Embed Size (px)
Citation preview
1
Lesson 11.1 & 11.2: Graphing Exponential & Logarithmic Functions
Learning Goals:
1) How do we graph an exponential function?
2) How do we find the domain and range of an exponential function?
3) How do we estimate the value of an exponential funciton using a graph?
4) How do we graph a logarithmic function?
5) What is the relationship between an exponential and a logarithmic
function?
6) How do we find the domain and range of a logarithmic function?
Standard Form of Exponential Functions: 𝑦 = 𝑏𝑥
1. Graph the functions:
a. 𝑦 = 2𝑥 b. 𝑦 = (1
2)
𝑥
What is the domain of the function? What is the domain of the function?
𝐷 = (−∞, +∞) 𝐷 = (−∞, +∞)
What is the range of the function? What is the range of the function?
𝑅 = (0, ∞) affected by asymptote at 𝑦 = 0 𝑅 = (0, ∞)
From the graph, estimate the value of From the graph, estimate the value of
2
21
2. (1
2)
2.5
𝑦 = 2𝑥 = 21
2 𝑦 = (1
2)
2.5= (
1
2)
𝑥
𝑥 =1
2 and 𝑦 ≈ 1.5 𝑥 = 2.5 and 𝑦 ≈ 0.2
c. What does the graph of 𝑒𝑥 look like? Graph 𝑦 = 𝑒𝑥.
State the domain and range of the
function.
𝐷 = (−∞, +∞)
𝑅 = (0, ∞)
From the graph, estimate the value of
𝑒2
≈ 7.3891
𝑒 is an irrational #!
Graph the decimals with accuracy.
3
Common Characteristics of Graphs of Exponential Functions of the form
𝑦 = 𝑏𝑥, 𝑏 > 0 and 𝑏 ≠ 1
Domain: (−∞, ∞)
Range: (0, ∞)
𝑦-intercept: (0, 1)
𝑥-axis is a horizontal asympotote
What is the equation of this horizontal asymptote? 𝑦 = 0
What is the end behavior of the graph due to the asympotote?
𝑥 → ∞, 𝑦 → ∞ and as 𝑥 → −∞, 𝑦 → 0 (going towards the asympotote)
1. Now let’s compare the graphs of the functions 𝑓2(𝑥) = 2𝑥 and 𝑓3(𝑥) = 3𝑥.
Sketch the graphs of the two exponential functions on the same set of axes;
then, answer the questions below.
4
a. Where do the two graphs intersect? (0, 1)
b. For which values of 𝑥 is 2𝑥 < 3𝑥? 0 < 𝑥 < ∞ or 𝑥 > 0
c. For which values of 𝑥 is 2𝑥 > 3𝑥? 𝑥 < 0
d. What happens to the values of the functions 𝑓2 and 𝑓3 as 𝑥 → ∞? 𝑦 → ∞
e. What happens to the values of the functions 𝑓2 and 𝑓3 as 𝑥 → −∞? 𝑦 → 0
f. Does either graph ever intersect the 𝑥- axis? Explain how you know. No
because of the asymptote at 𝑦 = 0.
When real-life quantity increases by a fixed percent each year (or other time
period), the amount 𝑦 of the quantity after 𝑡 years can be modeled by the
equation
𝑦 = 𝑎(1 + 𝑟)𝑡 The quantity 1 + 𝑟 is called the growth factor.
2. In January, 1993, there were about 1,313,000 internet hosts. During the next
five years, the number of hosts increased by about 100%
per year.
a. Write a model giving the number ℎ (in millions) of
hosts 𝑡 years after 1993. About how many hosts were there
in 1996?
b. Graph the model.
c. Use the graph to estimate the year when there
were 30 million hosts.
5
3: On the grid below graph the exponential function 𝑦 = 2𝑥 and its inverse.
𝑥 𝑦 = 2𝑥 𝑥 𝑦
−2 . 25 . 25 −2
−1 . 5 . 5 −1
0 1 1 0
1 2 2 1
2 4 4 2
3 8 8 3
a. What is the equation of the inverse of 𝑦 = 2𝑥? How would we solve it for 𝑦?
𝑥 = 2𝑦 Use logs: 𝑦 = log2𝑥
b. In what quadrants does the graph of the exponential function 𝑦 = 2𝑥 lie?
I & II
c. In what quadrants does the graph of its inverse lie?
I & IV
d. Is the graph of the inverse of 𝑦 = 2𝑥 a function? If so, what is its domain and
range? Yes, 𝐷 = (0, ∞) and 𝑅 = (−∞, ∞)… asymptote at 𝑥 = 0
6
4: On the grid below graph the exponential function 𝑦 = 𝑒𝑥 and its inverse.
a. What is the equation of the inverse of 𝑦 = 𝑒𝑥? How would we solve it for 𝑦?
𝑥 = 𝑒𝑦 Use logs: 𝑦 = log𝑒𝑥
b. In what quadrants does the graph of the exponential function 𝑦 = 𝑒𝑥 lie? I & II
c. In what quadrants does the graph of its inverse lie? I & IV
d. Is the graph of the inverse of 𝑦 = 𝑒𝑥 a function? If so, what is its domain and
range? Yes, 𝐷 = (0, ∞) and 𝑅 = (−∞, +∞)
A function 𝑓(𝑥) is an exponential function: 𝑓(𝑥) = 𝑏𝑥 or 𝑦 = 𝑏𝑥
Domain = (−∞, ∞) Range = (0, ∞) Asymptote at 𝑦 = 0
Its inverse 𝑓−1(𝑥) is a logarithmic function: 𝑓−1(𝑥) = log𝑏𝑥 or 𝑦 = log𝑏𝑥
Domain = (0, ∞) Range = (−∞, ∞) Asymptote at 𝑥 = 0
7
5: Graph the points in the table for the functions 𝑓(𝑥) = log(𝑥) and
𝑔(𝑥) = log2(𝑥), for the given values. Then, sketch smooth curves through those
points and answer the questions that follow.
a. What do the graphs indicate about the domain of your functions?
b. Describe the 𝑥-intercepts of the graphs.
c. Describe the 𝑦-intercepts of the graphs.
d. Find the coordinates of the point on the graph with 𝑦-value 1.
e. Describe the behavior of the function as 𝑥 → 0.
f. Describe the end behavior of the function as 𝑥 → ∞.
g. Describe the range of your function.
8
h. Does this function have any relative maxima or minima? Explain how you
know.
i. For which values of 𝑥 is log2(𝑥) < log (𝑥)
Key Features of the graph of 𝒚 = 𝐥𝐨𝐠𝒃(𝒙) 𝐟𝐨𝐫 𝒃 > 𝟏.
The domain is the positive real numbers, and the range is all real numbers.
The graphs all cross the 𝑥-axis at (1, 0).
A point on the graph is always (𝑏, 1).
None of the graphs intersect the 𝑦-axis. Asymptote at 𝑥 = 0
The have the same end behavior as 𝑥 → ∞, 𝑓(𝑥) → ∞, and they have the
same behavior as 𝑥 → 0, 𝑓(𝑥) → −∞.
The functions all increase quickly for 0 < 𝑥 < 1, then increase more and
more slowly.
As the value of 𝑏 increases, the graph will flatten as
𝑥 → ∞.
There are no relative maxima or minima.
Inverse of the exponential function 𝑦 = 𝑏𝑥.
9
6: Graph the points in the table for your assigned function
𝑟(𝑥) = log 1
10
(𝑥) and 𝑠(𝑥) = log1
2
(𝑥). Then sketch smooth curves through those
points, and answer the questions that follow.
a. What is the relationship between your graphs in Example 3 and your graphs
from this example.
b. Why does this happen? Use the change of base formula to justify what you
have observed in part (a).
c. For which values of 𝑥 is log1
2
(𝑥) < log 1
10
(𝑥)?
10
Conclusion: From what we have seen of these sets of graphs of functions, can
we state the relationship between the graphs of 𝑦 = log𝑏(𝑥) and 𝑦 = log1
𝑏
(𝑥),
for 𝑏 ≠ 1?
