I am faced with a problem that I cannot seem to solve. Here it is:
The graph $y = a^x + b$ is shown below, find the EXACT values of $a$ and $b$.
The graph is an exponential graph, it has a $y$-intercept at $(0,-4)$ and it also labels a point at $(2,5)$ in quadrant $1$.
Any help would be great, Thank you all!
Best Answer
The equations obtained by plugging in $(x,y)=(0,-4)$ is $-4=a^0+b=1+b$ (since $a^0=1$). This gives that $b=-5$. When the other point $(2,5)$ is plugged into the now updated $y=a^x-5$ it gives $5=a^2-5$ which is $a^2=10$. Usually the base of an exponential function is taken to be positive, so that the function is defined for all real $x$. If that is your convention, $a=\sqrt{10}$ is the only choice.
(Otherwise if one is content with using a negative base and restricting to only certain rational exponents, one could choose $-\sqrt{10}$ for $a$, but this causes problems when raised to real variable $x$...)
So this gives $y=(\sqrt{10})^x-5$ which may be rewritten to avoid the squareroot sign as $y=10^{x/2}-5.$
Check: at $x=0$ we get $y=10^0-5=-4$, and at $x=2$ it is $y=10^{2/2}-5=10-5=5.$