Question

which graph represents a function with an initial value of \\(\frac{1}{2}\\)?

Explanation:

Step1: Define initial value of function

An exponential function has the form $f(x) = a \cdot b^x$, where the initial value is $f(0) = a$. We need $a = \frac{1}{2}$, so $f(0) = \frac{1}{2}$. This means the graph passes through the point $(0, \frac{1}{2})$.

Step2: Analyze each graph

• Top graph: Passes through $(0, 2)$ → initial value 2.
• Second graph: Passes through $(0, \frac{1}{2})$ → initial value $\frac{1}{2}$.
• Third graph: Passes through $(0, 1)$ → initial value 1.
• Bottom graph: Passes through $(0, 1)$ → initial value 1.

Answer:

The second graph (the one with the curve passing through $(0, \frac{1}{2})$)