Question

which equation best represents the function graphed below? options: $f(x) = 2(.5)^x$, $f(x) = -(.3)^x$, $f(x) = -2(0.21)^x + 8$, $f(x) = 3^x - 1$, $f(x) = -3(2)^x + 7$

Explanation:

Step1: Analyze the graph's behavior

The graph shows an exponential growth (increasing rapidly as \( x \) increases) and passes near \( (0, 0) \) (since at \( x = 0 \), let's check the options).

Step2: Check \( x = 0 \) for each option

• For \( f(x)=2(0.5)^x \): \( f(0)=2(0.5)^0 = 2 \), not matching the graph (which is near 0 at \( x=0 \)).
• For \( f(x)=-(0.3)^x \): \( f(0)= -1 \), negative, graph is positive near \( x=0 \), so no.
• For \( f(x)=-2(0.21)^x + 8 \): \( f(0)= -2 + 8 = 6 \), too big.
• For \( f(x)=3^x - 1 \): \( f(0)=3^0 - 1 = 1 - 1 = 0 \), matches the y - intercept (near (0,0)). Also, \( 3^x \) is exponential growth, so \( 3^x - 1 \) will grow exponentially, matching the graph's shape.
• For \( f(x)=-3(2)^x + 7 \): \( f(0)= -3 + 7 = 4 \), too big.

Answer:

\( f(x) = 3^x - 1 \) (the fourth option: \( f(x) = 3^x - 1 \))