Do two points always determine an exponential function?

Task

An exponential function is a function of the form $f(x) = a b^x$ where $a$ is a real number and $b$ is a positive real number.

1. Suppose $P = (0,5)$ and $Q = (3,-3)$. For which real numbers $a$ and $b$ does the graph of the exponential function $f(x) = a b^x$ contain $P$? Explain. Do any of these graphs contain $Q$? Explain.
2. Suppose $R = (2,0)$. If $f(x) = a \cdot b^x$ is an exponential function whose graph contains $R$ what can you conclude about $a$? What is the graph of $f(x)$ in this case?

IM Commentary

This problem complements the problem ''Do two points always determine a linear function?'' There are two constraints on a pair of points $R_1$ and $R_2$ if there is an exponential function $f(x) = ab^x$, with $b>0$, whose graph contains $R_1$ and $R_2$. First, the $y$-coordinates of $R_1$ and $R_2$ cannot have different signs, that is it cannot be that one is positive while the other is negative. This is because the function $g(x) = b^x$ takes only positive values. Consequently, $f(x) = ab^x$ cannot take both positive and negative values. Furthermore, the only way $ab^x$ can be zero is if $a = 0$ and then the function is linear rather than exponential. As long as the $y$-coordinates of $R_1$ and $R_2$ are non-zero and have the same sign, there is a unique exponential function $f(x) = ab^x$ with $b>0$ whose graph contains $R_1$ and $R_2$.

Solution

1. If we evaluate $f(x)$ at $x = 0$ we find $$ f(0) = ab^{0} = ab^0 = a. $$ So the graph of $f(x) = ab^{x}$ contains $(0,5)$ when $a = 5$. In order to contain $Q$ we need $f(3) = -3$. If $f(x)$ goes through $(0,5)$ then it is of the form $f(x) = 5b^{x}$ for some $b$. When $x = 3$, $f$ takes the value $5b^{3}$. Since $b$ is the base of an exponential function, it must be positive. So regardless of what value $b$ is, $5b^{3}$ is a positive number and so can never be equal to $-3$. So none of the graphs of the exponential functions passing through $P$ also pass through $Q$. Below are graphs of the functions $f(x)=a\cdot b^x$ with $a=5$ and for $b$ equalling the values $\frac{1}{2}$, $\frac{2}{3}$, $1$, $\frac{3}{2}$, and $2$.

   
   
   

2. The graph of $f(x) = ab^{x}$ passes through $R = (2,0)$ when $f(2) = 0$. This is true when $$ ab^{x} = 0. $$ As we saw in part (a), $b^{x}$ only takes positive values so the only way $ab^{x}$ could be zero is if $a = 0$. Once $a = 0$ then, regardless of what value $b$ takes, $f(x) = 0$, a linear function. This is not considered an exponential function and so there is no exponential function whose graph contains the point $R$.