Question

which function transforms the graph of the parent function $f(x)=2^x$ by reflecting it across the $y$-axis and translating it up 5 units?
(1 point)
$\bigcirc$ $g(x)=-2^{x+5}$
$\bigcirc$ $g(x)=2^{-x+5}$
$\bigcirc$ $g(x)=-2^x + 5$
$\bigcirc$ $g(x)=2^{-x}+5$

Explanation:

Step1: Reflect across y - axis

To reflect a function \(y = f(x)\) across the \(y\) - axis, we replace \(x\) with \(-x\). For the parent function \(f(x)=2^{x}\), after reflecting across the \(y\) - axis, the function becomes \(y = 2^{-x}\).

Step2: Translate up 5 units

To translate a function \(y = h(x)\) up by \(k\) units, we use the transformation \(y=h(x)+k\). Here, \(h(x) = 2^{-x}\) and \(k = 5\). So after translating up 5 units, the function becomes \(g(x)=2^{-x}+5\).

Let's analyze the other options:

• For \(g(x)=-2^{x + 5}\): This is a reflection across the \(x\) - axis (because of the negative sign in front) and a horizontal translation, not what we want.
• For \(g(x)=2^{-x+5}=2^{-(x - 5)}\): This is a horizontal translation, not a vertical translation up 5 units.
• For \(g(x)=-2^{x}+5\): This is a reflection across the \(x\) - axis (negative sign in front of \(2^{x}\)) and a vertical translation up 5 units, not a reflection across the \(y\) - axis.

Answer:

\(g(x)=2^{-x}+5\) (the fourth option: \(g(x)=2^{-x}+5\))