A base form of an exponential function is $f\left(x\right)=a^x$f(x)=ax, where $a$a is a positive number and the variable is in the exponent. Unlike linear functions which increase or decrease by a constant, exponential functions increase or decrease by a constant multiplier. Let's first look at cases for $a>1$a>1, where we have exponential growth and identify key characteristics of such functions.
Let's create a table for the function $y=2^x$y=2x:
$x$x | $-4$−4 | $-3$−3 | $-2$−2 | $-1$−1 | $0$0 | $1$1 | $2$2 | $3$3 | $4$4 |
---|---|---|---|---|---|---|---|---|---|
$y$y | $\frac{1}{16}$116 | $\frac{1}{8}$18 | $\frac{1}{4}$14 | $\frac{1}{2}$12 | $1$1 | $2$2 | $4$4 | $8$8 | $16$16 |
We can see our familiar powers of two and as $x$x increases by one, the $y$y values are increasing by a constant multiplier - here they are doubling. This causes the differences between successive $y$y values to grow and hence, $y$y is increasing at an increasing rate. Let's look at what this function looks like when we graph it.
Key features:
How does this compare to other values of $a$a? Let's graph $y=2^x$y=2x, $y=3^x$y=3x and $y=5^x$y=5x on the same graph. You can create a table for each to confirm the values sketched in the graph below:
We can see all of the key features mentioned above were not unique to the graph of $y=2^x$y=2x.
Key features:
The difference is that for $x>0$x>0 the higher the $a$a value the faster the graph increases. Each graph goes through the point $\left(1,a\right)$(1,a) and we can see the larger the $a$a value the higher this point will be.
For $x<0$x<0 the higher the $a$a value the quicker the graph approaches the horizontal asymptote.
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
Decreasing
How would you describe the rate of increase of the function?
As $x$x increases, the function increases at a constant rate.
As $x$x increases, the function increases at a faster and faster rate.
As $x$x increases, the function increases at a slower and slower rate.
What is the domain of the function?
all real $x$x
$x\ge0$x≥0
$x<0$x<0
$x>0$x>0
What is the range of the function?
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.
The $y$y-value of every point on the graph is positive.
The $y$y-value of every point on the graph is an integer.
The $y$y-value of most points on the graph is positive, and the $y$y-value at one point is $0$0.
As 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
$-4$−4
$0$0
What 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.
An $x$x-intercept of the curve.
A $y$y-intercept of the curve.
Let's create a table for the function $y=\left(\frac{1}{2}\right)^x$y=(12)x:
$x$x | $-4$−4 | $-3$−3 | $-2$−2 | $-1$−1 | $0$0 | $1$1 | $2$2 | $3$3 | $4$4 |
---|---|---|---|---|---|---|---|---|---|
$y$y | $16$16 | $8$8 | $4$4 | $2$2 | $1$1 | $\frac{1}{2}$12 | $\frac{1}{4}$14 | $\frac{1}{8}$18 | $\frac{1}{16}$116 |
Again we can see our familiar powers of two but this time as $x$x increases by one the $y$y values are decreasing by a constant multiplier - here they are halving. The differences between successive $y$y values is shrinking and hence, $y$y is decreasing at a decreasing rate. Let's look at what this this function looks like when we graph it.
Key features:
And as before:
All graphs of the form $y=a^x$y=ax where $00<a<1 will have these similar key features. They will all be exponential decreasing (decaying) functions, since our multiplier is a fraction.
How did the graph and table of $y=\left(\frac{1}{2}\right)^x$y=(12)x compare that of $y=2^x$y=2x? Can you see they are a reflection of each other in the $y$y-axis? The values in the tables were reversed and the $y$y-value for $y=\left(\frac{1}{2}\right)^x$y=(12)x at $x=k$x=k was the same as $y=2^x$y=2x at $x=-k$x=−k. We can see why this is the case by using our index laws to rewrite $y=\left(\frac{1}{2}\right)^x$y=(12)x as follows:
Let $g(x)=\left(\frac{1}{2}\right)^x$g(x)=(12)x and $f(x)=2^x$f(x)=2x.
$g(x)$g(x) | $=$= | $\left(\frac{1}{2}\right)^x$(12)x |
$=$= | $\left(2^{-1}\right)^x$(2−1)x | |
$=$= | $\left(2\right)^{-x}$(2)−x | |
$=$= | $f\left(-x\right)$f(−x) |
In general, for $a>0$a>0 the graph of $g\left(x\right)=\left(\frac{1}{a}\right)^x$g(x)=(1a)x is equivalent to $g\left(x\right)=a^{-x}$g(x)=a−x, which is a decreasing exponential function and a reflection of the graph $f(x)=a^x$f(x)=ax in the $y$y-axis.
Consider the graphs of the functions $y=4^x$y=4x and $y=\left(\frac{1}{4}\right)^x$y=(14)x.
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Which function is an increasing function?
$y=\left(\frac{1}{4}\right)^x$y=(14)x
$y=4^x$y=4x
How would you describe the rate of increase of $y=4^x$y=4x?
$y$y is increasing at a constant rate
$y$y is increasing at a decreasing rate
$y$y is increasing at an increasing rate
Consider the function $y=\left(\frac{1}{2}\right)^x$y=(12)x
Which two functions are equivalent to $y=\left(\frac{1}{2}\right)^x$y=(12)x ?
$y=\frac{1}{2^x}$y=12x
$y=2^{-x}$y=2−x
$y=-2^x$y=−2x
$y=-2^{-x}$y=−2−x
Sketch a graph of $y=2^x$y=2x on the coordinate plane.
Using the result of the first part, sketch $y=\left(\frac{1}{2}\right)^x$y=(12)x on the same coordinate plane.
Consider the function $y=8^{-x}$y=8−x.
Can the value of $y$y ever be negative?
Yes
No
As the value of $x$x increases towards $\infty$∞ what value does $y$y approach?
$8$8
$-\infty$−∞
$\infty$∞
$0$0
As the value of $x$x decreases towards $-\infty$−∞, what value does $y$y approach?
$0$0
$\infty$∞
$8$8
$-\infty$−∞
Can the value of $y$y ever be equal to $0$0?
Yes
No
Determine the $y$y-value of the $y$y-intercept of the curve.
How many $x$x-intercepts does the curve have?
Which of the following could be the graph of $y=8^{-x}$y=8−x?