Exponential Functions From Two Points

Practice ii points ever determine an exponential function?

Alignments to Content Standards: F-LE.A.2

Chore

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

Suppose $P = (0,v)$ and $Q = (iii,-three)$. For which real numbers $a$ and $b$ does the graph of the exponential function $f(x) = a b^10$ contain $P$? Explain. Do whatsoever of these graphs contain $Q$? Explicate.
Suppose $R = (2,0)$. If $f(10) = a \cdot b^x$ is an exponential function whose graph contains $R$ what can you conclude well-nigh $a$? What is the graph of $f(x)$ in this example?

IM Commentary

This problem complements the problem ''Exercise 2 points always determine a linear role?'' In that location are two constraints on a pair of points $R_1$ and $R_2$ if at that place is an exponential function $f(10) = ab^x$, with $b>0$, whose graph contains $R_1$ and $R_2$. Kickoff, the $y$-coordinates of $R_1$ and $R_2$ cannot accept 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, $f(10) = ab^ten$ cannot take both positive and negative values. Furthermore, the only way $ab^ten$ 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, in that location is a unique exponential function $f(x) = ab^10$ with $b>0$ whose graph contains $R_1$ and $R_2$.

Solution

If nosotros evaluate $f(x)$ at $x = 0$ we find $$ f(0) = ab^{0} = ab^0 = a. $$ So the graph of $f(10) = ab^{ten}$ contains $(0,5)$ when $a = 5$. In order to contain $Q$ we need $f(iii) = -3$. If $f(10)$ goes through $(0,5)$ and so it is of the form $f(x) = 5b^{ten}$ for some $b$. When $x = iii$, $f$ takes the value $5b^{3}$. Since $b$ is the base of an exponential role, it must exist positive. And so regardless of what value $b$ is, $5b^{iii}$ is a positive number and so can never exist equal to $-iii$. Then none of the graphs of the exponential functions passing through $P$ as well pass through $Q$. Below are graphs of the functions $f(10)=a\cdot b^ten$ with $a=5$ and for $b$ equalling the values $\frac{one}{two}$, $\frac{2}{3}$, $1$, $\frac{iii}{ii}$, and $ii$.
The graph of $f(10) = ab^{x}$ passes through $R = (two,0)$ when $f(2) = 0$. This is true when $$ ab^{x} = 0. $$ Equally nosotros saw in function (a), $b^{x}$ only takes positive values so the merely style $ab^{10}$ could exist nothing is if $a = 0$. Once $a = 0$ then, regardless of what value $b$ takes, $f(ten) = 0$, a linear part. This is not considered an exponential function and and then there is no exponential function whose graph contains the point $R$.