By naming a function f, you can write the function using function notation. Function notation. ACTIVITY: Matching Functions with Their Graphs

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1 5. Function Notation represent a function? How can ou use function notation to B naming a function f, ou can write the function using function notation. f () = Function notation This is read as f of equals minus. The notation f () is another name for. When function notation is used, the parentheses do not impl multiplication. You can use letters other than f to name a function. The letters g, h, j, and k are often used to name functions. ACTIVITY: Matching Functions with Their Graphs Work with a partner. Match each function with its graph. a. f () = b. g() = + c. h() = d. j () = A. B. COMMON CORE Functions In this lesson, ou will evaluate, solve, and graph functions written in function notation. compare graphs of linear functions. Learning Standards F.BF. F.IF. F.IF. F.IF.7b C. D. Chapter 5 Linear Functions

2 ACTIVITY: Evaluating a Function Math Practice Evaluate Results Does our answer seem reasonable? How can ou check our answer? Work with a partner. Consider the function f () = +. Locate the points (, f ()) on the graph. Eplain how ou found each point. a. (, f ( )) f() b. (0, f (0)) c. (, f ()) d. (, f ()) ACTIVITY: Comparing Graphs of Functions Work with a partner. The graph of a function from trigonometr is shown at the right. Use the graph to sketch the graph of each function. Eplain our reasoning. f() a. g() = f () + b. g() = f () + c. g() = f () d. g() = f (). IN YOUR OWN WORDS How can ou use function notation to represent a function? How are standard notation and function notation similar? How are the different? Standard Notation Function Notation = + 5 f () = Use what ou discovered in Activit to write a general observation that compares the graphs of = f () and = f () + c. Use what ou learned about function notation to complete Eercises on page 9. Section 5. Function Notation 5

3 5. Lesson Lesson Tutorials Ke Vocabular function notation, p. In Section 5., ou learned that ou can write a linear function in the form = m + b. B naming a linear function f, ou can also write the function using function notation. f () = m + b Function notation The notation f () is another name for. If f is a function and is in its domain, then f () represents the output of f corresponding to the input. You can use letters other than f to name a function, such as g or h. Reading EXAMPLE The notation f () is read as the value of f at or f of. It does not mean f times. Evaluating a Function Evaluate f () = + 7 when =. f () = + 7 Write the function. f () = () + 7 Substitute for. = Multipl. = Add. When =, f () =. Eercises 9 Evaluate the function when =, 0, and.. f () = 5. g() = EXAMPLE Solving for the Independent Variable For h() = 5, find the value of for which h() = 7. h() = 5 Write the function. 7 = 5 Substitute 7 for h(). = Add 5 to each side. = Multipl each side b. When =, h() = 7. Eercises Find the value of so that the function has the given value.. f () = + 9; f () =. g() = + ; g() = Chapter 5 Linear Functions

4 EXAMPLE Graphing a Linear Function in Function Notation Graph f () = + 5. Step : Make a table of values. 0 f () Stud Tip The graph of f () consists of the points (, f ()). Step : Plot the ordered pairs. Step : Draw a line through the points f() 5 Eercises 7 Graph the linear function. 5. f () =. g() = + 7. h() = Vertical Translations The graph of f () + k is a vertical translation of the graph of f (), where k 0. f() k, k 0 f() f() k, k 0 In vertical translations of graphs of linear functions, the graphs have the same slope but different -intercepts. Section 5. Function Notation 7

5 EXAMPLE Comparing Graphs of Linear Functions Graph g() =. Compare the graph to the graph of f () =. f() 5 g() Use the slope and -intercept to graph the equations. g () = f () = = + ( ) = + 0 slope -intercept slope -intercept The graphs have the same slope but different -intercepts. The graph of g is a translation units down of the graph of f. EXAMPLE 5 Real-Life Application Distance (miles) Helicopter f() Hours The graph shows the number of miles a helicopter is from its destination after hours on its first flight. On its second flight, the helicopter travels at the same speed but 50 miles farther. Which statement is true about the graph of the function that represents the second flight compared to the graph of the function that represents the first flight? A The slope decreases. B The slope increases. C The graph is a translation D The graph is a translation 50 units down. 50 units up. The helicopter travels at the same speed on both flights. So, the graphs have the same slope. You can eliminate choices A and B. Because the helicopter travels 50 miles farther on the second flight, it is 50 miles farther from its destination when = 0. So, the graph of the function that represents the second flight is a vertical translation 50 units up of the graph of the function that represents the first flight. The correct answer is D. Eercises 9 Graph the function. Compare the graph to the graph of f () =. 8. g() = + 9. h() = 5 0. WHAT IF? In Eample 5, the helicopter travels the same distance but 50 miles per hour faster on the second flight. How does the graph of the function that represents the second flight compare to the graph of the function that represents the first flight? 8 Chapter 5 Linear Functions