𝑖𝑓 𝑏 ≠ 1, 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑔𝑟𝑎𝑝ℎ𝑠 𝑜𝑓 𝑦 = log𝑏(𝑥) 𝑎𝑛𝑑 𝑦 = log1𝑏
(𝑥) 𝑎𝑟𝑒 𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛𝑠
𝑜𝑓 𝑒𝑎𝑐ℎ 𝑜𝑡ℎ𝑒𝑟 𝑎𝑐𝑟𝑜𝑠𝑠 𝑡ℎ𝑒 𝑥 − 𝑎𝑥𝑖𝑠.
11
Key Features of the graph of 𝒚 = 𝐥𝐨𝐠𝒃(𝒙) 𝐟𝐨𝐫 𝟎 < 𝒃 < 𝟏.
The graph crosses the 𝑥-axis at (1, 0).
The graph does not intersect the 𝑦-axis.
The graph passes through the point (1
𝑏, −1).
As 𝑥 → 0, the function values increase quickly; that is, 𝑓(𝑥) → ∞.
As 𝑥 → ∞, the function values continue to decrease; that is, 𝑓(𝑥) → −∞
There are no relative maxima or minima.
Inverse of the exponential function 𝑦 = 𝑏𝑥.
12
Homework 11.1 & 11.2: Graphing Exponential & Logarithmic Functions
1. Sketch the graphs of the functions 𝑓1(𝑥) = (1
2)
𝑥 and 𝑓2(𝑥) = (
3
4)
𝑥 below and
answer the following questions.
a. Where do the two exponential graphs intersect?
b. For which values of 𝑥 is (1
2)
𝑥< (
3
4)
𝑥?
c. For which values of 𝑥is (1
2)
𝑥> (
3
4)
𝑥?
d. What happens to the values of the functions 𝑓1 and 𝑓2 as 𝑥 → ∞?
e. What are the domains of the two functions 𝑓1 and 𝑓2?
f. What are the ranges of the two functions 𝑓1 and 𝑓2?
13
2. Graph the function 𝑓(𝑥) = log3(𝑥) and identify its key features.
Domain:
Range:
End Behavior: As 𝑥 → 0, 𝑓(𝑥) → As 𝑥 → ∞, 𝑓(𝑥) →
𝑥-intercept
𝑦-intercept
Graph passes through point:
14
3. Consider the logarithmic functions 𝑓(𝑥) = log𝑏(𝑥) and 𝑔(𝑥) = log5(𝑥), where
𝑏 is a positive real number, and 𝑏 ≠ 1. The graph of 𝑓 is given.
a. Is 𝑏 > 5 or is 𝑏 < 5? Explain how you know.
b. Compare the domain and range of functions 𝑓 and 𝑔.
c. Compare the 𝑥-intercepts and 𝑦-intercepts of 𝑓 and 𝑔.
d. Compare the end behavior of 𝑓 and 𝑔.
15
4. On the same set of axes, sketch the functions
𝑓(𝑥) = log2(𝑥) and 𝑔(𝑥) = log2(𝑥3).
a. For which values of 𝑥 is log2(𝑥) < log2(𝑥3)?
b. At what point do the two graphs intersect?
c. Describe the end behavior of the function 𝑓(𝑥).
d. What is the domain and range for the function 𝑔(𝑥)?
16
Lesson 11.3: Transformations and Characteristics of Logarithmic and
Exponential Functions
Learning Goals:
1) How do we describe the transformations of a logarithmic or exponential
function?
2) What are the characteristics of a logarithmic or exponential function?
Warm-Up:
The figure below shows graphs of the functions 𝑓(𝑥) = log3(𝑥), 𝑔(𝑥) = log5(𝑥),
and ℎ(𝑥) = log11(𝑥).
a. Identify which graph corresponds to which function. Explain how you know.
b. Sketch the graph of 𝑘(𝑥) = log7(𝑥) on the same axes.
c. What happens to the values of the functions 𝑓(𝑥), 𝑔(𝑥), and ℎ(𝑥) as 𝑥 → ∞?
d. Do any of the graphs ever intersect the 𝑦-axis? Explain how you know.
17
Summary of Transformations
Function Transformation
𝑓(𝑥) + 𝑎 Vertical translation, up 𝑎
𝑓(𝑥) − 𝑎 Vertical translation, down 𝑎
𝑓(𝑥 + 𝑎) Horizontal translation, left 𝑎
𝑓(𝑥 − 𝑎) Horizontal translation, right 𝑎
−𝑓(𝑥) Reflection, negate 𝑦, 𝑟𝑥−𝑎𝑥𝑖𝑠
𝑓(−𝑥) Reflection, negate 𝑥, 𝑟𝑦−𝑎𝑥𝑖𝑠
𝑘 ∙ 𝑓(𝑥) or 1
𝑘∙ 𝑓(𝑥) Vertical, Dilation, multiply 𝑦 by either 𝑘 𝑜𝑟
1
𝑘
𝑓(𝑘 ∙ 𝑥) or 𝑓 (1
𝑘∙ 𝑥) Horizontal, Dilation, multiply 𝑥 by either 𝑘 𝑜𝑟
1
𝑘
18
Performing multiple transformations:
If a function has multiple transformations, they are applied in the following order:
H orizontal translation
D ilation
R eflection
V ertical translation
Sketching Log Equations
Example 1: Sketch the two logarithmic functions 𝑔2(𝑥) = log2(𝑥) and
𝑔3(𝑥) = log3(𝑥) on the axes below; then, answer the following questions.
a. For which value of 𝑥
is log2(𝑥) = log3(𝑥)?
𝑥 = 1
b. For which values of 𝑥
is log2(𝑥) < log3(𝑥)?
0 < 𝑥 < 1
c. For which values of 𝑥 is
log2(𝑥) > log3(𝑥)? 𝑥 > 1
d. What happens to the
values of the functions
𝑔2 and 𝑔3 as 𝑥 → ∞? 𝑦 →
∞
e. What happens to the values of the functions 𝑔2 and 𝑔3 as 𝑥 → 0?
𝑦 → −∞
f. Does either graph ever intersect the 𝑦-axis? Explain how you know. No
because the asymptote at 𝑥 = 0
19
Example 2: Which statement regarding the graphs of the functions below is
untrue?
𝑔(𝑥) = (𝑥 − .5)(𝑥 + 4)(𝑥 − 2)
ℎ(𝑥) = log2(𝑥) Asymptote is the 𝑦-axis!
𝑗(𝑥) = −|4𝑥 − 2| + 3
(1) 𝑓(𝑥) and 𝑗(𝑥) have a maximum 𝑦-value of 3.
(2) 𝑓(𝑥), ℎ(𝑥), and 𝑗(𝑥) have one 𝑦-intercept.
(3) 𝑔(𝑥) and 𝑗(𝑥) have the same end behavior as 𝑥 → −∞.
(4) 𝑔(𝑥), ℎ(𝑥), and 𝑗(𝑥) have rational zeros.
Example 3: Sketch the graph of 𝑓(𝑥) = log2(𝑥) by identifying and plotting at
least five key points. Use the table below to get started.
Given the above function 𝑓(𝑥) = log2(𝑥), describe each of the following:
a. The sequence of transformations that takes the graph of 𝑓 to the graph of
𝑔(𝑥) = log2(8𝑥). H orizontal D ilation (multiply 𝒙 by 𝟏
𝟖)
20
b. The sequence of transformations that takes the graph of 𝑓 to the graph of
ℎ(𝑥) = 3 + log2𝑥. V ertical translation (moves up 𝟑)
Example 4: Describe the graph of 𝑝(𝑥) = log2 (𝑥
4) as a vertical translation of the
graph of 𝑓(𝑥) = log2(𝑥). Justify your response.
log2𝑥 − log24 = log2𝑥 − 2 V ertical translation (moves down 𝟐)
Example 5: Describe the graph of 𝑔 as a transformation of the graph of 𝑓.
𝑓(𝑥) = log3(𝑥) and 𝑔(𝑥) = 2log3(𝑥 − 1). H orizontal translation (right 𝟏) and
Vertical D ilation (multiply 𝒚 by 𝟐)
Example 6: 𝑔(𝑥) is the image of 𝑓(𝑥) after a shift four units up followed by a
vertical stretch of 3. If 𝑓(𝑥) = log2(𝑥), which of the following gives the equation
of 𝑔(𝑥)?
