Ramon is graphing the function [tex]$f(x)=3(4)^x$[/tex]. He begins by plotting the initial value. Which graph represents his initial step?
Answer (2)
To find Ramon's initial step in graphing the function f ( x ) = 3 ( 4 ) x , we calculate the value of the function at x = 0 . This gives us f ( 0 ) = 3 , corresponding to the point (0, 3) on the graph. Thus, the point (0, 3) is the initial value that Ramon would plot.
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The problem asks for the initial value of the function f ( x ) = 3 ( 4 ) x .
The initial value is found by setting x = 0 in the function.
Calculate f ( 0 ) = 3 ( 4 ) 0 = 3 ( 1 ) = 3 .
The initial value is 3, corresponding to the point (0, 3) on the graph: 3 .
Explanation
Understanding the Problem The problem states that Ramon is graphing the function f ( x ) = 3 ( 4 ) x and begins by plotting the initial value. The initial value of a function is the value of the function when x = 0 .
Calculating the Initial Value To find the initial value, we need to calculate f ( 0 ) . This means we substitute x = 0 into the function: f ( 0 ) = 3 ( 4 ) 0
Simplifying the Expression Since any non-zero number raised to the power of 0 is 1, we have 4 0 = 1 . Therefore, f ( 0 ) = 3 ( 1 ) = 3
Determining the Initial Point The initial value of the function is 3. This corresponds to the point (0, 3) on the graph. Ramon would plot the point (0, 3) as his initial step.
Examples
Understanding initial values is crucial in many real-world scenarios. For example, in finance, the initial investment in an account can be modeled using an exponential function, similar to the one in the problem. If you invest 3000 inana cco u n tt ha t g ro w s a t a r a t eo f 4 f(x) = 3000(1.04)^x$. The initial value, $3000, represents the starting amount before any growth occurs. Plotting this initial value helps visualize the starting point of your investment's growth.
