. Comparing Linear, Eponential, and Quadratic Functions How can ou compare the growth rates of linear, eponential, and quadratic functions? ACTIVITY: Comparing Speeds Work with a partner. Three cars start traveling at the same time. The distance traveled in t minutes is miles. Complete each table and sketch all three graphs in the same coordinate plane. Compare the speeds of the three cars. Which car has a constant speed? Which car is accelerating the most? Eplain our reasoning. t = t t = t t = t................ COMMON CORE Linear, Quadratic, and Eponential Functions In this lesson, ou will identif linear, quadratic, and eponential functions using graphs or tables. Learning Standards F.IF. F.IF. F.IF.a F.LE. Distance (miles)........ Time (minutes). t Chapter Graphing Quadratic Functions
ACTIVITY: Comparing Speeds Math Practice Analze Relationships What is the relationship between the speeds of the cars? Work with a partner. Analze the speeds of the three cars over the given time periods. The distance traveled in t minutes is miles. Which car eventuall overtakes the others? a. t = t t = t t = t b. t = t t = t t = t. IN YOUR OWN WORDS How can ou compare the growth rates of linear, eponential, and quadratic functions? Which tpe of growth eventuall leaves the other two in the dust? Eplain our reasoning. Use what ou learned about comparing functions to complete Eercises on page. Section. Comparing Linear, Eponential, and Quadratic Functions
. Lesson Lesson Tutorials Linear Function Eponential Function Quadratic Function Line Curve Parabola = m + b = ab = a + b + c EXAMPLE Identifing Functions Using Graphs Plot the points. Tell whether the points represent a linear, an eponential, or a quadratic function. a. (, ), (, ), (, ), b. (, ), (, ), (, ), c. (, ), (, ), (, ), (, ) (, (, ) (, ), (, ) (, ),, ) Quadratic function Linear function Eponential function Eercises Plot the points. Tell whether the points represent a linear, an eponential, or a quadratic function.. (, ), (, ), (, ), (, ), (, ). (, ), (, ), (, ), (, ) (,, ). (, ), (, ), (, ), (, ), (, ) Chapter Graphing Quadratic Functions
Differences and Ratios of Functions Eponential Function: = () Linear Function: = + + + + + + + + + + + + + The -values have a common difference of. The -values have a common ratio of. Quadratic Function: = + + + + Stud Tip EXAMPLE + + For a linear function, the first differences are constant. + + + First differences + + Second differences For quadratic functions, the second differences are constant. Identifing Functions Using Differences or Ratios Tell whether the table of values represents a linear, an eponential, or a quadratic function. Stud Tip For a quadratic function, the -values will increase, then decrease, or the -values will decrease, then increase. Eercises + + + a. + + + + b. + + + The -values have a common difference of. So, the table represents a linear function.. Tell whether the table of values represents a linear, an eponential, or a quadratic function. Section. + + + + The second differences are constant. So, the table represents a quadratic function. Comparing Linear, Eponential, and Quadratic Functions
EXAMPLE Identifing and Writing a Function Stud Tip Tell whether the table of values represents a linear, an eponential, or a quadratic function. Then write an equation for the function using the form = m + b, = ab, or = a. Step : Step : Graph the data. The Check the -values. If there is no function appears to be common difference or ratio, check eponential or quadratic. the second differences. + + + To check our function in Eample, substitute the other points from the table to see if the satisf the function. Step : Use the form = a. + = a() Use the point (, ). Substitute for and for. = a Solve for a. + + + + So, an equation for the quadratic function is =. + + + The function is quadratic. Second differences are constant. Eercises. Tell whether the table of values represents a linear, an eponential, or a quadratic function. Then write an equation for the function using the form = m + b, = ab, or = a. Linear Function Eponential Function Quadratic Function = m + b = ab, a, b, and b > = a + b + c, a m a b : Eponential deca a b : Eponential growth a m a Chapter Graphing Quadratic Functions
. Eercises Help with Homework. VOCABULARY How can ou decide whether to use a linear, a quadratic, or an eponential function to model a data set?. WHICH ONE DOESN T BELONG? Which graph does not belong with the other three? Eplain our reasoning. f() g() m() n() +(-)= +(-)= +(-)= +(-)= Find the values of when f () is greater than g().. f() =. f() =. f() = g() = g() = g() = Match the function tpe with its graph.. Linear function. Eponential function. Quadratic function A. B. C. Plot the points. Tell whether the points represent a linear, an eponential, or a quadratic function.. (, ), (, ), (, ), (, ), (, ). (, ), (, ) (,, ), (, ), (, ). (, ), (, ), (, ), (, ), (, ). (, ), (, ), (, ), (, ), (, ). SUBWAY A student takes a subwa to a public librar. The table shows the distance d (in miles) the student travels in t minutes. Tell whether Time, t Distance, d..... the data can be modeled b a linear, an eponential, or a quadratic function. Section. Comparing Linear, Eponential, and Quadratic Functions
Tell whether the table of values represents a linear, an eponential, or a quadratic function............. REASONING Can the -values of a data set have both a common difference and a common ratio? Eplain our reasoning. Tell whether the data values represent a linear, an eponential, or a quadratic function. Then write an equation for the function using the form = m + b, = ab, or = a.. (, ), (, ), (, ), (, ), (, ). (, ), (, ), (, ), (, ), (,.)...... ERROR ANALYSIS Describe and correct the error in writing an equation for the function represented b the ordered pairs. (, ), (, ), (, ), (, ), (, ) + + + + + + + + + + First differences Second differences The ordered pairs represent a quadratic function. = a = a() = a So, an equation is =. Chapter Graphing Quadratic Functions
. HIGH SCHOOL FOOTBALL The table shows the number of people attending the first five football games at a high school. a. Plot the points. b. Does a linear, an eponential, or a quadratic function represent this situation? Eplain. Game, g Number of People, p. CRITICAL THINKING Is the graph of a set of points enough to determine whether the points represent a linear, an eponential, or a quadratic function? Justif our answer.. RECORDING STUDIO The table shows the amount of mone (in dollars) that a musician pas for using a recording studio. a. Plot the points. Then determine the tpe of function that best represents this situation. b. Write a function that models the data. Number of Hours, h Amount, m (dollars) c. How much does it cost to use the studio for hours?. TOURNAMENT At the beginning of a basketball tournament, there are teams. After each round, one-half of the remaining teams are eliminated. a. Make a table showing the number of teams remaining after each round. b. Determine the tpe of function that best represents this situation. c. Write a function that models the data. d. After which round do ou know the team that won the tournament?. Repeated Reasoning Write a function that has constant second differences of. Find the -intercept(s) of the graph. (Section. and Section.).... MULTIPLE CHOICE What is the factored form of? (Section.) A ( + ) B ( + )( ) C ( ) D ( + )( ) Section. Comparing Linear, Eponential, and Quadratic Functions