What constrains the following functional equation of exponents?

Question

If I am not incorrect,the standard (Is it he standard?) form of an exponential equation is $$y=ab^{x-h}+k$$ What are the constraints on this equation, or in other words, how do each of the variables make the equation graph on the coordinate plane? For instance, if we had $$y=2(3^{x-2})+ 10 $$ how would this graph flow,and why?

Answer 1

I assume you know what the graph of $$y=e^x$$ looks like.

The graph of $$y=b^x$$ is similar, only the slope of the tangent line at $x=0$ is $\log b$ instead of 1.

The graph of $$y=b^{x-h}$$ is similar to that of $y=b^x$, only pushed $h$ units to the right.

The graph of $$y=ab^{x-h}$$ is similar to that of $y=b^{x-h}$, just that the distances to the $x$-axis are magnified by a factor $a$.

And the graph of $$y=ab^{x-h}+k$$ is similar to that of $y=ab^{x-h}$, just pushed $k$ units up.

Answer 2

Just play a little with logarithms and you will quickly arrive to $$y=a e^{(x-h) \\log (b)}+k$$ which shows you a classical exponential function which will not make any problem to plot or evaluate (provided that $b$ be a real positive number).

Answer 3

Have you ever learned about transformations before (e.g. how the graph of $x^2-2x$ differs from the graph of $x^2-2x+3$). Many of the rules, if not all, of transformations applies to your question. Let's start with $b^x$. It looks like any other exponential equation would: staying very close to $0$ on the left, and then suddenly growing very fast on the right (this is what the phrase "growing exponentially" means). Adding a $-h$ to the exponent (now the expression looks like $b^{x-h}$) will shift the graph $h$ units to the right. Adding the $a$ (now the expression looks like $ab^{x-h}$) does not seem to change the graph, but it actually does. The $a$ makes the graph "grow" $a$ times faster. Finally, the $k$ (now the expression looks like $ab^{x-h}+k$) shifts the graph $k$ units upward.

To conclude, the graph of the equation you mentioned ($y=2(3^{x-2})+10$) will be shifted $2$ units to the right, grow $2$ times faster, and be shifted $10$ units upward. All these transformations will be applied to the original graph of $3^x$. I hope I helped you!