450. 12. a. Graph the three functions H(t). H down, and H up using x for t and the window settings Xmin= 10, Xmax= 10, Ymin= 50 and Ymax= 3000 b. What effect do the 40t and 40t terms seem to have upon the turning point of the graphs? If b = 0, the turning point of the parabola is located on the vertical axis. If b 0, the turning point will not be on the vertical axis. 13. For each of the following quadratic functions, identify the value of b and then, without graphing, determine whether or not the turning point is on the y-axis. Verify your conclusion by graphing the given function using desmos. a. y = x 2 b. y = x 2 4x c. y = x 2 + 4 d. y = x 2 + x e. y = x 2 3 7
14. Match each function with its corresponding graph below, and then verify using desmos. Explain how you chose your answer. a. f(x) = x 2 + 4x + 4 b. f(x) = 0.2x 2 + 4 c.h(x) = x 2 + 3x Explanations: Summary 1. The equation of a quadratic function with x as the input variable and y as the output variable has the standard form y = ax 2 + bx + c where a, b, and c represent real numbers and a 0. 2. The graph of a quadratic function is called a parabola. 3. For the quadratic function defined by f(x) = ax 2 + bx + c: a. If a > 0, the parabola opens upward. b. If a < 0, the parabola opens downward. The magnitude of a affects the width of the parabola. The larger the absolute value of a, the narrower the parabola. 4. If b = 0. the turning point of the parabola is located on the vertical axis. If b 0, the turning point will not be on the vertical axis. 5. The constant term c of a quadratic function f(x) = ax 2 + bx + c always indicates the vertical intercept of the parabola. The vertical intercept of any quadratic function is (0, c). 8
Practice 1. Complete the following table for f(x) = x 2 x 3 2 1 0 1 2 3 f(x) = x 2 b. Use the results of part a to sketch a graph y = x 2. Verify using desmos. c. What is the coefficient of the term x 2? d. From the graph, determine the domain and range of the function. e. Create a table similar to the one in Exercise 1a to show the output for g(x) = x 2. x 3 2 1 0 1 2 3 g(x) = x 2 f. Sketch the graph of g(x) = x 2 on the same coordinate axis in part b. Verify using desmos g. What is the coefficient of the term x 2? h. How can the graph of y = x 2 be obtained from the graph of y = x 2? 9
2. In each of the following functions defined by an equation of the form = ax 2 + bx + c, identify the value of a, b, and c. a. y = 2x 2 b. y = 2 5 x2 + 3 c. y = x 2 + 5x d. y = 5x 2 + 2x 1 3. Predict what the graph of each of the following quadratic functions will look like. Use desmos to verify your prediction. a. f(x) = 3x 2 + 5 b. g(x) = 2x 2 + 1 c. h(x) = 0.5x 2 3 4. Graph the following pairs of functions, and describe any similarities as well as any differences that you observe in the graphs. a. f(x) = 3x 2, g(x) = 3x 2 b. h(x) = 1 2 x2, f(x) = 2x 2 10
c. f(x) = 5x 2, g(x) = 5x 2 + 2 d. f(x) = 4x 2 3, g(x) = 4x 2 + 3 e. f(x) = 6x 2 + 1, h(x) = 6x 2 1 11
5. Use desmos to graph the two functions f1(x) = 3x 2 and f2(x) = 3x 2 + 2x 2. a. What is the vertical intercept of the graph of each function? b. Compare the two graphs to determine the effect of the linear term 2x and the constant term 2 on the graph of f1(x) = 3x 2. Discuss this below. For Exercises 6 10, determine a. whether the parabola opens upward or downward and b. the vertical intercept. 6. f(x) = 5x 2 + 2x 4 7. g(t) = 1 2 t2 + t 8. h(v) = 2v 2 + v + 3 a. a. a. b. b. b. 9. r(t) = 3t 2 + 10 10. f(x) = x 2 + 6x 7 a. a. b. b. 11. Does the graph of y = 2x 2 + 3x 4 have any horizontal intercepts? Explain. 12. a. Is the graph of y = 3 5 x2 wider or narrower than the graph of y = x 2? Why? b. For the same input value, which graph would have a larger output value? Prove this using an example. 13. Put the following in order from narrowest to widest. a. y = 0.5x 2 b, y = 8x 2 c. y = 2.3x 2 12