(1) 𝑔(𝑥) = 3log2(𝑥 + 4)
(2) 𝑔(𝑥) = 3log2𝑥 + 12
(3) 𝑔(𝑥) = 3log2𝑥 + 4
(4) 𝑔(𝑥) = 3log21
3𝑥 + 4
Example 7: Describe the given function as a transformation of the graph of a
function in the form 𝑓(𝑥) = 𝑏𝑥. So basically, 𝑓(𝑥) = 3𝑥
𝑔(𝑥) = 3𝑥−2
The exponent of 𝑥 − 2 means it is a horizontal translation that moves to the right
2 units.
Example 8: Using the function 𝑓(𝑥) = 2𝑥, create a new function 𝑔 whose graph
is a series of transformations of the graph of 𝑓 with the following characteristics:
21
The function 𝑔 is increasing for all real numbers.
The equation for the horizontal asymptote is 𝑦 = 5.
𝑔(𝑥) = 2𝑥 + 5
Example 9: Graph the following function and answer the questions below:
𝑓(𝑥) = 3𝑥 − 2
a. What is the domain of 𝑓(𝑥)? 𝐷 = (±∞)
b. What is the range of 𝑓(𝑥)? 𝑅 = (−2, ∞)
c. What is the equation of the asymptote? 𝑦 = −2
22
d. Describe the transformation that occur from 𝑔(𝑥) = 3𝑥 to 𝑓(𝑥). Vertical
translation down 2
Example 10: Graph the following function and answer the questions below:
𝑓(𝑥) = log3(𝑥 + 2)
a. What is the domain of 𝑓(𝑥)? 𝐷 = (−2, ∞)
b. What is the range of 𝑓(𝑥)? 𝑅 = (±∞)
c. What is the equation of the asymptote? 𝑥 = −2
d. Describe the transformation that occur from 𝑔(𝑥) = log3(𝑥) to 𝑓(𝑥). horizontal
translation left 2.
23
Homework 11.3: Transformations and Characteristics of Logarithmic and
Exponential Functions
1. Describe the graph of 𝑔 as a transformation of the graph of 𝑓.
a. 𝑓(𝑥) = log2𝑥 & 𝑔(𝑥) = log2(𝑥 − 3)
b. 𝑓(𝑥) = log2𝑥 & 𝑔(𝑥) = log2 (8
𝑥)
c. 𝑓(𝑥) = log5𝑥 & 𝑔(𝑥) = −log5(5(𝑥 + 2))
d. Describe the given function as a transformation of the graph of a function in
the form 𝑓(𝑥) = 4𝑥 for 𝑔(𝑥) = 4𝑥−2 + 5.
2. Relative to the graph of 𝑔(𝑥) = log2(𝑥), what is the shift of the graph of
𝑓(𝑥) = 3log2(𝑥 + 5)?
(1) 5 left (2) 5 up (3) 5 right (4) 5 down
24
Lesson 11.4: Solving Log Equations Graphically
Learning Goals:
1) How do we solve log equations graphically?
2) How do we graph a system of log equations?
Do Now: Answer the following questions in order to prepare for today’s lesson.
a) Sketch the graph of this pair of functions on the same coordinate axes.
𝑓(𝑥) = (1
2)
𝑥
𝑔(𝑥) = − (1
2)
𝑥
+ 4
b) Describe the graph of 𝑔(𝑥) as a series of transformations on the graph of
𝑓(𝑥). 𝑟𝑥−axis, then up 4 with a new asymptote at 𝑦 = 4
c) Describe the end behaviors of 𝑓(𝑥) and of 𝑔(𝑥).
𝑥 → −∞, 𝑦 → ∞ and 𝑥 → ∞, 𝑦 → 0 𝑥 → −∞, 𝑦 → −∞ and 𝑥 → ∞, 𝑦 → 4
d) Find the solution to the equation 𝑓(𝑥) = 𝑔(𝑥). (−1, 2) is where they intersect!
25
Solving a System of Log Equations Graphically
Review from the last unit:
Solve the following equation algebraically for all values of 𝑥: log(𝑥) = log (2
𝑥)
“drop a log”: 𝑥
1=
2
𝑥
𝑥2 = 2
𝑥 = ±√2 ≈ 1.4
Model Problem: Solve the following equations graphically for all values of
𝑥: log(𝑥) = log (2
𝑥)
26
1. Solve the following equation graphically for all values of 𝑥:
log(𝑥) = log (𝑥 − 2)
2. When 𝑓(𝑥) =2
𝑥+2 and ℎ(𝑥) = log(𝑥 + 1) + 3 are graphed on the same set of
axes, which coordinates best approximate their point of intersection?
(1) (−0.9, 1.8) (2) (−0.9, 1.9) (3) (1.4, 3.3) (4) (1.4, 3,4)
We can extend this idea to any unfamiliar equation. You can use this graphing method if:
The question does not specify to solve algebraically.
The question tells you to round your answers.
It is a multiple choice question where no work is required.
27
3. Given the two functions below:
𝑓(𝑥) =18
𝑥3 +78
𝑥2 −1511
𝑥 + 1
𝑔(𝑥) = −|0.5𝑥| + 4
State the solutions to the equation 𝑓(𝑥) = 𝑔(𝑥), rounded to the nearest
hundredth.
4. Solve for 𝑥: √30 − 2𝑥 = 𝑥 − 3 Justify your solution(s).
30 − 2𝑥 = (𝑥 − 3)2
30 − 2𝑥 = 𝑥2 − 6𝑥 + 9
0 = 𝑥2 − 4𝑥 − 21
0 = (𝑥 − 7)(𝑥 + 3)
𝑥 = −7 & 𝑥 = 3
No solution!
28
Homework 11.4: Solving Log Equations Graphically
1. Solve the following equation graphically 2. Solve the following equation
for all values of 𝑥: log(𝑥) = log(2𝑥) graphically for all values of 𝑥
to the nearest hundredth:
(1
4)
𝑥−2= log4(𝑥)
3. Given the graph of the following
function on the coordinate axes,
describe the intercepts and end
behavior of the graph 𝑓(𝑥) = (1
2)
𝑥+ 3
29
4. Use the relationship between the graphs of exponential and logarithmic
functions to sketch the graphs of the functions 𝑔1(𝑥) = log1
2
(𝑥) and 𝑔2 = log3
4
(𝑥)
on the same sheet of graph paper. Then, answer the following questions.
a. Where do the two logarithmic graphs intersect?
b. For which values of 𝑥 is
log1
2
(𝑥) < log3
4
(𝑥)?
c. For which values of 𝑥 is
log1
2
(𝑥) > log3
4
(𝑥)?
d. What happens to the
values of the functions
𝑔1 and 𝑔2 as 𝑥 → ∞?
e. What are the domains of
the two functions 𝑔1 and 𝑔2?
5. If 𝑓(𝑥) = 3(5)𝑥, determine the coordinates of the 𝑦-intercept of the function
defined by 𝑓(𝑥) + 4. (1) (0, 9) (2) (7, 0) (3) (0, 7) (4) (0, 5)
6. Given the two functions below:
𝑓(𝑥) =1
2𝑥2 −
2
5𝑥 + 3
𝑔(𝑥) = −4 √𝑥 − 23
State the solutions to the equation 𝑓(𝑥) = 𝑔(𝑥), rounded to the nearest
hundredth.
7. If 𝑓(𝑥) = 𝑥2 − 2𝑥 − 8 and 𝑔(𝑥) =1
4𝑥 − 1, for which value of 𝑥 is 𝑓(𝑥) = 𝑔(𝑥)?
(1) −1.75 and − 1.438 (2) −1.75 and 4 (3) −1.438 and 0 (4) 4 and 0
30
Lesson 11.5: Inverses of Logarithmic and Exponential Functions
Learning Goals:
1) How can we find the domain of a logarithmic function algebraically?
2) How do we find the inverse function of a logarithmic equation? How do we
find the inverse function of an exponential equation?
Do Now: How do you find the inverse of a function? Switch the 𝑥 and 𝑦-values!