6 5. Eercises Help with Homework. VOCABULARY What is function notation? Give an eample.. VOCABULARY Your height can be represented b a function h() where is our age. What does h() represent?. WRITING What tpe of graph is given b = m + b? How does changing the value of b affect the graph? 9+(-)= +(-)= +(-9)= 9+(-)= Evaluate the function when =, 0, and 5.. f () = + 5. g() =. h() = h() = 7 8. g() = 9. f () = ERROR ANALYSIS Describe and correct the error in evaluating the function g() = + when =. g( ) = ( ) + g = 8 + g = g = Find the value of so that the function has the given value.. h() = 7 + 0; h() =. t() = 5; t() =. n() = + 5; n() = 7. p() = ; p() = 8 5. q() = ; q() =. r() = + 7; r() = SUMMER JOB You earn $ per hour working at a grocer store during the summer. The function p() = represents the amount ou earn for working hours. a. You work 8 hours. How much do ou earn? b. How man hours do ou have to work to earn $75? 8. ORCHESTRA A group of friends are buing tickets to the orchestra. Each ticket costs $7.50 and one of the friends has a coupon for $0. The function C() = represents the total cost of buing tickets. a. How much does it cost to bu 5 tickets? b. How man tickets can ou bu with $0.00? 0? Section 5. Function Notation 9

7 Match the function with its graph. 9. f () = 0. g() = +. h() = A. B. C. Graph the linear function.. f () = +. g () = 5. h() = 5. f () = +. g () = 7 7. h() = ATMOSPHERIC TEMPERATURE Under normal conditions, the atmospheric temperature drops.5 F per 000 feet of altitude up to 0,000 feet. When the outside temperature is 80 F, the atmospheric temperature can be modeled b t() = , where is the altitude in thousands of feet. a. Graph the function and identif its domain and range. b. Find and interpret the value of so that t() = 5. Graph the function. Compare the graph to the graph of f () =. 9. g() = + 0. n() = 7. v() = 7. DECK The function C() = represents the labor cost for Jones Remodeling to build a deck, where is the number of hours. Sample labor costs from their main competitor, Premiere Remodeling, are shown in the table. a. Which cost function has the greater rate of change? What does the rate of change represent? b. The graph of which cost function has the greater -intercept? Interpret the -intercept. c. The job is estimated to take 8 hours. Which compan would ou hire? Eplain our reasoning. Hours Cost $0 $0 $90 0 Chapter 5 Linear Functions

8 Graph the functions f () and g () in the same coordinate plane. Use the graph to solve f () = g ().. f () =. f () = 5 5. f () = 7 g() = 8 g() = + 8 g() = +. CHOOSE TOOLS What tool would ou use to solve f () = g() when f () = and g () = 0.8? Eplain. Then solve f () = g(). Given f () = +, find the value of k so that the graph is f () + k f() PERIMETER The graph shows the relationship between the width and length of a rectangle in inches. A second rectangle has a perimeter that is 0 inches less than the perimeter of the first rectangle. How does the graph relating the width and length of the second rectangle compare to the graph shown?. CRITICAL THINKING The graph of = + is a translation units up of the graph of =. How can ou obtain the graph of = + from the graph of = using a horizontal translation?. Given that f () = 5 and g() =, write a function that represents f (g()) and a function that represents g( f ()). Write in slope-intercept form an equation of the line that passes through the given points. (Section.). (0, 0), (, ). (, 9), (, ) 5. (, ), (, ). MULTIPLE CHOICE You bu a pair of gardening gloves for $.5 and packets of seeds for $0.88 each. Which equation represents the total cost? (Skills Review Handbook) A = B = C = D = Section 5. Function Notation