# 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 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 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.