Comparing Exponential 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.

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

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

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.