Find the inverse of each of the following:
1. 𝑓(𝑥) = 3 log(𝑥2) for 𝑥 > 0
𝑦 = 3 log(𝑥2)
𝑥 = 3 log(𝑦2) inverse
𝑥 = 3(2)log𝑦
𝑥
6=
6log𝑦
6
𝑥
6= log 𝑦 “loop” and your base = 10
𝑦 = 10𝑥
6
2. 𝑓(𝑥) = 2𝑥−3
𝑦 = 2𝑥−3 “anti-loop”
𝑥 = 2𝑦−3 inverse log2𝑥 = 𝑦 − 3
log𝑥 = log2𝑦−3 add logs to both sides 𝑦 = log2𝑥 + 3
log𝑥
log 2=
(𝑦−3)log2
log2
𝑦 − 3 =log𝑥
log 2
𝑦 =log𝑥
log 2+ 3
31
3. Consider the function 𝑓(𝑥) = 2𝑥 + 1, whose graph is shown below.
a. What are the domain and range of 𝑓?
𝐷 = (−∞, ∞) and 𝑅 = (1, ∞)
b. Sketch the graph of the inverse function 𝑔 on
the graph. What type of function do you expect 𝑔
to be? Find points on 𝑓(𝑥) and switch 𝑥 and 𝑦-
values. Log Function!
c. What are the domain and range of 𝑔? How does that relate to your answer in
part (a)? 𝐷 = (1, ∞) and 𝑅 = (−∞, ∞), really just swap domain and range of 𝑓
d. Find the formula for 𝑔. 𝑔(𝑥) is the inverse of 𝑓(𝑥).
𝑦 = 2𝑥 + 1
𝑥 = 2𝑦 + 1 “anti-loop”
𝑥 − 1 = 2𝑦 log2(𝑥 − 1) = 𝑦
log(𝑥−1)
log2=
𝑦log2
log2 add logs to both sides.
𝑦 =log(𝑥−1)
log2
32
4. The graph of a function 𝑓 is shown below. Sketch the graph of its inverse
function 𝑔 on the same axes.
a. Explain how you made your sketch.
Switch 𝑥 and 𝑦-value points on 𝑓(𝑥).
b. The function 𝑓 graphed above is the function 𝑓(𝑥) = log2(𝑥) + 2 for 𝑥 > 0.
Find a formula for the inverse of this function.
𝑦 = log2(𝑥) + 2
𝑥 = log2(𝑦) + 2
𝑥 − 2 = log2(𝑦) “loop”
2𝑥−2 = 𝑦
5. Find the inverse of the following function. Then find the domain and range of
both the original function and its inverse.
𝑓(𝑥) = 𝑒𝑥−5 𝐷 = (−∞, ∞) and 𝑅 = (0, ∞)
𝑦 = 𝑒𝑥−5
𝑥 = 𝑒𝑦−5 add logs
ln𝑥 = (𝑦 − 5)ln𝑒
ln𝑥 + 5 = 𝑦 𝐷 = (0, ∞) and 𝑅 = (−∞, ∞)
33
Practice: Find the inverse of each of the following functions:
6*. 𝑓(𝑥) = 7 log(1 + 9𝑥) 7*. 𝑓(𝑥) = ln(𝑥) − ln(𝑥 + 1)
𝑦 = 7log (1 + 9𝑥) 𝑦 = ln(𝑥) − ln(𝑥 + 1)
𝑥 = 7log (1 + 9y) 𝑥 = ln(𝑦) − ln(𝑦 + 1)
𝑥
7= log (1 + 9𝑦) 𝑥 = ln (
𝑦
𝑦+1)
10𝑥
7 = 1 + 9𝑦 𝑒𝑥 =𝑦
𝑦+1
10𝑥
7 − 1 = 9𝑦 𝑒𝑥(𝑦 + 1) = 𝑦
10𝑥7−1
9= 𝑦 𝑒𝑥𝑦 + 𝑒𝑥 = 𝑦
𝑒𝑥 = 𝑦 − 𝑒𝑥𝑦
𝑒𝑥 = 𝑦(1 − 𝑒𝑥)
𝑒𝑥
1−𝑒𝑥 = 𝑦 or 𝑒𝑥 − 1 = 𝑦
If two functions whose domain and range are a subset of the real numbers
are inverses, then their graphs are reflections of each other across the
diagonal line given by 𝑦 = 𝑥 in the Cartesian plane.
If 𝑓 and 𝑔 are inverses of each other, then
o The domain of 𝑓 is the same set as the range of 𝑔.
o The range of 𝑓 is the same set as the domain of 𝑔.
In general, to find the formula for an inverse function 𝑔 of a given function 𝑓:
o Write 𝑦 = 𝑓(𝑥) using the formula for 𝑓.
o Interchange the symbols 𝑥 and 𝑦 to get 𝑥 = 𝑓(𝑦).
o Solve the equation for 𝑦 to write 𝑦 as an expression in 𝑥.
o Then, the formula for 𝑔 is the expression in 𝑥 found in step (iii).
The functions 𝑓(𝑥) = log𝑏(𝑥) and 𝑔(𝑥) = 𝑏𝑥 are inverses of each other.
34
8*. 𝑓(𝑥) =2𝑥
2𝑥+1 9. 𝑓(𝑥) = 2𝑥 + 1
𝑦 =2𝑥
2𝑥+1 𝑦 = 2𝑥 + 1
𝑥 =2𝑦
2𝑦+1 𝑥 = 2𝑦 + 1
𝑥(2𝑦 + 1) = 2𝑦 𝑥 − 1 = 2𝑦
𝑥2𝑦 + 𝑥 = 2𝑦 log(𝑥 − 1) = 𝑦log2
𝑥 = 2𝑦 − 2𝑦𝑥 log (𝑥−1)
log2= 𝑦
𝑥 = 2𝑦(1 − 𝑥)
𝑥
𝑥−1= 2𝑦
log (𝑥
𝑥−1) = 𝑦log2
log(𝑥
𝑥−1)
log2= 𝑦
35
Homework 11.5: Inverses of Logarithmic and Exponential Functions
1. Find the inverse of each of the following functions. In each case, indicate the
domain and range of both the original function and its inverse.
a. 𝑓(𝑥) = 6log (1 + 2𝑥) b. 𝑓(𝑥) = ln(𝑥 + 5) − ln (𝑥 + 1)
c. 𝑓(𝑥) = 8 + ln (5 + √𝑥3
) d. 𝑓(𝑥) = 25−8𝑥
36
Lesson 11.6: Identifying Exponential Growth and Decay
Learning Goal: How do we determine if a situation is exponential growth or
decay and what the growth/decay rate is?
Do Now: Answer the following questions in order to prepare for today’s lesson.
1. A population of wolves in a county is represented by the equation
𝑃(𝑡) = 80(1.12)𝑡, where 𝑡 is the number of years since 1998. How many wolves
will there be in the year 2010? 𝑡 = 2010 − 1998 = 12
𝑃(12) = 80(1.12)12 = 311, so it is a growth!
2. A population of wolves in a county is represented by the equation
𝑃(𝑡) = 80(0.90)𝑡, where 𝑡 is the number of years since 1998. How many wolves
will there be in the year 2010? 𝑡 = 2010 − 1998 = 12
𝑃(12) = 80(0.90)12 = 22, so it is a decay!
Which question above represents growth? Which question represents
decay? Why? See above
What part of the formula causes the growth or decay?
𝑃(𝑡) = 80(1.12)𝑡 𝑃(𝑡) = 80(0.90)𝑡
In the formula, what does the "80" represent? Initial number of wolves!
37
𝑦 = final amount 𝑡 = time 𝑎 = initial amount
(1 + 𝑟) or (1 − 𝑟) = rate of change
Directions: Determine whether each function represents exponential growth or
decay. Then identify the rate of change.
a. 𝑦 = 5(1.07)𝑡 growth! 1.07 > 1 b. 𝑓(𝑡) = 2(0.98)𝑡 decay! .98 < 1
1 + 𝑟 = 1.07 1 − 𝑟 = .98
𝑟 = .07 = 7% −𝑟 = −.02
𝑟 = .02 = 2%
38
c. 𝑦 = (1
2)
𝑡 decay!
