To determine the graph that represents Chelsea's first step, we need to find the initial value of the function she is graphing. The function given is:
[tex]\[ f(x) = 20 \left( \frac{1}{4} \right)^x \][/tex]
The initial value of a function is the value of the function when [tex]\( x = 0 \)[/tex]. To find this, substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ f(0) = 20 \left( \frac{1}{4} \right)^0 \][/tex]
Recall that any non-zero number raised to the power of 0 is 1. Thus:
[tex]\[ \left( \frac{1}{4} \right)^0 = 1 \][/tex]
Now, substitute this back into the function:
[tex]\[ f(0) = 20 \times 1 \][/tex]
[tex]\[ f(0) = 20 \][/tex]
So, the initial value of the function is 20.
When Chelsea begins plotting the graph, she will plot the point where [tex]\( x = 0 \)[/tex] and [tex]\( y = 20 \)[/tex]. This point is [tex]\((0, 20)\)[/tex]. The graph she will start with will have this point, [tex]\((0, 20)\)[/tex], clearly marked.
