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### Course: Integrated math 1>Unit 14

Lesson 5: Exponential growth & decay

# Graphing exponential growth & decay

Graphing the exponential functions y=27⋅(⅓)ˣ and y=-30⋅2ˣ.

## Want to join the conversation?

• Would the asymptote for an exponential function ever be anything but zero?
• Yes. for example y=(2^x)+3
The horizontal asymptote is now y=3
• Is there previous lesson about asymptotes, or was it introduced it in this video the first time ?
• he didn't explain the asymptote at all?
• He probably assumed that everyone watching this knew what that was from a previous lesson, but I couldn't find anything myself about intro. to asymptotes in Algebra I.
An asymptote is basically a point where the line approaches but never meets, no matter how close it gets. For example, this is sort of unrelated to a graph situation but, if you take the number 1 and keep dividing it by 1/2 you will never get to zero.
• How does this relate to the real world
• JFails,

Exponential growth comes up a lot in business (investments), in social science (population growth), and in virology (how quickly a virus spreads).

Exponential decay comes up in audio engineering (sound levels decrease exponentially over longer distance), sports tournaments (if a tournament starts with 32 teams, how many rounds are there until the championship), archaeology(radiocarbon dating), and climate studies (atmospheric pressure decreases exponentially with increased altitude.)
• Is the asymptote always horizontal and on the zero line? Also how do you figure where the asymptote lies. Lastly, what is the asymptote and is this asked or used on the SAT. Whoever answers my questions will get an upvote from me! Please help me out here.
• No, there are vertical and other asymptotes as well. For exponential functions, the basic parent function is y=2^x which has a asymptote at x=0, but if it is shifted up or down by adding a constant (y = 2^x + k), the asymptote also shifts to x=k. I do not know what all is on the SAT, but if you have a rational function whose parent function is y = 1/x, you have a horizontal asymptote at x=0 and a vertical asymptote at y=0. These both can be shifted by adding (h,k) so that y = 1/(x-h) + k.
• In other higher level math courses, lines can occasionally cross a horizontal asymptote. Is this possible when graphing exponential functions?
• how would you graph a function with a negative base like y= -5^x
• I'm assuming you mean y=-(5)^x (the positioning of the negative sign is very important). In that case, you'll plot the graph of y=5^x and then reflect it over the y axis.

In the case where you want to take the exponent of a negative value, it gets a lot more tricky, because sometimes, the function is not defined for negative values. for x=0.5, y=(-5)^x is not defined on the real line (you're taking the square root of a negative number).
• Is it possible to move the asymptote so that it's not zero? What changes to the equation do we have to make? And also, in the video with negative 30, the graph is decaying. How might we have a graph that is growing but the value of the y-intercept is still negative?
• All functions can be shifted horizontally and/or vertically.
In general 𝑓(𝑥 − ℎ) + 𝑘 shifts 𝑓(𝑥) ℎ units to the right and 𝑘 units up.

Also, 𝑓(−𝑥) reflects 𝑓(𝑥) over the 𝑥-axis
and −𝑓(𝑥) reflects 𝑓(𝑥) over the 𝑦-axis.

– – –

As you mention, ℎ(𝑥) = 27⋅(1∕3)^𝑥 has a horizontal asymptote at 𝑦 = 0.
If we want to move that to, say, 𝑦 = −4,
then instead of graphing 𝑦 = ℎ(𝑥)
we should graph 𝑦 = ℎ(𝑥) − 4 = 27⋅(1∕3)^𝑥 − 4.

– – –

To make 𝑔(𝑥) = −30⋅2^𝑥 growing instead of decaying,
we can reflect it over the 𝑥-axis
by graphing 𝑦 = −𝑔(𝑥) = 30⋅2^𝑥

This of course changes the 𝑦-intercept to (0, 30), so if we still want it to have a negative 𝑦-intercept we could move it down maybe 40 units by graphing
𝑦 = −𝑔(𝑥) − 40 = 30⋅2^𝑥 − 40
• How do I find where the asymptote goes?
Thanks! :)