8.5 Quadratic Functions and Their Graphs

Transcription

1 CHAPTER 8 Quadratic Equations and Functions 8. Quadratic Functions and Their Graphs S Graph Quadratic Functions of the Form f = + k. Graph Quadratic Functions of the Form f = - h. Graph Quadratic Functions of the Form f = - h + k. Graph Quadratic Functions of the Form f = a. Graph Quadratic Functions of the Form f = a - h + k. Graphing f() k We first graphed the quadratic equation = in Section.. In Section., we learned that this graph defines a function, and we wrote = as f =. In these sections, we discovered that the graph of a quadratic function is a parabola opening upward or downward. In this section, we continue our stud of quadratic functions and their graphs. (Much of the contents of this section is a review of shifting and reflecting techniques from Section., but specific to quadratic functions.) First, let s recall the definition of a quadratic function. Quadratic Function A quadratic function is a function that can be written in the form f = a + b + c, where a, b, and c are real numbers and a 0. Notice that equations of the form = a + b + c, where a 0, define quadratic functions, since is a function of or = f. Recall that if a 7 0, the parabola opens upward and if a 0, the parabola opens downward. Also, the verte of a parabola is the lowest point if the parabola opens upward and the highest point if the parabola opens downward. The ais of smmetr is the vertical line that passes through the verte. f() a b c, a 0 Verte f() a b c, a 0 Verte Ais of Smmetr Ais of Smmetr EXAMPLE Graph f = and g = + on the same set of aes. Solution First we construct a table of values for f() and plot the points. Notice that for each -value, the corresponding value of g() must be more than the corresponding value of f() since f = and g = +. In other words, the graph of g = + is the same as the graph of f = shifted upward units. The ais of smmetr for both graphs is the -ais. f () g() n Each -value is increased b. f() (0, ) (0, 0) Graph f = and g = - on the same set of aes. g()

2 Section 8. Quadratic Functions and Their Graphs In general, we have the following properties. Graphing the Parabola Defined b f () k If k is positive, the graph of f = + k is the graph of = shifted upward k units. If k is negative, the graph of f = + k is the graph of = shifted downward 0 k 0 units. The verte is (0, k), and the ais of smmetr is the -ais. EXAMPLE Graph each function. a. F = + b. g = - Solution a. F = + The graph of F = + is obtained b shifting the graph of = upward units. F() (0, ) b. g = - The graph of g = - is obtained b shifting the graph of = downward units. g() (0, ) Graph each function. a. f = - b. g = + Graphing f() ( h) Now we will graph functions of the form f = - h.

3 CHAPTER 8 Quadratic Equations and Functions EXAMPLE Graph f = and g = - on the same set of aes. Solution B plotting points, we see that for each -value, the corresponding value of g() is the same as the value of f() when the -value is increased b. Thus, the graph of g = - is the graph of f = shifted to the right units. The ais of smmetr for the graph of g = - is also shifted units to the right and is the line =. f () g() ( ) n Each -value increased b corresponds to same -value. f() g() ( ) (0, 0) (, 0) Graph f = and g = + on the same set of aes. In general, we have the following properties. Graphing the Parabola Defined b f ( ) ( h) If h is positive, the graph of f = - h is the graph of = shifted to the right h units. If h is negative, the graph of f = - h is the graph of = shifted to the left 0 h 0 units. The verte is (h, 0), and the ais of smmetr is the vertical line = h. EXAMPLE Graph each function. a. G = - b. F = + Solution a. The graph of G = - is obtained b shifting the graph of = to the right units. The graph of G() is below on the left. b. The equation F = + can be written as F = [ - -]. The graph of F = [ - -] is obtained b shifting the graph of = to the left unit. The graph of F() is below on the right. G() ( ) F() ( ) (, 0) (, 0)