1
2< 1 d. 𝑝(𝑡) = 100(2)𝑡 growth! 2 > 1
1 − 𝑟 =1
2 1 + 𝑟 = 2
−𝑟 = −1
2 𝑟 = 1 = 100%
𝑟 =1
2= 50%
Rewriting Exponential Functions to Determine Growth or Decay
𝑦 = 𝑎(1 + 𝑟)2 𝑡 = 𝑎((1 + 𝑟)2)𝑡
Directions: Rewrite each function to determine whether each function represents
exponential growth or exponential decay. Then identify the rate of change.
a. 𝑓(𝑡) = 3(1.02)10𝑡 b. 𝑦 = 100(0.92)𝑡
4
𝑓(𝑡) = 3(1.0210)𝑡 𝑦 = 100 ((0.92)1
4)𝑡
𝑓(𝑡) = 3(1.21899442)𝑡 𝑦 = 100(.9793703613)𝑡
𝑓(𝑡) = 3(1.22)𝑡 growth! 1.22 > 1 𝑦 = 100(.98)𝑡 decay! .98 < 1
1 + 𝑟 = 1.22 1 − 𝑟 = .98
𝑟 = .22 = 22% −𝑟 = −.02
𝑟 = .02 = 2%
39
c. 𝑝(𝑡) = 80 ∙ (49
64)
1
2 𝑡 d. 𝑦 = (1.02)−2 𝑡
𝑝(𝑡) = 80 ((49
64)
1
2)
𝑡
𝑦 = (1.02−2)𝑡
𝑝(𝑡) = 80 (7
8)
𝑡 𝑦 =. 9611687812𝑡
𝑝(𝑡) = 80(. 88)𝑡 decay! .88 < 1 𝑦 =. 96𝑡 decay! .96 < 1
1 − 𝑟 = .88 1 − 𝑟 = .96
−𝑟 = −.12 −𝑟 = −.04
𝑟 = .12 = 12% 𝑟 = .04 = 4%
Rewriting Exponential Functions in Terms of
Monthly Rate of Growth or Decay
𝑦 = 𝑎(1 + 𝑟)𝑡 = 𝑎(1 + 𝑟)1212 𝑡 = 𝑎 ((1 + 𝑟)
112)
12 𝑡
Directions: Rewrite each function in terms of monthly rate of growth or decay.
Round decimals to the nearest thousandth.
a. 𝑃(𝑡) = 500(1.12)𝑡 b. 𝑦 = (1
2)
𝑡
𝑃(𝑡) = 500(1.12)12
12 𝑡
𝑦 = (1
2)
12
12 𝑡
𝑃(𝑡) = 500 (1.121
12)12𝑡
𝑦 = (1
2
1
12)
12 𝑡
𝑃(𝑡) = 500(1.009)12 𝑡 (monthly) 𝑦 = (.944)12 𝑡 (monthly)
40
Determining Growth or Decay of Difficult Exponential Functions
Based on today’s lesson, how do you think we can answer the following
questions?
1) identify the initial amount (𝑡 = 0); 2) pick another value for time (𝑡 = 1)
1. The function 𝑀(𝑡) represents the mass of radium over time, 𝑡, in years.
𝑀(𝑡) = 100𝑒(ln
12)𝑡
1590
Determine if the function 𝑀(𝑡) represents growth or decay. Explain your
reasoning.
𝑡 = 0, 𝑀 = 100 are the initial amounts
use 𝑡 = 1
𝑀(𝑡) = 100𝑒(ln
12
)(1)
1590
𝑀(𝑡) = 99.95641534 Decay because it is less than the initial amount
2. The function 𝑓(𝑡) represents a colony of bacteria over time, 𝑡, in years.
𝑓(𝑡) = 10𝑒ln200
4 𝑡
Determine if the function 𝑓(𝑡) represents growth or decay. Explain your
reasoning.
𝑡 = 0, 10 are the initial amounts
use 𝑡 = 1
𝑓(𝑡) = 10𝑒(ln200)
4 (1)
𝑀(𝑡) = 37.60603093 Growth because it is more than the initial amount
41
Homework 11.6: Identifying Exponential Growth and Decay
1. Use the properties of exponents to identify the percent rate of change (to the
nearest hundredth) of the functions below, and classify them as representing
exponential growth or decay.
a. 𝑓(𝑡) = 100(1.02)𝑡 b. 𝑓(𝑡) = 2(1.01)12 𝑡
c. 𝑓(𝑡) = (0.97)𝑡 d. 𝑓(𝑡) = 50 ∙ (2
3)
1
2 𝑡
2. Milton has his money invested in a stock portfolio. The value, 𝑣(𝑥), of his
portfolio can be modeled with the function 𝑣(𝑥) = 30,000(0.78)𝑥, where 𝑥 is the
number of years since he made his investment. Which statement describes the
rate of change of the value of his portfolio?
(1) It decreases 78% per year.
(2) It decreases 22% per year.
(3) It increases 78% per year.
(4) It increases 22% per year.
3. The growth of a certain organism can be modeled by 𝐶(𝑡) = 10(1.029)24 𝑡,
where 𝐶(𝑡) is the total number of cells after 𝑡 hours. Which function is
approximately equivalent to 𝐶(𝑡)?
(1) 𝐶(𝑡) = 240(. 083)24 𝑡
(2) 𝐶(𝑡) = 10(. 083)𝑡
(3) 𝐶(𝑡) = 10(1.986)𝑡
(4) 𝐶(𝑡) = 240(1.986)𝑡
24
42
4. A study of the annual population of the red-winged blackbird in Ft. Mill, South
Carolina, shows the population, 𝐵(𝑡), can be represented by the function
𝐵(𝑡) = 750(1.16)𝑡, where the 𝑡 represents the number of years since the study
began. In terms of the monthly rate of growth, the population of red-winged
blackbirds can be best approximated by the function.
(1) 𝐵(𝑡) = 750(1.012)𝑡
(2) 𝐵(𝑡) = 750(1.012)12 𝑡
(3) 𝐵(𝑡) = 750(1.16)12 𝑡
(4) 𝐵(𝑡) = 750(1.16)𝑡
12
5. The function 𝑓(𝑡) represents a heard of llamas over time, 𝑡, years.
𝑓(𝑡) = 1000𝑒ln2
4 𝑡
Determine if the function 𝑓(𝑡) represents growth or decay. Explain your reasoning.
43
Lesson 11.7 & 11.8: Writing Exponential Growth and Decay Functions &
Exponential and Logarithmic Regression
Learning Goals:
1) How can we write an exponential growth/decay equation and define its
variables?
2) How do we write an equation that models an exponential or logarithmic
function.
Do Now: Answer the following questions in order to prepare for today’s lesson.
1. Given the exponential functions below, identify the initial amount, the rate of
change, and if it is growth or decay.
a. 𝑓(𝑥) = 125(1.035)𝑥 b. 𝑓(𝑥) = 5000(0.8)2𝑥
Initial Amount = 125 Initial Amount = 5000
Rate of Change = 3.5% Rate of Change = 36%
1 + 𝑟 = 1.035 𝑓(𝑥) = 5000(. 82)𝑥 = 5000(.64)𝑥
𝑟 = .035 = 3.5% (Growth) 1 − 𝑟 = .64
−𝑟 = −.36
𝑟 = .36 = 36% (Decay)
Writing Exponential Growth and Decay Functions
44
𝑦 = final amount
𝑎 = initial amount (1 + 𝑟) or (1 − 𝑟) = rate of change
𝑡 = time
1. The inaugural attendance of an annual music festival is 150,000. The
attendance increases by 8% each year.
Write an exponential function that represents the attendance after 𝑡 years.
How many people will attend the festival in the fifth year? Round your
answer to the nearest thousand.
Find a formula…Growth! 𝑦 = 𝑎(1 + 𝑟)𝑡 𝑦 = 150000(1 + .08)𝑡
𝑦 = 150000(1 + .08)5
𝑦 = 150000(1.08)5
𝑦 = 220399.2115 ≈ 220,000
2. The value of a car is $21,500. Its value depreciates by 12% each year.
Write a function that represents the value, in dollars, of the car after 𝑡
years.
Use this function to find how long it will take, to the nearest tenth of a year,
for the car to be valued at $15,000
Find a formula…Decay! 𝑦 = 𝑎(1 − 𝑟)𝑡 𝑦 = 21500(1 − .12)𝑡
15000 = 21500(1 − .12)𝑡
15000 = 21500(.88)𝑡
15000
21500=
21500(.88)𝑡
21500
30
43= (.88)𝑡
log (30
43) = 𝑡log(.88) or “anti-loop” log.88 (
30
43) = 𝑡
𝑡 = 2.8
45
3. An apartment purchased 4 years ago for $80,000 was just sold for $105,000.
Assuming exponential growth, approximate the annual growth rate, to the
nearest percent.
