6.01 Graphs and characteristics of exponential functions
Identify characteristics of exponential functions
The mathematical function which models natural growth and decay phenomena is known as the exponential model. What distinguishes it from other functions you may have encountered is that the independent variable is in the exponent, (the exponent is also called the power or index).
Examples include $y=2^x$y=2x, $y=3\times2^{-x}$y=3×2−x, $y=8\times\left(0.5\right)^x+1$y=8×(0.5)x+1, $y=4-5^x$y=4−5x, etc.
If we first consider the function $y=2^x$y=2x, and substitute the consecutive integer value of $x$x into it, starting from $x=-3$x=−3 and finishing at $x=3$x=3, we could develop a table of values for the function.
So for $x=-3$x=−3, $y=2^{-3}=\frac{1}{2^3}=\frac{1}{8}$y=2−3=123=18. For $x=-2$x=−2, $y=\frac{1}{4}$y=14 and continuing:
| $x$x | $-3$−3 | $-2$−2 | $-1$−1 | $0$0 | $1$1 | $2$2 | $3$3 |
|---|---|---|---|---|---|---|---|
| $y$y | $\frac{1}{8}$18 | $\frac{1}{4}$14 | $\frac{1}{2}$12 | $1$1 | $2$2 | $4$4 | $8$8 |
Typical of exponential growth we see that initially there are small increases in $y$y , but as we continue to increase $x$x, the rate of increase in the size of $y$y is increasing. That is to say the $y$y values are getting larger faster as $x$x steadily increases. Populations, by and large, behave like this - the more people that live in a city, the faster the total population grows. Investments in bank accounts are similar - the more money you have in the bank, the more interest you earn.
Things can work in reverse as well. Consider the function given by $y=120\times2^{-x}$y=120×2−x. This table shows what happens to $y$y for integer values between $0$0 and $6$6.
| $x$x | $0$0 | $1$1 | $2$2 | $3$3 | $4$4 | $5$5 | $6$6 |
|---|---|---|---|---|---|---|---|
| $y$y | $120$120 | $60$60 | $30$30 | $15$15 | $7.5$7.5 | $3.75$3.75 | $1.875$1.875 |
We can use the table to model depreciation. For example, imagine an expensive car, say $$120000$120000, that halves its value every $4$4 years. After the first four years ($x=1$x=1), it would be worth $$60000$60000. After the second $4$4 years ($x=2$x=2) it would be worth $$30000$30000. After $24$24 years ($x=6$x=6), the car's value is reduced to $$1875$1875.
Understanding the shape of the exponential graph.
Below is the graph of $y=2^x$y=2x. Here are some key features worth noting:
- The function is increasing over its domain
- The rate of change increases as $x$x gets larger and larger
- It has a $y$y-intercept of $1$1 because,at $x=0$x=0, $y=2^0=1$y=20=1
- The point $\left(1,2\right)$(1,2) can help us graph the function
- End behavior
- As the $x$x tends to $-\infty$−∞, the curve draws closer and closer to the $x$x-axis the further it is from the origin. The $x$x-axis becomes what is known as a horizontal asymptote - a line that a curve approaches without ever touching it.
- As the $x$x tends to $\infty$∞, the curve moves upward away from the $x$x-axis. As it moves upward the bend becomes more pronounced as the rate of change in the y values becomes larger and larger.
-
The domain is all of the real numbers and the range is $y>0$y>0
The function $y=2^{-x}$y=2−x is graphed below. Here are some key features worth noting:
- The function is decreasing over its domain
- The rate of change approaches $0$0 as $x$x gets larger and larger
- It has a $y$y-intercept of $1$1 because,at $x=0$x=0, $y=2^{-0}=1$y=2−0=1
- The point $\left(-1,2\right)$(−1,2) can help us graph the function
- End behavior
- As the $x$x tends to $-\infty$−∞, the curve moves upward away from the $x$x-axis. As it moves upward the bend becomes more pronounced as the rate of change in the $y$y values becomes steeper.
