Question

on a piece of paper, graph $f(x) = 3 \cdot (0.5)^x$. then determine which answer choice matches the graph you drew. four graphs labeled a, b, c, d click here for long description. \\(\bigcirc\\) a. graph a \\(\bigcirc\\) b. graph b \\(\bigcirc\\) c. graph c \\(\bigcirc\\) d. graph d

Explanation:

Step1: Identify Function Type

The function \( f(x) = 3 \cdot (0.5)^x \) is an exponential function. Exponential functions have the form \( f(x)=ab^x \), where \( a = 3 \) (positive) and \( b = 0.5 \) (between 0 and 1), so it's a decreasing exponential function.

Step2: Analyze Key Points

• Y - intercept: When \( x = 0 \), \( f(0)=3\cdot(0.5)^0 = 3\cdot1 = 3 \). So the graph passes through \( (0, 3) \).
• End - behavior: As \( x

ightarrow+\infty \), \( (0.5)^x
ightarrow0 \), so \( f(x)
ightarrow0 \). As \( x
ightarrow-\infty \), \( (0.5)^x=(2)^{-x}
ightarrow+\infty \), so \( f(x)
ightarrow+\infty \) (but since \( x
ightarrow-\infty \), the left - hand side of the graph will rise, and the right - hand side will approach the x - axis).

Step3: Eliminate Options

• Option A: The graph in A seems to be an increasing exponential (since it rises on the right), but our function is decreasing, so A is out.
• Option B: Graph B is a parabola (quadratic function), not an exponential function, so B is out.
• Option D: Graph D is an increasing exponential (rises on the right), but our function is decreasing, so D is out.
• Option C: Graph C has a y - intercept around 3, is a decreasing exponential (approaches x - axis as \( x

ightarrow+\infty \) and rises as \( x
ightarrow-\infty \)), which matches our function.

Answer:

C. Graph C