1. The problem is to match each given function with its corresponding graph based on the description of their shapes. 2. The functions are: - $t(x) = \frac{2}{3}3^x$ - $h(x) = -\frac{2}{3}3^x$ - $q(x) = \frac{2}{3}\left(\frac{1}{2}\right)^x$ - $v(x) = -\frac{2}{3}\left(\frac{1}{2}\right)^x$ 3. Important rules: - Exponential growth occurs when the base of the exponent is greater than 1. - Exponential decay occurs when the base is between 0 and 1. - A negative coefficient reflects the graph across the x-axis. 4. Analyze each function: - $t(x) = \frac{2}{3}3^x$: base 3 > 1, positive coefficient, so exponential growth starting near 0 for negative $x$ and increasing rapidly. - $h(x) = -\frac{2}{3}3^x$: base 3 > 1, negative coefficient, so exponential growth reflected below x-axis, starting near 0 for negative $x$ and decreasing rapidly to negative values. - $q(x) = \frac{2}{3}\left(\frac{1}{2}\right)^x$: base $\frac{1}{2} < 1$, positive coefficient, so exponential decay starting high for negative $x$ and approaching 0 as $x$ increases. - $v(x) = -\frac{2}{3}\left(\frac{1}{2}\right)^x$: base $\frac{1}{2} < 1$, negative coefficient, so exponential decay reflected below x-axis, starting low (negative) for negative $x$ and approaching 0 from below as $x$ increases. 5. Match with graphs: - Top-left graph: exponential growth near 0 for negative $x$, increasing rapidly to 10 at $x=2$ matches $t(x)$. - Top-right graph: exponential growth reflected below x-axis, decreasing to about -10 at $x=10$ matches $h(x)$. - Bottom-left graph: exponential decay starting near 10 at $x=-10$, approaching 0 matches $q(x)$. - Bottom-right graph: exponential decay reflected below x-axis, starting near -10 at $x=-10$, approaching 0 matches $v(x)$. Final answer: - Top-left: $t(x) = \frac{2}{3}3^x$ - Top-right: $h(x) = -\frac{2}{3}3^x$ - Bottom-left: $q(x) = \frac{2}{3}\left(\frac{1}{2}\right)^x$ - Bottom-right: $v(x) = -\frac{2}{3}\left(\frac{1}{2}\right)^x$