4 Section 8. Quadratic Functions and Their Graphs Graph each function. a. G = + b. H = - 7 Graphing f () ( h) k As we will see in graphing functions of the form f = - h + k, it is possible to combine vertical and horizontal shifts. Graphing the Parabola Defined b f( ) ( h) k The parabola has the same shape as =. The verte is (h, k), and the ais of smmetr is the vertical line = h. EXAMPLE Graph F = - +. Solution The graph of F = - + is the graph of = shifted units to the right and unit up. The verte is then (, ), and the ais of smmetr is =. A few ordered pair solutions are plotted to aid in graphing. F () ( ) F() ( ) (, ) Graph f = + +. Graphing f () a Net, we discover the change in the shape of the graph when the coefficient of is not. EXAMPLE Graph f =, g =, and h = on the same set of aes. Solution Comparing the tables of values, we see that for each -value, the corresponding value of g() is triple the corresponding value of f(). Similarl, the value of h() is half the value of f(). f () (Continued on net page) g() h()

5 CHAPTER 8 Quadratic Equations and Functions The result is that the graph of g = is narrower than the graph of f =, and the graph of h = is wider. The verte for each graph is (0, 0), and the ais of smmetr is the -ais g() f() h() q Graph f =, g =, and h = on the same set of aes. Graphing the Parabola Defined b f() a If a is positive, the parabola opens upward, and if a is negative, the parabola opens downward. If 0 a 0 7, the graph of the parabola is narrower than the graph of =. If 0 a 0, the graph of the parabola is wider than the graph of =. EXAMPLE 7 Graph f = -. Solution Because a = -, a negative value, this parabola opens downward. Since 0-0 = and 7, the parabola is narrower than the graph of =. The verte is (0, 0), and the ais of smmetr is the -ais. We verif this b plotting a few points. f () (0, 0) f() Graph f = -. Graphing f() a( h) k Now we will see the shape of the graph of a quadratic function of the form f = a - h + k.

6 Section 8. Quadratic Functions and Their Graphs 7 EXAMPLE 8 smmetr. Graph g = + +. Find the verte and the ais of Solution The function g = + + ma be written as g() = [ - (-)] +. Thus, this graph is the same as the graph of = shifted units to the left and units up, and it is wider because a is. The verte is -,, and the ais of smmetr is = -. We plot a few points to verif. g() ( ) (, ) g() q( ) 8 Graph h = - -. In general, the following holds. Graph of a Quadratic Function The graph of a quadratic function written in the form f = a - h + k is a parabola with verte (h, k). If a 7 0, the parabola opens upward. If a 0, the parabola opens downward. The ais of smmetr is the line whose equation is = h. f() a( h) k a 0 (h, k) (h, k) h h a 0 CONCEPT CHECK Which description of the graph of f = is correct? a. The graph opens downward and has its verte at -,. b. The graph opens upward and has its verte at -,. c. The graph opens downward and has its verte at -, -. d. The graph is narrower than the graph of =. Answer to Concept Check: c

7 8 CHAPTER 8 Quadratic Equations and Functions Graphing Calculator Eplorations Use a graphing calculator to graph the first function of each pair that follows. Then use its graph to predict the graph of the second function. Check our prediction b graphing both on the same set of aes.. F = ; G = +. g = ; H = -. H = 0 0 ; f = 0-0. h = + ; g = - +. f = ; F = G = - ; g = - - Vocabular, Readiness & Video Check Use the choices below to fill in each blank. Some choices will be used more than once. upward highest parabola downward lowest quadratic. A(n) function is one that can be written in the form f = a + b + c, a 0.. The graph of a quadratic function is a(n) opening or.. If a 7 0, the graph of the quadratic function opens.. If a 0, the graph of the quadratic function opens.. The verte of a parabola is the point if a The verte of a parabola is the point if a 0. State the verte of the graph of each quadratic function. 7. f = 8. f = - 9. g = - 0. g = +. f = +. h = -. g = + +. h = Martin-Ga Interactive Videos See Video 8. Watch the section lecture video and answer the following questions.. From Eamples and and the lecture before, how do graphs of the form f = + k differ from =? Consider the location of the verte (0, k) on these graphs of the form f = + k b what other name do we call this point on a graph?. From Eample and the lecture before, how do graphs of the form f = - h differ from =? Consider the location of the verte (h, 0) on these graphs of the form f = - h b what other name do we call this point on a graph? 7. From Eample and the lecture before, what general information does the equation f = - h + k tell us about its graph? 8. From the lecture before Eample, besides the direction a parabola opens, what other graphing information can the value of a tell us? 9. In Eamples and 7, what four properties of the graph did we learn from the equation that helped us locate and draw the general shape of the parabola?

