![Page 1: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/1.jpg)
Transformations of Functions andExponential Functions
January 24, 2017
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 1 / 35
![Page 2: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/2.jpg)
Review of Section 1.2
Reminder: Week-in-Review, Help Sessions, O�ce Hours
Mathematical Models
Linear Regression
Function classes
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 2 / 35
![Page 3: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/3.jpg)
Review of Section 1.2
Alice's parents recorded her height every 3 years when she was a child.
Find the linear regression. Estimate her height at age 8.
Age (years) Height (in)
3 36
6 42
9 48
12 60
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 3 / 35
![Page 4: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/4.jpg)
Review of Section 1.2
Classify the function as polynomial, power, rational, algebraic,
trigonometric, exponential, or logarithmic.
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 4 / 35
![Page 5: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/5.jpg)
Review of Section 1.2
Classify the function as polynomial, power, rational, algebraic,
trigonometric, exponential, or logarithmic.
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 5 / 35
![Page 6: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/6.jpg)
Outline of Section 1.3 (New Functions from Old)
Vertical and horizontal shifts
Vertical and horizontal stretching
Composition
Commonly seen classes of functions
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 6 / 35
![Page 7: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/7.jpg)
Horizontal shifts
To shift the graph of y = f (x) to the left by a units, use
y = f (x − a)
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 7 / 35
![Page 8: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/8.jpg)
Horizontal shifts
To shift the graph of y = f (x) to the right by a units, use
y = f (x + a)
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 8 / 35
![Page 9: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/9.jpg)
Vertical shifts
To shift the graph of y = f (x) up by a units, use
y = f (x) + a
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 9 / 35
![Page 10: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/10.jpg)
Vertical shifts
To shift the graph of y = f (x) down by a units, use
y = f (x)− a
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 10 / 35
![Page 11: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/11.jpg)
Question
Why is the sign reversed for horizontal shifts?
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 11 / 35
![Page 12: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/12.jpg)
Horizontal stretching
To stretch the graph of y = f (x) out horizontally by a factor c , use
y = f (x/c)
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 12 / 35
![Page 13: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/13.jpg)
Vertical stretching
To stretch the graph of y = f (x) out vertically by a factor c , use
y = cf (x)
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 13 / 35
![Page 14: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/14.jpg)
Question
Why do you divide by c for horizontal stretching?
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 14 / 35
![Page 15: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/15.jpg)
Re�ection across the y -axis
To re�ect the graph of y = f (x) across the y axis, use
y = f (−x)
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 15 / 35
![Page 16: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/16.jpg)
Re�ection across the x-axis
To re�ect the graph of y = f (x) across the x axis, use
y = −f (x)
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 16 / 35
![Page 17: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/17.jpg)
Important note
Remember: To re�ect across the x-axis, multiply the y-value by −1. Tore�ect across the y-axis, multiply the x-value by −1
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 17 / 35
![Page 18: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/18.jpg)
Summary
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 18 / 35
![Page 19: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/19.jpg)
Function composition
The composition (f ◦ g)(x) = f (g(x)) means plugging g(x) into f (x)
Think of functions like machines
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 19 / 35
![Page 20: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/20.jpg)
Function composition
Example:
f (x) = sin(x), g(x) = x2 +1
x
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 20 / 35
![Page 21: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/21.jpg)
Function composition
Example:
f (x) =√x + 1, g(x) =
x + 1
x − 1, h(x) = x9
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 21 / 35
![Page 22: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/22.jpg)
Function composition
Example:
f (x) = sin2(x + 2)
Find a functions g , h, k such that f = g ◦ h ◦ k .
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 22 / 35
![Page 23: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/23.jpg)
Function composition
Example:
f (x) = −2x2 + 2x − 1, g(x) = −2x + 1
Find a function h(x) such that (g ◦ h)(x) = f (x)
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 23 / 35
![Page 24: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/24.jpg)
Function composition
Example:
f (x) = x2 − 4x + 1, g(x) = −x + 2
Find a function h(x) such that (h ◦ g)(x) = f (x)
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 24 / 35
![Page 25: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/25.jpg)
Function composition
Example:
f (x) = 3x2 + 6x − 2, g(x) = 3x + 1
Find a function h(x) such that (g ◦ h)(x) = f (x)
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 25 / 35
![Page 26: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/26.jpg)
Domain review
Find the domain of
f (x) = log(2−√x + 1)
.
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 26 / 35
![Page 27: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/27.jpg)
Outline of Section 1.5
Exponential functions
Rules for manipulating exponential functions
Applications: Exponential growth and decay
Compound interest
Rate of growth of exponential functions
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 27 / 35
![Page 28: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/28.jpg)
Exponential functions revisited
Recall
An exponential function is a function of the form
f (x) = ax ,
where a is a postive constant.
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 28 / 35
![Page 29: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/29.jpg)
Exponential rules
Exponential rules
If a and b are positive numbers then,
ax+y = axay ax−y =ax
ay(ax)y = axy (ab)x = axbx
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 29 / 35
![Page 30: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/30.jpg)
Exponential rules
30
811/2
84/3
4−1/2
53 · 5−5
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 30 / 35
![Page 31: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/31.jpg)
Exponential rules
(32)3
72 · 42
(2+ 3)2
√4+ 9
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 31 / 35
![Page 32: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/32.jpg)
Exponential rules
Simplify the expression
(4x6)−1/2 · x2
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 32 / 35
![Page 33: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/33.jpg)
Exponential decay
The half-life of Carbon-14 is 5730 years. If a sample initially contains 5mg
of Carbon-14 at time t = 0, calculate the amount of Carbon-14 in the
sample at an arbitrary time t ≥ 0.
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 33 / 35
![Page 34: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/34.jpg)
Exponential growth
Suppose that a population of bacteria doubles every two hours, and the
initial population of a sample is 300. Find the population of the sample 5
hours later.
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 34 / 35
![Page 35: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/35.jpg)
Exponential growth
A country has an annual population growth rate of 3%. Assuming
exponential growth, how many years will it take for the population to
double?
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 35 / 35
![Page 36: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/36.jpg)
Converting to base e
Convert the equation P = 121(0.89)t to the form P = P0ekt .
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 36 / 35
![Page 37: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/37.jpg)
Compound Interest
If P dollars are invested at an annual rate r compounded n times per year,
then the amount accumulated after t years is
A(t) = P(1+r
n)nt
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 37 / 35
![Page 38: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/38.jpg)
Continuously compounded Interest
If P dollars are invested at an annual rate r compounded continuously, then
the amount accumulated after t years is
A(t) = Pert
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 38 / 35
![Page 39: Transformations of Functions and Exponential Functionspgustafs/math131/Section1_3.pdf · · 2017-01-19ransfoTrmations of Functions and Exponential Functions January 24, ... Vertical](https://reader033.vdocument.in/reader033/viewer/2022042620/5b0349a47f8b9a2e228c293a/html5/thumbnails/39.jpg)
Rate of growth of exponential functions
Out of all the function classes we've discussed so far, exponential functions
of positive base grow the fastest.
Transformations of Functions and Exponential FunctionsJanuary 24, 2017 39 / 35