Question

what is the initial value of the exponential function shown on the graph? 0 1 2 4

Explanation:

Step1: Recall initial value definition

The initial value of an exponential function \( y = a \cdot b^x \) is the value when \( x = 0 \), i.e., \( y(0)=a \). So we need to find the \( y \)-intercept (where \( x = 0 \)) of the graph.

Step2: Find the \( y \)-intercept from the graph

Looking at the graph, when \( x = 0 \) (the \( y \)-axis), the graph intersects the \( y \)-axis at \( y = 4 \)? Wait, no, wait. Wait, let's check the grid. Wait, the graph: when \( x = 0 \), the point is at \( y = 4 \)? Wait, no, the options include 4. Wait, let's re-examine. The graph: at \( x = 0 \), the \( y \)-coordinate is 4? Wait, the options are 0,1,2,4. So when \( x = 0 \), the value of the exponential function (the \( y \)-intercept) is 4? Wait, no, wait the graph: let's see the grid. The \( y \)-axis: the point at \( x = 0 \) is at \( y = 4 \)? Wait, the graph starts at \( x = 0 \) with \( y = 4 \)? Wait, no, looking at the graph, when \( x = 0 \), the \( y \)-value is 4? Wait, the options have 4 as an option. Wait, maybe I misread. Wait, the initial value is when \( x = 0 \), so we find the \( y \)-intercept. From the graph, at \( x = 0 \), the \( y \)-coordinate is 4. So the initial value is 4.

Wait, but let's confirm. The exponential function's initial value is \( a \) in \( y = a b^x \), so when \( x = 0 \), \( y = a \cdot 1 = a \). So the \( y \)-intercept is the initial value. Looking at the graph, when \( x = 0 \), the graph is at \( y = 4 \). So the initial value is 4.

Answer:

D. 4 (assuming the options are labeled as A. 0, B. 1, C. 2, D. 4) Wait, the original options are: 0, 1, 2, 4. So the correct option is the one with 4. So the answer is 4 (the option with 4, e.g., if the options are A. 0, B. 1, C. 2, D. 4, then D. 4).