8 Section 8. Quadratic Functions and Their Graphs 9 8. Eercise Set MIXED Sketch the graph of each quadratic function. Label the verte and sketch and label the ais of smmetr. See Eamples through.. f = -. g = +. h = +. h = -. g = + 7. f = - 7. f = - 8. g = + 9. h = + 0. H = -. G = +. f = -. f = - +. g = - +. h = + +. G = g = h = + - Sketch the graph of each quadratic function. Label the verte, and sketch and label the ais of smmetr. See Eamples and H = 0. f =. h =. f = -. g = -. g = - Sketch the graph of each quadratic function. Label the verte and sketch and label the ais of smmetr. See Eample 8.. f = - +. g = h = f = H = G = + + MIXED Sketch the graph of each quadratic function. Label the verte and sketch and label the ais of smmetr.. f = - -. g = - +. F = - +. H = F = -. g = - 7. h = f = F = a + b - 0. H = a + b -. F = g = f = - 9. H = -. h = h = f = G = REVIEW AND PREVIEW Add the proper constant to each binomial so that the resulting trinomial is a perfect square trinomial. See Section z - z z - z Solve b completing the square. See Section =. + = -. z + 0z - = = 0. z - 8z =. - 0 = CONCEPT EXTENSIONS Solve. See the Concept Check in this section. 7. Which description of f = is correct? Graph Opens Verte a. upward (0.,.) b. upward -,. c. downward (0.,.) d. downward -0.,. 8. Which description of f = a + b + is correct? Graph Opens Verte a. upward a, b b. upward a -, b c. downward a, - b d. downward a -, - b. G = a + b. F = a - b Write the equation of the parabola that has the same shape as f = but with the following verte. 7. h = f = g = G = (, ) 70. (, ) 7. -, 7., -

9 0 CHAPTER 8 Quadratic Equations and Functions The shifting properties covered in this section appl to the graphs of all functions. Given the graph of = f below, sketch the graph of each of the following. f() 7. = f + 7. = f - 7. = f - 7. = f = f = f The quadratic function f = 8-0,9 + 7 approimates the number of tet messages sent in the United States each month between 000 and 008, where is the number of ears past 000 and f is the number of tet messages sent in the U.S. each month in millions. (Source: cellsigns) a. Use this function to find the number of tet messages sent in the U.S. each month in 00. b. Use this function to predict the number of tet messages sent in the U.S. each month in Use the function in Eercise 79. a. Use this function to predict the number of tet messages sent in the U.S. each month in 08. b. Look up the current number of cell phone subscribers in the U.S. c. Based on our answers for parts a. and b., discuss some possible limitations of using this quadratic function to predict data. 8. Further Graphing of Quadratic Functions S Write Quadratic Functions in the Form = a - h + k. Derive a Formula for Finding the Verte of a Parabola. Find the Minimum or Maimum Value of a Quadratic Function. Writing Quadratic Functions in the Form a h k We know that the graph of a quadratic function is a parabola. If a quadratic function is written in the form f = a - h + k we can easil find the verte (h, k) and graph the parabola. To write a quadratic function in this form, complete the square. (See Section 8. for a review of completing the square.) EXAMPLE Graph f = - -. Find the verte and an intercepts. Solution The graph of this quadratic function is a parabola. To find the verte of the parabola, we will write the function in the form = - h + k. To do this, we complete the square on the binomial -. To simplif our work, we let f =. = - - Let f =. + = - Add to both sides to get the -variable terms alone. Now we add the square of half of - to both sides. - = - and - = + + = = - = - - f = - - Add to both sides. Factor the trinomial. Subtract from both sides. Replace with f. From this equation, we can see that the verte of the parabola is, -, a point in quadrant IV, and the ais of smmetr is the line =. Notice that a =. Since a 7 0, the parabola opens upward. This parabola opening upward with verte, - will have two -intercepts and one -intercept. (See the Helpful Hint after this eample.)