5.5 Transforming & Modeling Functions CC.9-12.F.BF.3 Identify the effect on the graph of replacing...

5.5 Transforming & Modeling FunctionsCC.9-12.F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. CC.9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Warm UpA rental car costs $45 per day plus $0.10 for every mile over 200.1. Find the cost of renting the car for a day and driving

250 miles. $50

quadratic: y ≈ 2.13x2 – 2x + 6.12

x 3 5 8 13

y 19 50 126 340

2. Use a calculator to perform quadratic and exponential regressions on the following data.

exponential: y ≈ 10.57(1.32)x


Transform functions.

Recognize transformations of functions.Apply functions to problem situations.

Use mathematical models to make predictions.

Objectives


Given f(x) = write the

rule g(x), a vertical stretch by a factor of 3.

– x if x < 0

Example 1: Transforming Functions

x2 if x ≥ 0

1 2

1 2


Example 1 Continued

g(x) = 3f(x) =

– x if x < 0 3 2

3( x2) if x ≥ 0 1 2

3(– x) if x < 0 1 2

x2 if x ≥ 0 3 2

Each piece of f(x) must be vertically stretched by a factor of 3. Replace every y in the function by 3y, and simplify.

=


Check Graph both functions to support your answer.

Example 1 Continued


When functions are transformed, the intercepts may or may not change. By identifying the transformations, you can determine the intercepts, which can help you graph a transformed function.


Identify the x- and y-intercepts of f(x). Without graphing g(x), identify its x- and y-intercepts.

Example 2 Transforming Functions

f(x) =–2x – 4 ; g(x) =

Find the intercepts of the original function.

f(0) = –2(0) – 4 = – 4

f(0) = –4

0 = –2x – 4

–2 = x


Example 2 Continued

The y-intercept is –4, and the x-intercept is –2. Note that g(x) is a horizontal stretch of f(x) by a factor of 2. So the y-intercept of g(x) is also –4. The x-intercept is 2(–2), or –4.

Check A graph supports your answer.


Identify the x- and y-intercepts of f(x). Without graphing g(x), identify its x- and y-intercepts.

Example 3 Transforming Functions

f(x) = x + 4 and g(x) = –f(x) 2 3

Find the intercepts of the original functions.

f(0) = 4 –6 = x

f(0) = (0) + 4 2 3

0 = x + 4 2 3


Example 3 Continued

The y-intercept is 4, and the x-intercept is –6. Note that g(x) is a reflection of f(x) across the x-axis. So the x-intercept of g(x) is also –6. The y-intercept is –1(4), or –4.

Check A graph supports your answer.


Use constant differences or ratios to determine which parent function would best model the given data set.

Example 1 Modeling Functions

Notice that the time data are evenly spaced. Check the first differences between the heights to see if the data set is linear.

Time (yr) 5 10 15 20 25

Height (in.) 58 93 128 163 198

First differences 35 35 35 35 .


Example 2 Modeling Functions

Time (yr) 4 8 12 16 20

Population 10,000 9,600 9,216 8,847 8,493

First differences –400 –384 –369 –354

Notice that the time data are evenly spaced. Check the first differences between the populations.

Population 10,000 9,600 9,216 8,847 8,493

Second differences 16 15 15


and ≈ 0.96. 8493 8847

Neither the first nor second differences are constant. Check ratios between the volumes.

9216 9600

8847 9216

= 0.96, 9600 10,000

= 0.96, ≈ 0.96,

Example 2 Cont’ Modeling Functions


Because the ratios between the values of the dependent variable are constant, an exponential function would best model the data.

Check A scatter plot reveals a shape similar to an exponential decay function.

Example 2 cont’ Modeling Functions


Example 3

Time (s) 1 2 3 4 5

Height (m) 132 165 154 99 0

First differences 33 –11 –55 –99

Notice that the time data are evenly spaced. Check the first differences between the heights.

Height (m) 132 165 154 99 0

Second differences –44 –44 –44


Example 3 Continued

Because the second differences of the dependent variables are constant when the independent variables are evenly spaced, a quadratic function will best model the data.

Check A scatter reveals a shape similar to a quadratic parent function f(x) = x2.


Independent Practice

Due Tomorrow at Beginning of Class

p. 436-437 # 8-14 all

p. 462 # 6-9 all


Lesson Quiz: Part I

Consider the functions

f(x): x-ints. = –2 and 2, y-int. = –2

1. Identify the intercepts of f(x) and g(x).

g(x): x-ints. = –2 and 2, y-int. = –8

f(x) = x2 – 2, g(x) = 4f(x), and h(x) = f( x) + 3 .

1 3

1 2

5.5 Transforming & Modeling Functions CC.9-12.F.BF.3 Identify the effect on the graph of replacing...

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