Match each exponential function to its graph. $$ \begin{array}{l}d(x)=3\left(\frac{1}{4}\right)^{x-2}-5 \ s(x)=3(4)^{x+2}-5 \ z(x)=3(4)^{x+2}+5 \ p(x)=3\left(\frac{1}{4}\right)^{x-2}+5\end{array} $$

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$$ d(x)=3\left(\frac{1}{4} ight)^{x-2}-5 $$ • Compared to a standard exponential $$ f(x)=A\cdot b^{x-h}+k $$, here • $$ A=3 $$ • $$ b=\frac{1}{4}<1 $$ (implying a “decay‐type” behavior as $$ x $$ increases) • Horizontal shift: $$ h=2 $$ (shifted right 2) • Vertical shift: $$ k=-5 $$ (asymptote at $$ y=-5 $$) Thus, its graph has a horizontal asymptote at $$ y=-5 $$. As $$ x o+\infty $$, $$ d(x) o-5 $$; as $$ x o-\infty $$, $$ d(x) o+\infty $$. $$ s(x)=3(4)^{x+2}-5 $$ • Here, • $$ A=3 $$ • $$ b=4>1 $$ (an increasing or “growth‐type” exponential) • Horizontal shift: $$ h=-2 $$ (shifted left 2, since the exponent is $$ x+2 $$) • Vertical shift: $$ k=-5 $$ (asymptote at $$ y=-5 $$) So, its graph has asymptote $$ y=-5 $$, with $$ s(x) o-5 $$ as $$ x o-\infty $$ and $$ s(x) o+\infty $$ as $$ x o+\infty $$. $$ z(x)=3(4)^{x+2}+5 $$ • Here, • $$ A=3 $$ • $$ b=4>1 $$ (growth) • Horizontal shift: $$ h=-2 $$ (shifted left 2) • Vertical shift: $$ k=+5 $$ (asymptote at $$ y=5 $$) Thus, its graph has an asymptote at $$ y=5 $$ with $$ z(x) o5 $$ as $$ x o-\infty $$ and $$ z(x) o+\infty $$ as $$ x o+\infty $$. $$ p(x)=3\left(\frac{1}{4} ight)^{x-2}+5 $$ • Here, • $$ A=3 $$ • $$ b=\frac{1}{4}<1 $$ (decay) • Horizontal shift: $$ h=2 $$ (shifted right 2) • Vertical shift: $$ k=+5 $$ (asymptote at $$ y=5 $$) Thus, its graph has asymptote $$ y=5 $$. As $$ x o+\infty $$, $$ p(x) o5 $$; as $$ x o-\infty $$, $$ p(x) o+\infty $$. Summary of matching: - The graph with horizontal asymptote $$ y=-5 $$ that **decreases** as $$ x $$ increases is the graph of $$ d(x)=3\left(\frac{1}{4} ight)^{x-2}-5 $$. - The graph with horizontal asymptote $$ y=-5 $$ that **increases** as $$ x $$ increases is the graph of $$ s(x)=3(4)^{x+2}-5 $$. - The graph with horizontal asymptote $$ y=5 $$ that **increases** as $$ x $$ increases is the graph of $$ z(x)=3(4)^{x+2}+5 $$. - The graph with horizontal asymptote $$ y=5 $$ that **decreases** as $$ x $$ increases is the graph of $$ p(x)=3\left(\frac{1}{4} ight)^{x-2}+5 $$. - $$ d(x)=3\left(\frac{1}{4} ight)^{x-2}-5 $$ decreases and has a horizontal asymptote at $$ y=-5 $$. - $$ s(x)=3(4)^{x+2}-5 $$ increases and has a horizontal asymptote at $$ y=-5 $$. - $$ z(x)=3(4)^{x+2}+5 $$ increases and has a horizontal asymptote at $$ y=5 $$. - $$ p(x)=3\left(\frac{1}{4} ight)^{x-2}+5 $$ decreases and has a horizontal asymptote at $$ y=5 $$.

Match each exponential function to its graph.

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