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In this lesson, we will use graphs to illustrate some of the laws of exponents, and compare exponential graphs to linear and quadratic ones.
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.
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.
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 $c$.
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 $b$.
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.
The equations $y_1=x^2$ and $y_2=5x$ are graphed to the left.
Now click on the Zoom Out button below the graph. The right edge of the graph now occurs at $x=6$.
Click on the Zoom Out button again, so the right edge of the graph now occurs at $x=8$.
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 $x^2$.
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.
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.
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.