1. ## exponential equation transformation....

Hi,

I have the following problem that I am stuck on. I know it is probably something basic that I am missing and I would appreciate any and all help.

Thanks,
TC

Write 2 different exponential equations with different bases if there is a y intercept of 5 and an asymptote at y=3.

I have gotten this far....

y = a^x + 4

a is the base, I know that anything to the power of 0 equals 1 so this equation satisfies the y-intercept part but I am at a loss about finding some way to have an asymptote at y=3

2. Hello, TC!

This is trickier than you think . . .

Write two different exponential equations with different bases
which have a y-intercept of 5 and an asymptote at y=3.

An exponential function can have a coefficient in front: . $y \:=\:k\!\cdot\!a^x$

This equation has a horizontal asymptote at $y = 0$ (the x-axis).

To have an asymptote at $y = 3$, we must "raise" the function 3 units.
. . So we have: . $y \:=\:k\!\cdot\!a^x + 3$

To have a y-intercept of 5, we want: . $x = 0$ and $y = 5$
. . Substitute: . $5 \:=\:k\!\cdot\!a^0 + 3\quad\Rightarrow\quad k = 2$

Hence, the function is: . $y \:=\:2\!\cdot\!a^x + 3
$

This equation has a graph with a y-intercept of 5 and an asymptote at $y = 3$.
It already satisfies the problem . . . the base does not matter.

So pick any two positive bases (other than 1).

I'd offer: . $\begin{Bmatrix}y \;= \;2\!\cdot\!10^x + 3 \\ y \;= \;2\!\cdot\!e^x + 3\end{Bmatrix}$

. . but you can use your imagination . . .

Examples

. . $y \;=\;2\!\cdot\!\pi^x + 3$

. . $y \;=\;2\!\cdot\left(\frac{1}{4}\right)^x + 3 \;=\;2\!\cdot\left(4^{-1}\right)^x + 3\quad\Rightarrow\quad y \;=\;2\!\cdot\!4^{-x} + 3$

3. Thank you so much! I was approaching it from the y-intercept not the asymptote.

I appreciate the assistance.

TC

4. we could also just change the x to a -x

so two would be: $y = 2 \cdot e^x + 3 \mbox { and } y = 2 \cdot e^{-x} + 3$

the asymptotes and intercepts would remain the same, but you would reflect one graph in the y-axis to get the other