- As the $x$x tends to $\infty$∞, the curve draws closer and closer to the $x$x-axis the further it is from the origin. The $x$x-axis becomes what is known as a horizontal asymptote - a line that a curve approaches without ever touching it.
- The domain is all of the real numbers and the range is $y>0$y>0
One of the most commonly used bases for exponential functions is a number represented by the letter $e$e. Named after Leonard Euler, it is defined as $e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n$e=limn→∞(1+1n)n. For really large values of $n$n, this is approximately $2.71$2.71. However, its digits are nonrepeating and neverending, so this is merely an approximation. The number $e$e is an irrational number.
Practice questions
QUESTION 1
Consider the expression $3^x$3x.
Evaluate the expression when $x=4$x=4.
Evaluate the expression when $x=-4$x=−4. Leave your answer in fractional form.
What happens to the value of $3^x$3x as $x$x gets larger?
it stays the same
Ait gets smaller, approaching zero
Bit gets larger, approaching infinity
CWhat happens to the value of $3^x$3x as $x$x gets smaller?
it gets larger, approaching infinity
Ait gets smaller, approaching zero
Bit stays the same
C
QUESTION 2
Consider the graph of the equation $y=4^x$y=4x.
What can we say about the $y$y-value of every point on the graph?
The $y$y-value of most points of the graph is greater than $1$1.
AThe $y$y-value of every point on the graph is positive.
BThe $y$y-value of every point on the graph is an integer.
CThe $y$y-value of most points on the graph is positive, and the $y$y-value at one point is $0$0.
DAs the value of $x$x gets large in the negative direction, what do the values of $y$y approach but never quite reach?
$4$4
A$-4$−4
B$0$0
CWhat do we call the horizontal line $y=0$y=0, which $y=4^x$y=4x gets closer and closer to but never intersects?
A horizontal asymptote of the curve.
AAn $x$x-intercept of the curve.
BA $y$y-intercept of the curve.
C
QUESTION 3
Consider the function $y=3^x$y=3x.
Complete the table of values.
$x$x $-5$−5 $-4$−4 $-3$−3 $-2$−2 $-1$−1 $0$0 $1$1 $2$2 $3$3 $5$5 $10$10 $y$y $\frac{1}{243}$1243 $\frac{1}{81}$181 $\frac{1}{27}$127 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Is $y=3^x$y=3x an increasing function or a decreasing function?
Increasing
ADecreasing
BHow would you describe the rate of increase of the function?
As $x$x increases, the function increases at a constant rate.
AAs $x$x increases, the function increases at a faster and faster rate.
BAs $x$x increases, the function increases at a slower and slower rate.
CWhat is the domain of the function?
all real $x$x
A$x\ge0$x≥0
B$x<0$x<0
C$x>0$x>0
DWhat is the range of the function?
Transformations of exponential functions
The exponential curve given by $y=a\times b^{n(x-h)}+k$y=a×bn(x−h)+k represents a transformation of the basic curve $y=b^x$y=bx.
Whilst the general form is a comprehensive tool for sketching exponential curves, there are a few simpler observations to keep in mind. We can summarize them using examples as shown in this table for the parent function $y=3^x$y=3x:
| Specific Example | Observation |
|---|---|
| $y=-3^x$y=−3x | Reflect $y=3^x$y=3x across the $x$x-axis |
| $y=3^{x-5}$y=3x−5 | Translate $y=3^x$y=3x horizontally to the right by $5$5 units |
| $y=3^x-5$y=3x−5 | Translate $y=3^x$y=3x vertically downward by $5$5 units |
| $y=2\times3^x$y=2×3x | Vertical stretch away from $x$x-axis, double every $y$y value of $y=3^x$y=3x |
More complex forms of the exponential require more thought. For example, the function $y=3^{2x-5}$y=32x−5 is quite interesting to think about. The applet below can produce the graph as a plot of points, but we can think about what the curve might look like without it.
