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Problem 9 Medium Difficulty

Graph the given functions on a common screen. How are these graphs related?

$ y = 3^x $ , $ y = 10^x $ , $ y = (\frac{1}{3})^x $ , $ y = (\frac{1}{10})^x $

Answer

The functions with bases greater than $1\left(3^{x} \text { and } 10^{x}\right)$ are increasing, while those
with bases less than $1\left[\left(\frac{1}{3}\right)^{x} \text { and }\left(\frac{1}{10}\right)^{x}\right]$ are decreasing. The graph of $\left(\frac{1}{3}\right)^{x}$ is the reflection of that of $3^{x}$ about the $y$ -axis, and the graph of $\left(\frac{1}{10}\right)^{x}$ is the reflection of that of $10^{x}$ about the $y$ -axis. The graph of $10^{x}$ increases more quickly than that of $3^{x}$ for $x>0,$ and approaches 0 faster as $x \rightarrow-\infty$.

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Video Transcript

in this problem. We're using a graphing calculator to compare some graphs. We want to see what they have in common and how they were, how they are related. So we go to the y equals menu and we type the four equations in their Michael's three to the X 10 to the X 1/3 to the X and 1/10 to the X for a window. I've chosen to use negative 10 to 10 on my X axis and negative 2 to 20 on my Y axis. And those numbers can be different. Just fiddle around with until you find something you like. Okay, here are all four graphs together. So the blue one is y equals three to the X, and the black one is y equals 1/3 to the X, so noticed that the blue one and the black one are reflections Across the Y axis. The red one is y equals 10 to the X, and the pink one is y equals 1/10 to the X, so noticed that the red one and the pink one are also reflections across the y axis. So if you have the reciprocal of your base, you're going to get a reflection. The two functions that had a base greater than 13 to the X and tend to the X. Those were the exponential growth graphs, and the two that had a base between zero and one, 1/3 and 1/10. Those were the exponential decay graphs. One other thing we can note is that when you have a greater base such as Y equals tend to the X, it's going to be a steeper curve compared with when you have a smaller base, such as Michael's three to the X.

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