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| 11 Theory slides |
| 16 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
The rate of change for linear functions is constant. For each step in the x-direction, the change between y-values is the same. Therefore, the difference between y-values for consecutive x-values is the same. Conversely, the rate of change for exponential functions is not constant. This means that the differences between y-values for consecutive x-values are not the same.
In the table on the left, the difference between y-values for consecutive x-values is 2. This means that the rate of change is constant. Conversely, in the table on the right, the differences between y-values for consecutive x-values are not the same. This means that for this table the rate of change is not constant.
The table on the left represents a linear function because it shows a constant rate of change. Conversely, the table on the right represents a non-linear function. Notice how the y-values, for the table on the right, are doubled for each step in the x-direction. To obtain a y-value, the previous value is multiplied by 2.
Does the table below correspond to an exponential function? If yes, write only the constant multiplier. If not, write only no.
The graphs of f(x)=ex and g(x)=e-x are shown.
The initial value of an exponential function, y=abx, is the number without an exponent, or the value of a. The initial value is also known as the y-intercept (0,a). In this example, the value of a is 1, and thus the y-intercept of the graph of y=ex is (0,1).
In natural base functions, the constant multiplier — the number with an exponent — is the number e. This means that when x-values increase by 1, the y-values are multiplied by e. With this information, more points can be plotted on the coordinate plane. It is advised to have not less than three points.
Lastly, the graph can be drawn by connecting the points with a smooth curve.
It is worth remembering that, in general, the graph of a natural base function is always a smooth curve.
When interest is compounded infinitely many times, it is said to be continuously compounded. Let A be the balance of an account that is continuously compounded, P the initial amount, r the interest rate, and t the time. These values are connected by the following formula.
A=Pert
Keep in mind that, in this formula, the value of r must be written as a decimal and the time t must be in years. Also, the initial amount P is usually called principal.
Zosia wants to visit family in Argentina, and while there she hopes to climb Mount Aconcagua! This is going to cost a pretty penny so she needs to increase her savings. Zosia knows that banks do not offer continuously compounded interest, but she is daydreaming about opening an account that does use it.
The constant multiplier is e. This means that when x-values increase by 1, the y-values are multiplied by e.
Finally, the curve can be graphed by connecting these points.
t=1.75
Use a calculator
Round to 2 decimal place(s)
The second coordinate of this point is a bit less than 100. Therefore, it is reasonable to have a balance of $97.83 after 1 year and 9 months.
The first coordinate of the point is almost equal to 2. Therefore, the balance would be $120 after about 2 years. Zosia is realizing that it might take her quite some time to meet her financial goal, but she is making a valiant effort and certainly on her way!
Zosia finally saved enough money to visit family in Argentina and set out to climb Mount Aconcagua while there! She is as excited as ever.
x=8849
(-a)b=-ab
Use a calculator
Round to nearest integer
x | 1013e-0.000128x | y=1013e-0.000128x |
---|---|---|
0 | 1013e-0.000128(0) | 1013 |
1000 | 1013e-0.000128(1000) | ≈891 |
2000 | 1013e-0.000128(2000) | ≈784 |
3000 | 1013e-0.000128(3000) | ≈690 |
4000 | 1013e-0.000128(4000) | ≈607 |
5000 | 1013e-0.000128(5000) | ≈534 |
6000 | 1013e-0.000128(6000) | ≈470 |
7000 | 1013e-0.000128(7000) | ≈414 |
8000 | 1013e-0.000128(8000) | ≈364 |
9000 | 1013e-0.000128(9000) | ≈320 |
10000 | 1013e-0.000128(10000) | ≈282 |
The obtained points will now be plotted on a coordinate plane and connected with a smooth curve. Since only non-negative values are considered, the graph just needs to be drawn in the first quadrant.
Tracing a pen or pencil vertically down from the point of the graph that is even, horizontally, with y-value of 700 leads to the x-value of 3. This means that the altitude with an atmospheric pressure of 700 hecto Pascals is about 3000 meters. Zosia feels comfortable knowing she will be able to rest for a few nights at this level.
Logarithmic functions are functions that involve logarithms.
f(x)=logbx, b>0 and b=1
The function f(x)=logbx is the parent function of logarithmic functions. Since logarithms are defined for positive numbers, the domain of the function is x>0 and its range is all real numbers. If b is less than 1, the graph of the function is decreasing over its entire domain. Conversely, if b is greater than 1, the graph is increasing over its entire domain.