Write an exponential growth formula! 𝑦 = 𝑎(1 + 𝑟)𝑡
105000 = 80000(1 + 𝑟)4
105000
80000=
80000(1+𝑟)4
80000
1.3125 = (1 + 𝑟)4 raise both sides to 1
4 or √
4 both sides!
(1.3125)1
4 = ((1 + 𝑟)4)1
4
1.070347571 = 1 + 𝑟
𝑟 = .070347571 ≈ 7%
Writing More Difficult Exponential Growth and Decay Functions
4. An intial population of 20 mice triples each year. Write an equation that
models this population after 𝑡 years.
Exponential growth (triples each year)! 𝑦 = 𝑎(1 + 𝑟)𝑡
1 + 𝑟 = 3 Initial Amount = 20
𝑦 = 20(3)𝑡
5. The half-life of a medication is the time it takes for the medication to reduce to
half of its original amount in a patient’s bloodstream. A certain antibiotic has a
half-life of one hour. A patient takes 500 milligrams of the medication. Write a
function 𝐴(𝑡) that models the amount of the medication in the patient’s
bloodstream over time.
Exponential decay (half-life)! 𝑦 = 𝑎(1 − 𝑟)𝑡
What if the word problem does not give you the rate of change as a percent?
Exponential growth – look for keywords such as doubles, triples, quadruples, …
Exponential decay – look for keywords such as halves (half-life), a third, ...
46
1 − 𝑟 =1
2 Initial Amount = 500
𝐴(𝑡) = 500 (1
2)
𝑡
What if the word problem mentions the growth or decay rate over a specific
period of time!
a. The half-life of a medication is the time it takes for the medication to reduce to
half of its original amount in a patient’s bloodstream. A certain antibiotic has a
half-life of one hour. A patient takes 500 milligrams of the medication.
Divide by 1
𝐴(𝑡) = 500 (1
2)
𝑡
b. The half-life of a medication is the time it takes for the medication to reduce to
half of its original amount in a patient’s bloodstream. A certain antibiotic has a
half-life of about 3 hours. A patient takes 500 milligrams of the medication.
Divide by 3
𝐴(𝑡) = 500 (1
2)
𝑡
3
6. Suppose that a water tank is infested with a colony of 1,000 bacteria. In this
tank the colony doubles in number every 4 days. Determine a formula, 𝐴(𝑡), for
the number of bacteria present in the tank after 𝑡 days.
Exponential growth (doubles in number)! 𝑦 = 𝑎(1 + 𝑟)𝑡
Every four days means to divide the time by 4
1 + 𝑟 = 2 Initial Amount = 1000
𝐴(𝑡) = 1000(2)𝑡
4
47
7. Lead- 209, a radioactive isotope, decays to nonradioactive lead over time.
The half-life of lead-209 is 8 days. Suppose that 20 milligrams of lead-209 are
created by a particle physics experiment. Write an equation for the amount of
lead-209 present 𝑡 days after the experiment.
Exponential decay (half-life)! 𝑦 = 𝑎(1 − 𝑟)𝑡
Every eight days means to divide the time by 8
1 + 𝑟 =1
2 Initial Amount = 20
𝑦 = 20 (1
2)
𝑡
8
8. Titanium-44 is a radioactive isotope such that every 63 years, its mass
decreases by half. For a sample of titanium-44 with an initial mass of 100 grams,
write a function that will give the mass of the sample remaining after any amount
of time.
Exponential decay (decreases by half)! 𝑦 = 𝑎(1 − 𝑟)𝑡
Every 63 years means to divide the time by 63
1 + 𝑟 =1
2 Initial Amount = 100
𝐴(𝑡) = 100 (1
2)
𝑡
63
LOCATION OF REGRESSIONS IN THE CALCULATOR
These non-linear regressions are also found using the graphing calculator.
All types of regressions on the calculator are prepared in a similar manner.
Your regression options can be found under 𝐒𝐓𝐀𝐓 → 𝐂𝐀𝐋𝐂 (scroll for more
choices)
48
9. A box containing 1,000 coins is shaken, and the coins are emptied onto a
table. Only the coins that land heads up are returned to the box, and then the
process is repeated. The accompanying table shows the number of trials and
the number of coins returned to the box after each trial. Write an exponential
regression equation, rounding the calculated values to the nearest ten-
thousandth.
stat → edit → 𝐿1,𝐿2
stat → calc → ∅: ExpReg
𝑦 = 𝑎(𝑏)𝑥
𝑦 = 1018.2839(.5969)𝑥 Decay!
49
10. An object at a temperature of 160° is removed from a furnace and placed in
a room at 20℃. The table shows the temperatures 𝑑 at selected times 𝑡 (in
hours) after the object was removed from the furnace. Write a logarithmic
regression equation for this set of data, rounding coefficients to 3 decimal places.
stat → edit → 𝐿1,𝐿2 then stat → calc → 9: LnReg
𝑦 = 𝑎 + 𝑏ln𝑥
𝑡 = .863 − 6.447ln𝑑
11. Jean invested $380 in stocks. Over the next 5 years, the value of her
investment grew, as shown in the
accompanying table.
Write the exponential regression
equation for this set of data, rounding all
values to two decimal places.
𝑦 = 𝑎(𝑏)𝑥
𝑦 = 379.92(1.04)𝑥 Growth!
Using this equation, find the value of her
stock, to the nearest dollar, 10 years
after her initial purchase.
𝑦 = 379.92(1.04)10 = $562 𝑥 = 10 and 𝑦 =?
Using this equation, also find how many years, to the nearest tenth, it will take for
her investment to grow to reach $900. 𝑥 =? and 𝑦 = 900
900 = 379.92(1.04)𝑥
900
379.92=
379.92(1.04)𝑥
379.92
2.368919773 = 1.04𝑥 “anti-loop” or log both sides
𝑥 = 22.0
50
12. The accompanying table show wind speed and the corresponding wind chill
factor when the air temperature is 10℉.
Write the logarithmic regression
equation for this set of data, rounding
coefficients to the nearest ten
thousandth.
𝑦 = 𝑎 + 𝑏ln𝑥
𝑦 = 13.0134 − 7.3135ln𝑥
Based on your equation, if the wind chill
factor is 0, what is the wind speed, to
the nearest mile per hour? 𝑦 = 0 and 𝑥 =?
0 = 13.0134 − 7.3135ln𝑥
−13.0134 = −7.3135ln𝑥
−13.0134
−7.3135=
−7.3135ln𝑥
−7.3135
ln𝑥 = 1.779366924 “loop” with a base = 𝑒
𝑒1.779366924 = 𝑥 𝑥 = 6
13. The following set of data shows U.S. gas prices in
recent years. The table below represents the U.S. price of
gas from 2005 to 2015, where 𝑡 = 1 represents year 2005.
a) Based on the table what was the average rate of change
in the price of gasoline from 2005 to 2014, to the nearest
thousandth?
b) What is the exponential regression for the data in the
table, rounding coefficients to the nearest thousandth?
c) Based upon your regression, what is the average rate of
change in the price of gasoline from 2005 to 2014, to the
nearest thousandth?
d) Why is there a difference between your answers using the table and using the
regression equation?
51
Homework 11.7 & 11.8: Writing Exponential Growth and Decay Functions &
Exponential and Logarithmic Regression
1. Doug drank a cup of tea with 130 mg of caffeine. Each hour, the caffeine in
Doug’s body diminishes by about 12%. Write a formula to model the amount of
caffeine remaining in Doug’s system after each hour. How long, to the nearest
hour, should it take for the level of caffeine in Doug’s system to drop below 30
mg.
2. A tree planted 10 years ago at 3 feet tall now measures 25 feet tall.
Assuming exponential growth, approximate the annual growth rate, of the tree to
the nearest percent.
3. Which function represents exponential decay:
(1) 𝑦 = 20.3𝑡 (2) 𝑦 = 1.23𝑡 (3) 𝑦 = (1
2)
−𝑡 (4) 𝑦 = 5−𝑡
4. The number of carbon atoms in a fossil is given by the function
𝑦 = 5100(0.95)𝑥, where 𝑥 represents the number of years since being
discovered. What is the percent of change each year? Explain how you arrived
at your answer.