For example, we can rewrite the function as follows:
$y=3^{2\left(x-\frac{5}{2}\right)}=\left(3^2\right)^{\left(x-\frac{5}{2}\right)}=9^{\left(x-\frac{5}{2}\right)}.$y=32(x−52)=(32)(x−52)=9(x−52).
Hence, the function could be thought of as the function $y=9^x$y=9x translated to the right by $2\frac{1}{2}$212 units.
Exploration
The applet below is extremely versatile, but we need to keep in mind that it is a learning tool exploring the effects of the different constants involved.
Try to create the four graphs shown in the table below and note the key features of the graphs.
| Function | $y$y-intercept | Asymptote | Increasing/decreasing |
|---|---|---|---|
| $y=2^x$y=2x | |||
| $y=3^{-x+1}$y=3−x+1 | |||
| $y=3\times4^x-2$y=3×4x−2 | |||
| $y=\left(0.5\right)^x$y=(0.5)x |
After experimenting with these, try other combinations of constants. What can you learn?
Bases between 0 and 1
One final point that should be noted is that a curve like $y=\left(0.5\right)^x$y=(0.5)x is none other than $y=2^{-x}$y=2−x in disguise. Thus:
$y=\left(0.5\right)^x=\left(\frac{1}{2}\right)^x=\frac{1}{2^x}=2^{-x}$y=(0.5)x=(12)x=12x=2−x
In a similar way we can say that $y=\left(\frac{1}{b}\right)^x=b^{-x}$y=(1b)x=b−x, and so every exponential curve of the form $y=b^x$y=bx, with a base $b$b in the interval $0
Practice questions
Question 4
Consider the function $y=-3^x+2$y=−3x+2.
Find the $y$y-intercept of the curve $y=-3^x+2$y=−3x+2.
Find the equation of horizontal asymptote of the curve $y=-3^x+2$y=−3x+2.
Hence plot the curve $y=-3^x+2$y=−3x+2.
Loading Graph...
Question 5
Consider the function $y=3^{-x}-1$y=3−x−1.
Find the $y$y-intercept of the curve $y=3^{-x}-1$y=3−x−1.
Find the horizontal asymptote of the curve $y=3^{-x}-1$y=3−x−1.
Now use your previous answers to plot $y=3^{-x}-1$y=3−x−1.
Loading Graph...
Finding and using an exponential equation
Under certain circumstances, we are able to deduce a function's equation from a few known points the associated curve passes through. For example it may be possible when the form of the equation is known.
Worked examples
question 6
Consider the curve of a function with the exponential form $y=a\left(b^x\right)$y=a(bx) which passes through $\left(2,18\right)$(2,18) and $\left(5,486\right)$(5,486). Find the equation of the curve.
Think: These points must satisfy the equation (that is, when substituted, will make the equation true). Thus we know that:
| $18$18 | $=$= | $ab^2$ab2 (1) |
| $486$486 | $=$= | $ab^5$ab5 (2) |
Do:
Dividing equation (2) by equation (1) gives:
| $\frac{486}{18}$48618 | $=$= | $\frac{ab^5}{ab^2}$ab5ab2 |
Performing the division |
| $27$27 | $=$= | $b^3$b3 |
Simplifying (assume $a\ne0$a≠0) |
| $3$3 | $=$= | $b$b |
Take the cube root of both sides |
Thus since $ab^2=18$ab2=18, we have $9a=18$9a=18 and $a=2$a=2. So we identify the function as given by $y=2\left(3\right)^x$y=2(3)x.
Reflect: Could you have solved this using a different strategy?