A logarithm and a power with the same base undo
each other.
logbbx=xandblogbx=x
In particular, the above equations also hold true for common and natural logarithms.
The general equations will be proved one at a time.
Zosia is getting everything ready for the multi-day hike in Mount Aconcagua. Because her extra battery pack is quite limited, she wants her smartphone and camera to have their batteries fully charged.
The night before the hike, Zosia charges her smartphone and camera. She does not want to leave her devices plugged all night long because it is a waste of energy and could damage the batteries. She read in the user manuals of both devices that the charge in the batteries can be modeled by two logarithmic functions.Phone's Battery:
Camera's Battery:
To graph p(x)=lnx, first graph its inverse function g(x)=ex and then reflect the curve across the line y=x. To graph c(x)=2log(x−0.5), make a table of values. Consider its domain first.
The logarithmic functions will be graphed one at a time.
x | 2log(x−0.5) | c(x)=2log(x−0.5) |
---|---|---|
1 | 2log(1−0.5) | ≈-0.6 |
2 | 2log(2−0.5) | ≈0.35 |
3 | 2log(3−0.5) | ≈0.8 |
4 | 2log(4−0.5) | ≈1.1 |
5 | 2log(5−0.5) | ≈1.3 |
The expedition was a complete success! What is more, some members of the hiking team felt so inspired by Zosia’s math skills in helping them reach the summit, that they decided to practice some logarithmic functions. They believe this knowledge will also help them in their next expedition to the Amazon Rainforest.
Touched by their passion to learn, Zosia found a few interesting exercises about exponential and logarithmic functions in her online textbook to share with the team.
LHS+1=RHS+1
Rearrange equation
y=f-1(x)
x | 2log(x−1) | f(x) |
---|---|---|
1.5 | 2log(1.5−1) | ≈-0.6 |
5 | 2log(5−1) | ≈1.2 |
10 | 2log(10−1) | ≈1.9 |
12 | 2log(12−1) | ≈2.1 |
18 | 2log(18−1) | ≈2.5 |
Next, the table that corresponds to f-1(x) will be constructed.
x | 1021x+1 | f-1(x) |
---|---|---|
-2 | 1021(-2)+1 | 1.1 |
-1 | 1021(-1)+1 | ≈1.3 |
0 | 1021(0)+1 | 2 |
1 | 1021(1)+1 | ≈4.2 |
2 | 1021(2)+1 | 11 |
Now the points can be plotted on a coordinate plane and connected with smooth curves. Also, the line y=x will be graphed.
The graphs are each other's reflection across the line y=x. Therefore, f-1(x)=1021x+1 is the inverse function of f(x)=2log(x−1).
LHS/3=RHS/3
ba=b1⋅a
Rearrange equation
y=g-1(x)
x | 51e3x | g(x) |
---|---|---|
-1 | 51e3(-1) | ≈0.01 |
-0.5 | 51e3(-0.5) | ≈0.04 |
0 | 51e3(0) | 0.2 |
0.5 | 51e3(0.5) | ≈0.9 |
1 | 51e3(1) | ≈4 |
Next, the table that corresponds to g-1(x) will be constructed.
x | 31ln5x | g-1(x) |
---|---|---|
0.1 | 31ln5(0.5) | ≈-0.2 |
1 | 31ln5(1) | ≈0.5 |
2 | 31ln5(2) | ≈0.8 |
3 | 31ln5(3) | ≈0.9 |
4 | 31ln5(4) | ≈1 |
The obtained points can be plotted on a coordinate plane and connected with smooth curves. Also, the line y=x will be graphed.
It can be seen that the graphs are each other's reflection across the line y=x. Therefore, g-1(x)=31ln5x is the inverse function of g(x)=51e3x.
f-1(x)=1021x+1
Subtract term
log(10m)=m
Associative Property of Multiplication
2⋅2a=a
Identity Property of Multiplication
f(x)=2log(x−1)
Associative Property of Multiplication
2a⋅2=a
Identity Property of Multiplication
10log(m)=m
Add terms
Definition of the First Function | Substitute the Second Function | Simplify | |
---|---|---|---|
g(g-1(x)) | 51e3g-1(x) | 51e3(31ln5x) | x ✓ |
g-1(g(x)) | 31ln5g(x) | 31ln5(51e3x) | x ✓ |
Since both compositions simplify to the identity function, g-1(x)=31ln5x is the inverse function of g(x)=51e3x.