52
5. Last year, the total revenue for Home Style, a national restaurant chain,
increased 5.25% over the previous year. If this trend were to continue, which
expression could the company’s chief financial officer use to approximate their
monthly percent increase in revenue?
(1) (1.0525)𝑚 (2) (1.0525)12
𝑚 (3) (1.00427)𝑚 (4) (1.00427)𝑚
12
6. The price of a postage stamp in the years
since the end of World War I is shown in the
scatterplot. The equation that best models
the price, in cents, of a postage stamp based
on these data is:
(1) 𝑦 = 0.59𝑥 − 14.82
(2) 𝑦 = 1.04(1.43)𝑥
(3) 𝑦 = 1.43(1.04)𝑥
(4) 𝑦 = 1.04(0.96)𝑥
7. The number of ticks in an infested field is decreasing with the use of a new
pesticide. Below is a table which shows the number of ticks in the field after 𝑥
applications of pesticide.
𝑥, number of applications 0 1 2 3 4
𝑦, number of ticks 5000 4000 3200 2560 2048
a) Determine an exponential equation of the form 𝑦 = 𝑎 ∙ 𝑏𝑥 that models this
data, rounding all values to two decimal places.
b) Using this equation, determine the number of ticks, to the nearest whole
number, in the field after 9 applications of the pesticide.
c) Using this equation, find the number of applications, to the nearest whole
number, necessary to reduce the number of ticks to 1000.
53
8. The height of a tree increases with age. At the time of planting, the tree was 1
year old. The table below shows the height ℎ (in feet) of the tree at specific ages
𝑡 (in years).
a) Use a graphing calculator to find a logarithmic model of the form
ℎ = 𝑎 + 𝑏 ln 𝑡 that represents the data. Round all values to the nearest
hundredth.
b) Estimate the height, to the nearest tenth, when the tree is 3 years old.
54
Lesson 11.9: Calculating Monthly Payments
Learning Goal: How do we determine a monthly payment for a loan using any
given formula?
Warm-Up: You are 25 years old and hold a steady job. One day on the way to
work, you see that your dream car is on sale at a car dealership. You go to the
dealer to see about buying it. The car dealer tells you that he has this great deal
on the car. Apparently, only for today, he is selling the car for $45,000 with a
down payment of $3,000. He says he did some calculations and gives you three
car loan options which are shown below:
What will be the monthly payments for each option? Which will you take? Can
you afford any of them? If you end up buying it, how much spending money will
you have left over each month for each option? How much will you end up
paying back (with interest) after the loan is paid off? Is there a better deal out
there? Since we are talking math here, there definitely is a better way to figure
all this out than just signing the agreement and hoping for the best.
Many people take out a loan to purchase a car and then repay the loan on a
monthly basis. Today we will figure out how banks determine the monthly
payment for a loan in today’s class. As seen in the scenario above, when
deciding to get a car loan, there are many things to consider. For car loans, a
down payment is a very common requirement. A down payment is the amount of
money the person will pay towards the cost of the car before the loan is taken.
For example: For the scenario above, he is selling the car for $45,000 with a
down payment of $3,000. What will the loan amount be if he decides to choose
one of the three loan options?
55
When calculating loans and monthly payments, there are a few different formulas
that can be used. The formula depends on the type of loan and what is included
in the loan. Below are some examples of monthly loan formulas.
Compounding periods indicate how often the interest is calculated:
Annually (𝑛 = 1), Semi-annually (𝑛 = 2), Quarterly (𝑛 = 4), Monthly (𝑛 = 12)
1. Monthly mortgage payments can be calculated according to the formula,
𝐴 =𝑀𝑝𝑛 𝑡(1−𝑝)
(1−𝑝𝑛 𝑡), where 𝑀 is the size of the mortgage, 𝑛 is the number of
compounds per year, 𝑡 is the length of the mortgage, in years, and 𝑝 = (1 +𝑟
𝑛)
where 𝑟 is the interest rate as a decimal.
Tips when working with monthly loan formulas:
Read the question twice. Be sure to pay attention to what each variable
stands for.
The second time, write down the numbers that correspond to each variable.
Determine the variable that they are asking you to find.
Substitute the values in the given formula.
Try to simplify any expressions within the formula before solving for the
remaining variable.
If you are solving for an exponent, you will have to use logs.
A down payment is the amount you would pay prior to taking out a loan. This
amount gets subtracted from the total purchase price.
56
a. What would the monthly mortgage payments be on a $180,000, 15 year
mortgage with 6% interest, compounded monthly, to the nearest dollar?
b. Determine the length of the mortgage, to the nearest year, in order for the
monthly payment to be $1,080.
𝐴=?𝑀=180000
𝑛=12𝑡=15𝑟=.06
𝑝 = (1 +𝑟
𝑛) = 1 +
.06
12= 1.005 𝐴 =
𝑀𝑝𝑛 𝑡(1−𝑝)
(1−𝑝𝑛 𝑡)
𝐴 =180000(1.005)12∙15(1−1.005)
(1−1.00512∙15)
𝐴 =−2208.684206
−1.454093562= $1519
2. Monthly mortgage payments can be found using the formula below:
𝑀 =𝑃 (
𝑟12
) (1 +𝑟
12)
𝑛
(1 +𝑟
12)
𝑛− 1
𝑀 = monthly payment, 𝑃 = amount borrowed, 𝑟 = annual interest rate, 𝑛 =
number of monthly payments
The Banks family would like to borrow $120,000 to purchase a home. They
qualified for an annual interest rate of 4.8%. Algebraically determine the fewest
number of whole years the Banks family would need to include in the mortgage
agreement in order to have a monthly payment of no more than $720.
𝑛=?(𝑚𝑜𝑛𝑡ℎ𝑙𝑦)𝑀=720
𝑃=120000𝑟=.048
720 =120000(
.048
12)(1+
.048
12)
𝑛
(1+.048
12)
𝑛−1
720 =480(1.004)𝑛
(1.004)𝑛−1 “cross multiply”
720((1.004)𝑛 − 1) = 480(1.004)𝑛
720((1.004)𝑛−1)
720=
480(1.004)𝑛
720
57
(1.004)𝑛 − 1 =2
3(1.004)𝑛 let 𝑥 = 1.004𝑛
𝑥 − 1 =2
3𝑥
1
3𝑥 = 1
𝑥 = 3 sub the 𝑥 back in
1.004𝑛 = 3 add logs or “anti-loop”
𝑛log1.004 = log3 or log1.0043 = 𝑛
𝑛 = 275.2020158 (months)
Fewest number of years!! Must divide by 12
𝑛
12= 22.93350 always round up so 23 years!
3. Using the formula below, determine the monthly payment on a 3-year car loan
with a monthly percentage rate of 0.75% for a car with an original cost of $25,000
and a $2,000 down payment, to the nearest cent.
𝑃𝑛 = 𝑃 𝑀 𝑇 ((1 − (1 + 𝑖)−𝑛)
𝑖)
𝑃𝑛 = present amount borrowed, 𝑛 = number of monthly payments, 𝑃 𝑀 𝑇 =
monthly payment, and 𝑖 = interest rate per month
𝑃𝑀𝑇=?𝑃𝑛=25000−2000=23000
𝑛=3∗12=36
𝑖=.75
100=.0075
23000 = 𝑃 𝑀 𝑇 ((1−(1+.0075)−36)
.0075)
23000 = 𝑃 𝑀 𝑇 (31.44680525)
731.3938512 = $731.39 = 𝑃 𝑀 𝑇
The affordable monthly payment is $350 for the same time period. Determine an
appropriate down payment, to the nearest dollar.
58
𝑃𝑀𝑇=350𝑃𝑛=25000−𝐷𝑃
𝑛=3∗12=36
𝑖=.75
100=.0075
25000 − 𝐷𝑃 = 350 ((1−(1+.0075)−36)
.0075)
25000 − 𝐷𝑃 = 11006.38184
𝐷𝑃 = $13994
4. Jim is looking to buy a vacation home for $172,600 near his favorite southern
beach. The formula to compute a mortgage payment, 𝑀, is 𝑀 = 𝑃 ∙𝑟(1+𝑟)𝑁
(1+𝑟)𝑁−1
where 𝑃 is the principal amount of the loan, 𝑟 is the monthly interest rate, and 𝑁
is the number of monthly payments. Jim’s bank offers a monthly interest rate of
0.305% for a 15-year mortgage. With no down payment, determine Jim’s
mortgage payment, rounded to the nearest dollar.
