Question

match each graph below with the appropriate function.
a)
options: ( f(x) = -2^{x - 2} ), ( f(x) = -2^x ), ( f(x) = 2^{x - 2} ), ( f(x) = 2^x )

Explanation:

Step1: Analyze the y-intercept

To find the y-intercept, set \( x = 0 \) in each function.

• For \( f(x)=-2^{x - 2} \), \( f(0)=-2^{-2}=-\frac{1}{4} \) (not matching the graph's y-intercept which is -1? Wait, no, let's re - calculate. Wait, the graph crosses the y - axis at \( x = 0 \). Let's look at the graph: when \( x = 0 \), the y - value is - 1? Wait, no, let's check the functions again.
• For \( f(x)=-2^{x} \), when \( x = 0 \), \( f(0)=-2^{0}=-1 \).
• For \( f(x)=2^{x - 2} \), \( f(0)=2^{-2}=\frac{1}{4} \) (positive, but the graph is negative at \( x = 0 \), so eliminate).
• For \( f(x)=2^{x} \), \( f(0)=2^{0}=1 \) (positive, eliminate as the graph is negative at \( x = 0 \)).

Step2: Analyze the shape (exponential decay with reflection)

The function \( y = 2^{x} \) is an exponential growth function. The function \( y=-2^{x} \) is a reflection of \( y = 2^{x} \) over the x - axis, so it is an exponential decay (since as \( x\) increases, \( 2^{x}\) increases, and with the negative sign, it decreases towards \( -\infty \)), which matches the shape of the graph (as \( x\) increases, the graph goes down, and as \( x\) decreases, it approaches 0 from below? Wait, no, when \( x\) is negative, \( -2^{x}=-\frac{1}{2^{|x|}} \), so as \( x\to-\infty \), \( -2^{x}\to0^{-} \), and as \( x\to+\infty \), \( -2^{x}\to-\infty \), which matches the graph. The function \( f(x)=-2^{x - 2}=- \frac{2^{x}}{4} \), when \( x = 0 \), it's \( -\frac{1}{4} \), but our graph at \( x = 0 \) is - 1, so \( f(x)=-2^{x} \) is correct.

Answer:

\( f(x)=-2^{x} \)