In this lesson, exponential functions, logarithmic functions, and the relationship between their graphs have been discussed. One more characteristic of these graphs is that they have asymptotes.
Every logarithmic graph has a vertical asymptote. In the applet below, both graphs have the vertical line x=0, which is the y-axis, as an asymptote.
An asymptote of a graph is an imaginary line that the graph gets close to as x tends towards positive or negative infinity or a particular number. Find the asymptotes of the graphs.
This given graph is one of an exponential function. The domain of an exponential function is the set of all real numbers, which means that the graph cannot have a vertical asymptote.
Exponential Function | y = e^x -1 |
---|---|
Domain | All real numbers |
Asymptote | Horizontal |
If we pay close attention, we can see that when x tends to negative infinity, the graph approaches the horizontal line y=- 1.
Therefore, y=- 1 is a horizontal asymptote.
This graph is the graph of a logarithmic function. Since the argument of a logarithm must be positive, the domain of this type of function is not the set of all real numbers. This means that the graph has a vertical asymptote.
Logarithmic Function | y = log(x+2) |
---|---|
Domain | Not all real numbers |
Asymptote | Vertical |
If we pay close attention, we can see that x approaches - 2, but its value never is - 2.
Therefore, x=- 2 is a vertical asymptote.
Considering the equations and graphs below. Order the functions from least to greatest average rate of change over the interval that goes from 1 to 10.
Let's find the rate of change for each function in the interval 1≤ x ≤ 10. We will consider each function one at a time.
To find the rate of change for this logarithmic function in the interval from 1 to 10, we need to recall that the rate of change is the ratio of the change in y to the change in x. Rate of Change=y_2-y_1/x_2-x_1 Next, we will substitute x_1=1, x_2=10, y_1=log_6 1, and y_2=log_6 10 into the formula. Recall that the calculator only calculates logarithms with base 10 or base e — common and natural logarithms. Therefore, we will use the Change of Base Formula to convert the logarithms to common logarithms.
For the function in option A, the rate of change in the interval 1≤ x ≤ 10 is about 0.143.
We will find the rate of change for this function in the interval from 1 to 10 by following the same procedure as in option A. Let's do it!
Rate of change for y=log_(35) x in the interval 1≤ x ≤ 10 | ||
---|---|---|
Formula | Substitute values | Evaluate |
y_2-y_1/x_2-x_1 | log_(35) 10- log_(35)1/10- 1 | ≈ - 0.501 |
Therefore, for the function in option B, the rate of change in the interval 1≤ x ≤ 10 is about - 0.501.
By using the graph, we can find the y-coordinates for x=1 and x=2.
We see in the graph that when x=1, the value of y is 1. Also, when x=10, the value of y is - 2. Let's use these values to find the rate of change in the interval 1≤ x≤ 10.
Rate of change for the graph in the interval 1≤ x ≤ 10 | ||
---|---|---|
Formula | Substitute values | Evaluate |
y_2-y_1/x_2-x_1 | - 2- 1/10- 1 | ≈ - 0.333 |
Therefore, for the function in option C, the rate of change in the interval 1≤ x ≤ 10 is - 0.333.
Finally, to find the rate of change for the curve shown in option D, we will follow the same procedure as in option C. Let's do it!
We see that when x=1, the value of y is 0. Also, when x=10, the value of y is 8. We can use these values to find the rate of change for this curve in the interval that goes from 1 to 10.
Rate of change for the graph in the interval 1≤ x ≤ 10 | ||
---|---|---|
Formula | Substitute values | Evaluate |
y_2-y_1/x_2-x_1 | 8- 0/10- 1 | ≈ 0.889 |
Therefore, for the function in option D, the rate of change in the interval 1≤ x ≤ 10 is about 0.889.
Now that we have the four rates of change in the interval 1≤ x ≤ 10, we can write the options in order from least to greatest.
Option | B | C | A | D |
---|---|---|---|---|
Rate of change | ≈ - 0.501 | ≈ - 0.333 | ≈ 0.143 | ≈ 0.889 |
The domain of the function contains x=0 in its domain. |
Let's use the definition of a logarithm to rewrite the equation. Definition:& c=log_b a ⇔ b^c= a Equation:& y=log_b x ⇔ b^y= x We found that x is equal to b^y. It is given that b is a positive real number. We know that any power of a positive number is always a positive number. Therefore, for any value of y, the value of b^y, or x, will always be positive. x > 0 Since the domain is the set of x-values and x>0, the domain of the function never contains 0.