Question

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Explanation:

Analyze the base function features

Assume the base function is an exponential growth function \(f(x) = 2^x\), which has a \(y\)-intercept at \((0,1)\), an asymptote at \(y=0\), and increases as \(x\) increases.

Match each transformation

• For \(j(x) = f(x+2)\):

$$ j(0) = f(2) = 2^2 = 4 \implies y\text{-intercept at } (0,4) $$

• For \(h(x) = f(x) + 2\):

$$ \text{Vertical shift up by 2} \implies \text{asymptote at } y = 2 $$

• For \(g(x) = 2f(x)\):

$$ g(0) = 2f(0) = 2(1) = 2 \implies y\text{-intercept at } (0,2) $$

• For \(m(x) = -f(x)\):

$$ \text{Reflection across the } x\text{-axis} \implies \text{function decreases as } x \text{ increases} $$

Answer:

Transformation	Feature
\(h(x) = f(x) + 2\)	asymptote of \(y = 2\)
\(g(x) = 2f(x)\)	\(y\)-intercept at \((0,2)\)
\(m(x) = -f(x)\)	function decreases as \(x\) increases