Question 7
Fill in the table for values of a function of the form $y=b^{-x}+c$y=b−x+c as shown here:
| $x$x | $-2$−2 | $-1$−1 | $0$0 | $1$1 | $2$2 |
|---|---|---|---|---|---|
| $y$y | $5$5 | $3.5$3.5 |
Think: We know the two points $\left(-1,5\right)$(−1,5) and $\left(1,3.5\right)$(1,3.5), so we form two equations as before:
| $5$5 | $=$= | $b^1+c$b1+c $(1)$(1) |
| $3.5$3.5 | $=$= | $b^{-1}+c$b−1+c $(2)$(2) |
Do: This means that from equation (2) we have $c=3.5-\frac{1}{b}$c=3.5−1b and so substituting this into equation (1) we have that $5=b+\left(3.5-\frac{1}{b}\right)$5=b+(3.5−1b) or when simplified $1.5=b-\frac{1}{b}$1.5=b−1b.
Solving for $b$b we have:
| $1.5$1.5 | $=$= | $b-\frac{1}{b}$b−1b |
| $1.5b$1.5b | $=$= | $b^2-1$b2−1 |
| $3b$3b | $=$= | $2b^2-2$2b2−2 |
| $2b^2-3b-2$2b2−3b−2 | $=$= | $0$0 |
| $\left(2b+1\right)\left(b-2\right)$(2b+1)(b−2) | $=$= | $0$0 |
| $b$b | $=$= | $-\frac{1}{2},2$−12,2 |
So there seem to be two possibilities. By substituting back into equation (1) above we find that if $b=-\frac{1}{2}$b=−12, we find $c=5\frac{1}{2}$c=512, and for $b=2$b=2 we find $c=3$c=3.
However, the first pairing, delivering $y=\left(-\frac{1}{2}\right)^{-x}+5\frac{1}{2}$y=(−12)−x+512 needs to be discarded, because the base b is defined to be a positive number.
So the correct solution becomes $y=2^{-x}+3$y=2−x+3.
| $x$x | $-2$−2 | $-1$−1 | $0$0 | $1$1 | $2$2 |
|---|---|---|---|---|---|
| $y$y | $7$7 | $5$5 | $4$4 | $3.5$3.5 | $3.25$3.25 |
For an simple exponential form like $y=b^x$y=bx, one point will suffice. However, we need to take care to only allow valid equations. For example, given the point $\left(2,25\right)$(2,25), we can simply substitute the values into the equation. Thus $25=b^2$25=b2, and $b=\pm5$b=±5, but we must reject $b=-5$b=−5 since $b>0$b>0.
Practice questions
Question 8
Find the equation of the curve in the form $y=a^x$y=ax.
Question 9
Consider the given table of values.
| $x$x | $1$1 | $2$2 | $3$3 | $4$4 |
|---|---|---|---|---|
| $y$y | $10$10 | $100$100 | $1000$1000 | $10000$10000 |
Identify the common ratio between consecutive $y$y values.
State the equation relating $x$x and $y$y.
Find the value of $y$y when $x=10$x=10.
Question 10
If the electricity bill is not paid by the due date, the company charges a fee for each day that it is overdue. The table shows the fees.
| number of days after bill due ($x$x) | $1$1 | $2$2 | $3$3 | $4$4 | $5$5 | $6$6 |
|---|---|---|---|---|---|---|
| overdue fee in dollars ($y$y) | $4$4 | $8$8 | $16$16 | $32$32 | $64$64 | $128$128 |
At what rate is the fee increasing each day?
Select the two correct answers.
increasing by $2$2 each day
Adoubling each day
Btripling each day
Cincreasing by a factor of $2$2 each day
DIf the bill is paid $5$5 days overdue, what overdue fee will it incur?
Which function below models the overdue fee $y$y as a function of the number of days overdue $x$x?
$y=2\left(4^x\right)$y=2(4x)
A$y=4\left(2^{x-1}\right)$y=4(2x−1)
B$y=4^x$y=4x
C$y=4\left(2^x\right)$y=4(2x)
D