𝑀=?𝑃=172,600
𝑟=.305%=.00305𝑁=15∗12=180
𝑀 = 172600.00305(1+.00305)180
(1+.00305)180−1= 1247.49339
Algebraically determine the fewest number of whole months that Jim needs to
make in order for his mortgage payment to be no more than $1,100.
𝑀=1100𝑁=?
𝑃=172600𝑟=.00305
1100 = 172600.00305(1.00305)𝑁
(1.00305)𝑁−1
1100
172600=
. 00305(1.00305)𝑁
(1.00305)𝑁 − 1 cross multiply
5.2643(1.00305)𝑁 = 11(1.00305𝑁 − 1)
Let 𝑥 = 1.00305𝑁
5.2643𝑥 = 11(𝑥 − 1)
5.2643𝑥 = 11𝑥 − 11
−5.7357𝑥 = −11
𝑥 = 1.917812996
1.917812996 = 1.00305𝑁
𝑁 = 213.828861 so 214 months
59
Homework 11.9: Calculating Monthly Payments
1. Monthly mortgage payments can be found using the formula below:
𝑀 =𝑃 (
𝑟12
) (1 +𝑟
12)
𝑛
(1 +𝑟
12)
𝑛− 1
𝑀 = monthly payment, 𝑃 = amount borrowed, 𝑟 = annual interest rate, 𝑛 =
number of monthly payments
The Ranallo family just bought a house. They qualified for an annual interest
rate of 5.2% and their monthly payment was $800 for a 30 year mortgage.
Algebraically determine the amount of their loan to buy the house.
2. Using the formula below, determine the original loan value, to the nearest
dollar, Sam would have taken out for a 5 year car loan with a monthly percentage
rate of 2.5% if he is paying $300 a month.
𝑃𝑛 = 𝑃 𝑀 𝑇 ((1 − (1 + 𝑖)−𝑛)
𝑖)
𝑃𝑛 = present amount borrowed, 𝑛 = number of monthly payments, 𝑃 𝑀 𝑇 =
monthly payment, and 𝑖 = interest rate per month
60
Lesson 11.10: Compound Interest Formula
Learning Goals:
1) How do we determine the value of an investment using the compound
interest formula?
2) How do we determine the amount of time it takes for an investment to
reach a certain value?
Warm-Up: Use the formula 𝐴 = 𝑃 𝑒𝑟 𝑡 to answer the following question.
Jackie’s parents invested money at an interest rate of 6% compounded
continuously, to save toward Jackie’s college education.
How much money did the parents invest when Jackie started first grade to
accumulate $20,000 after 12 years? 𝐴 = 𝑃 𝑒𝑟 𝑡 for compounded continuously
𝐴 = 2000 𝐴 = 𝑃 𝑒𝑟 𝑡
𝑃 =? 2000 = 𝑃 𝑒 .06∙12
𝑟 = .06 2000 = 2.054433211 𝑃
𝑡 = 12 𝑃 = $9735.05
Compound Interest Formula
𝐴 = 𝑃 (1 +𝑟
𝑛)
𝑛 𝑡(memorize) Compounding periods
𝐴 = final amount Annually (𝑛 = 1)
𝑃 = initial amount (investment) Semi-annually (𝑛 = 2)
𝑟 = interest rate Quarterly (𝑛 = 4)
𝑛 = compound periods Monthly (𝑛 = 12)
𝑡 = time
61
1. Robert invests $800 in an account at 1.8% interest compounded annually. He
will make no deposits or withdrawals on this account for 3 years. Which formula
could be used to find the balance, 𝐴, in the account after the 3 years.
(1) 𝐴 = 800(1 − .18)3 (2) 𝐴 = 800(1 + .18)3
(3) 𝐴 = 800(1 − .018)3 (4) 𝐴 = 800(1 + .018)3
𝐴 = 𝑃 (1 +𝑟
𝑛)
𝑛 𝑡 𝐴 = 800 (1 +
.018
1)
1∙3(
𝑃 = 800 𝐴 = 800(1 + .018)3
𝑟 = .018
𝑛 = 1
𝑡 = 3
2. Duane plans to make a one-time investment and wants to reach $10,000 in
thirty-six months for Kira’s wedding. He finds an investment with 3.5% interest
compounded monthly. How much should he invest?
𝐴 = 10000 𝐴 = 𝑃 (1 +𝑟
𝑛)
𝑛 𝑡
𝑃 =? 10000 = 𝑃 (1 +.035
12)
12∙3
𝑟 = .035 10000 = 𝑃(1.002916667)36
𝑛 = 12 10000 = 𝑃(1.110540876)
𝑡 = 3 years 𝑃 = 9004.62
3. A home owner invests $15,000 at an annual interest rate of 4%. What will be
the value of this investment after 6 years if the investment is compounded
quarterly?
𝐴 = 𝑃 (1 +𝑟
𝑛)
𝑛 𝑡
= 15000 (1 +. 04
4)
4∙6
= 19046.02
62
4. Linda invests $6,000 into an account. After 5 years her investment reaches
$7,200. If the interest was compounded annually, determine the rate to the
nearest tenth of a percent.
𝐴 = 7200 𝐴 = 𝑃 (1 +𝑟
𝑛)
𝑛 𝑡
𝑃 = 6000 7200 = 6000 (1 +𝑟
1)
1∙5
𝑟 =? 7200 = 6000(1 + 𝑟)5
𝑛 = 1 7200
6000=
6000(1+𝑟)5
6000
𝑡 = 5 years 1.2 = (1 + 𝑟)5 get rid of exponent
1.21
5 = ((1 + 𝑟)5)1
5
1.037137289 = 1 + 𝑟
𝑟 = .037137289 = 3.7%
5. Jack and Jill are twins that received $2,000 each for their 16th birthday. They
decided to separately invest their money. Jack is going to invest his money at
3.7% interest compounded quarterly. Jill is going to invest her money at 3.6%
interest compounded monthly.
a. Write a function of Jack’s option and Jill’s option that calculates the value of
each account after 𝑡 years.
𝐴 = 𝑃 (1 +𝑟
𝑛)
𝑛 𝑡
Jack: 𝐴 = 2000 (1 +.037
4)
4 𝑡 Jill: 𝐴 = 2000 (1 +
.036
12)
12 𝑡
b. Determine who will have more money by the time they graduate high school
in two years, to the nearest cent. 𝑡 = 2
Jack: 𝐴 = 2000 (1 +.037
4)
4 ∙ 2 Jill: 𝐴 = 2000 (1 +
.036
12)
12 ∙ 2
63
Jack: 𝐴 = $2152.88 Jill: 𝐴 = $2149.08
c. Algebraically determine, to the nearest tenth of a year, how long it would take
for Jill’s initial investment to triple.
𝐴 = 2000 × 3 = 6000 6000 = 2000 (1 +.036
12)
12 ∙ 𝑡
𝑡 =? 6000
2000=
2000(1.003)12 ∙ 𝑡
2000
3 = 1.00312𝑡
log3 = 12𝑡log1.003
log3
12log1.003=
12𝑡log1.003
12log1.003
𝑡 = 30.6
Finding Growth Rate
6*. Find the rate of growth to the nearest ten-thousandth for the amount of
money accumulated if $6,000 is invested at 8% interest compounded quarterly.
7*. If $10,000 is invested in an account at 6.5% compounded monthly, determine
the growth rate to the nearest thousandth.
64
Homework 11.10: Compound Interest Formula
1. Krystal was given $3,000 when she turned 2 years old. Her parents invested
it at a 2% interest rate compounded annually. No deposits or withdrawals were
made. Which expression can be used to determine how much money Krystal
had in the account when she turned 18?
(1) 3000(1 + 0.02)16 (2) 3000(1 − 0.02)16
(3) 3000(1 + 0.02)18 (4) 3000(1 − 0.02)18
2. Nelson’s financial goal is to invest a sum of money at 3.5% interest
compounded monthly. If his goal is to have $10,000 at the end of 3 years, how
much will he need to invest? (Assume rounding to nearest penny.)
3. Katie invests $4,500 in an account that pays 3.75% interest compounded
quarterly. How long, to the nearest tenth of a year, will it take for Katie’s
investment to double?
