Comparing Exponential Graphs

In this lesson, we will use graphs to illustrate some of the laws of exponents, and compare exponential graphs to linear and quadratic ones.


Exponential laws and graphs

The equation $y=m ⋅ 2^x$ is graphed on the grid to the left, with a slider for $m$. Notice that changing $m$ from 1 scales (multiplies) $y$ by $m$, moving the points on the graph vertically.

When $m=2$, what is the $y$-intercept of the graph? (That is, where does the graph cross the $y$-axis?)
When $m=3$, what is the $y$-intercept of the graph?
What point do you think is the $y$-intercept of the graph when $m=32$?

Now the equation $y=2^{x+c}$ is graphed on the grid to the left, with a slider for $c$. Notice that changing $c$ moves the graph horizontally.

When $c=1$, what is the $y$-intercept of the graph?
When $c=2$, what is the $y$-intercept of the graph?
What will the $y$-intercept of the graph be when $c=5$? (You can find $y$-intercepts algebraically by setting $x=0$ in the equation.)

Now both of the equations $y_1=m ⋅ 2^x$ and $y_2 = 2^{x+c}$ are graphed to the left, with sliders for both $m$ and $c$.

Slide $c$ to 3, so the blue graph corresponds to the equation $y_2=2^{x+3}$. Now slide $m$ until the red graph completely overlaps the blue graph. When $c=3$ and the two graphs overlap completely, what is the equation for the red graph?
To see why the graphs overlap completely, use the laws of exponents to rewrite $2^{x+3}$ in the form $m ⋅ 2^x$.
For these values of $c$ and $m$, and any value of $x$, does $y_1$ equal $y_2$?
Now slide $c$ to $-1$. When $c=-1$ and the two graphs overlap completely, what is the equation for the red graph?
Using the laws of exponents, rewrite $2^{x-1}$ in the form $m ⋅ 2^x$, and then write the fraction you get as a decimal.
For these values of $c$ and $m$, and any value of $x$, does $y_1$ equal $y_2$?

The last question showed you some graphs that illustrated the law of exponents $a^{c+d}=a^c⋅ a^d$. In this question, we’ll look at graphs which illustrate the law $a^{cd}=(a^c)^d$.

The equations $y_1=a^{rx}$ and $y_2=b^x$ are graphed to the left, with sliders for $a$, $r$, and $b$.

Slide $a$ and $r$ until the red graph corresponds to the equation $y_1=3^{2x}$. Now slide $b$ until the two graphs overlap completely. What is the resulting equation for the blue graph?
To see why the graphs overlap completely, use the law of exponents $a^{cd}=(a^c)^d$ to rewrite $3^{2x}$ in the form $b^x$.
For these values of $a$, $r$, and $b$, and any value of $x$, does $y_1$ equal $y_2$?
Now, slide $a$ and $r$ until the red graph corresponds to the equation $y_1=2^{-x}$, and again slide $b$ until the two graphs overlap completely. What is the resulting equation for the blue graph?
Using the law of exponents $a^{cd}=(a^c)^d$, rewrite $2^{-x}$ in the form $b^x$ and then write the fraction you get as a decimal. (Remember that $-x=(-1) ⋅ x$.)
For these values of $a$, $r$, and $b$, and any value of $x$, does $y_1$ equal $y_2$?

Equations such as $y_1=a^{rx}$ and $y_2=b^x$ are called exponential because $x$ appears in an exponent. When $r > 0$ or $b > 1$ we have exponential growth. When $r < 0$ or $b < 1$ we have exponential decay.

Comparing growth rates

The equations $y_1=x^2$ and $y_2=5x$ are graphed to the left.

At the right edge of the graph (when $x=4$), is $x^2$ larger or smaller than $5x$?

Now click on the Zoom Out button below the graph. The right edge of the graph now occurs at $x=6$.

When $x=6$, is $x^2$ larger or smaller than $5x$?

Click on the Zoom Out button again, so the right edge of the graph now occurs at $x=8$.

When $x=8$, is $x^2$ larger or smaller than $5x$?

In fact, $x^2$ will be larger than $5x$ whenever $x>5$. To see this, notice that if $x>5$ then $x$ is positive, so we can multiply both sides of the inequality $x>5$ by $x$, to get the inequality $x^2>5x$.

This works for any linear expression, not just $5x$.

If $m$ is a positive real number, the value of $x^2$ is larger than the value of $mx$ whenever $x>m$.

That is, positive quadratic expressions always get bigger than positive linear expressions if $x$ is big enough.

In the previous question, you learned that quadratic expressions are always bigger than linear expressions when $x$ is large. We’ll now compare the size of exponential expressions to quadratic expressions when $x$ is large, starting by comparing $2^x$ to $x^2$.

The equations $y_1=2^x$ and $y_2=x^2$ are graphed on the grid to the left. At the right edge of the graph (when $x=6$), is $2^x$ larger or smaller than $x^2$?
By zooming out on the graph, can you find any value of $x$ with $x>6$ where $2^x$ is smaller than $x^2$?

In the last question, you compared $2^x$ to $x^2$. In this question, you’ll compare a much smaller exponential expression ($1.2^x$) to a much larger quadratic expression ($5x^2$).

The equations $y_1=1.2^x$ and $y_2=5x^2$ are graphed to the left. Notice that $y_1$ looks like it is growing much more slowly than $y_2$ over the entire range of the graph.

Use the Zoom Out button until the right edge of the graph occurs where $x=20$. When $x=20$, is $1.2^x$ larger or smaller than $5x^2$?
Now, use the Zoom Out button repeatedly until the right edge of the graph occurs where $x=60$. When $x=60$, is $1.2^x$ larger or smaller than $5x^2$?

In fact, just as $x^2$ eventually gets bigger than any expression $mx$, exponential expressions all eventually get bigger than any quadratic expression:

If $a>1$ and $x$ is large enough, then $a^x > mx^2$ for any real number $m$.

However, in this case it is more difficult to determine exactly how large $x$ must be in order for this inequality to be true.

Fill in the tables below, using a calculator as necessary.

$x$$5x$$x^2$$2^x$

$x$$10x$$10x^2$$10^x$

Notice that when $x$ is large, the exponential expressions $2^x$ and $10^x$ are much larger than the quadratic and linear expressions in the table.