Answer :
To solve this problem, we need to identify the exponential function that fits the given table data and then determine its y-intercept.
### Step-by-Step Solution:
1. Identify the Form of the Exponential Function:
An exponential function can generally be written in the form:
[tex]\[ f(x) = a \cdot b^x \][/tex]
where [tex]\(a\)[/tex] is the initial value (y-intercept when [tex]\(x=0\)[/tex]), and [tex]\(b\)[/tex] is the base of the exponential function.
2. Determine the Relationship Between the Provided Points:
Given the points:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 2 & 3 & 4 & 5 \\ \hline f(x) & 16 & 8 & 4 & 2 \\ \hline \end{array} \][/tex]
Notice that from [tex]\(x = 2\)[/tex] to [tex]\(x = 3\)[/tex], [tex]\(f(x)\)[/tex] is halved: [tex]\( \frac{16}{2} = 8 \)[/tex]
Similarly, from [tex]\(x = 3\)[/tex] to [tex]\(x = 4\)[/tex], [tex]\(f(x)\)[/tex] is halved again: [tex]\( \frac{8}{2} = 4 \)[/tex]
This pattern continues for each increment of [tex]\(x\)[/tex] leading to the values halving every step.
3. Mathematically Determine the Base [tex]\(b\)[/tex]:
Since the function's values halve for each unit increase in [tex]\(x\)[/tex], the base [tex]\(b\)[/tex] of the exponential function is [tex]\( \frac{1}{2} \)[/tex].
With [tex]\( b = \frac{1}{2} \)[/tex], the function's structure is:
[tex]\[ f(x) = a \cdot \left( \frac{1}{2} \right)^x \][/tex]
4. Calculate the Coefficient [tex]\(a\)[/tex]:
Use one of the provided points to solve for [tex]\(a\)[/tex]. Let's use the point [tex]\((x = 2, f(x) = 16)\)[/tex].
[tex]\[ f(2) = a \cdot \left( \frac{1}{2} \right)^2 = 16 \][/tex]
[tex]\[ a \cdot \frac{1}{4} = 16 \][/tex]
[tex]\[ a = 16 \cdot 4 = 64 \][/tex]
The exponential function is:
[tex]\[ f(x) = 64 \cdot \left( \frac{1}{2} \right)^x \][/tex]
5. Identify the Y-Intercept:
The y-intercept occurs when [tex]\( x = 0 \)[/tex].
[tex]\[ f(0) = 64 \cdot \left( \frac{1}{2} \right)^0 = 64 \cdot 1 = 64 \][/tex]
6. Graph [tex]\(f(x)\)[/tex]:
To graph the exponential function [tex]\( f(x) = 64 \cdot \left( \frac{1}{2} \right)^x \)[/tex], plot the given points and the computed y-intercept:
[tex]\[ \begin{align*} \text{At } x = 0, & \quad f(0) = 64 \\ \text{At } x = 2, & \quad f(2) = 16 \\ \text{At } x = 3, & \quad f(3) = 8 \\ \text{At } x = 4, & \quad f(4) = 4 \\ \text{At } x = 5, & \quad f(5) = 2 \end{align*} \][/tex]
The plot should display these points and reveal the exponential decay pattern, starting at the y-intercept 64 when [tex]\(x = 0\)[/tex].
### Conclusion:
The y-intercept for the exponential function represented by the table is [tex]\( \boxed{64} \)[/tex].
### Step-by-Step Solution:
1. Identify the Form of the Exponential Function:
An exponential function can generally be written in the form:
[tex]\[ f(x) = a \cdot b^x \][/tex]
where [tex]\(a\)[/tex] is the initial value (y-intercept when [tex]\(x=0\)[/tex]), and [tex]\(b\)[/tex] is the base of the exponential function.
2. Determine the Relationship Between the Provided Points:
Given the points:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 2 & 3 & 4 & 5 \\ \hline f(x) & 16 & 8 & 4 & 2 \\ \hline \end{array} \][/tex]
Notice that from [tex]\(x = 2\)[/tex] to [tex]\(x = 3\)[/tex], [tex]\(f(x)\)[/tex] is halved: [tex]\( \frac{16}{2} = 8 \)[/tex]
Similarly, from [tex]\(x = 3\)[/tex] to [tex]\(x = 4\)[/tex], [tex]\(f(x)\)[/tex] is halved again: [tex]\( \frac{8}{2} = 4 \)[/tex]
This pattern continues for each increment of [tex]\(x\)[/tex] leading to the values halving every step.
3. Mathematically Determine the Base [tex]\(b\)[/tex]:
Since the function's values halve for each unit increase in [tex]\(x\)[/tex], the base [tex]\(b\)[/tex] of the exponential function is [tex]\( \frac{1}{2} \)[/tex].
With [tex]\( b = \frac{1}{2} \)[/tex], the function's structure is:
[tex]\[ f(x) = a \cdot \left( \frac{1}{2} \right)^x \][/tex]
4. Calculate the Coefficient [tex]\(a\)[/tex]:
Use one of the provided points to solve for [tex]\(a\)[/tex]. Let's use the point [tex]\((x = 2, f(x) = 16)\)[/tex].
[tex]\[ f(2) = a \cdot \left( \frac{1}{2} \right)^2 = 16 \][/tex]
[tex]\[ a \cdot \frac{1}{4} = 16 \][/tex]
[tex]\[ a = 16 \cdot 4 = 64 \][/tex]
The exponential function is:
[tex]\[ f(x) = 64 \cdot \left( \frac{1}{2} \right)^x \][/tex]
5. Identify the Y-Intercept:
The y-intercept occurs when [tex]\( x = 0 \)[/tex].
[tex]\[ f(0) = 64 \cdot \left( \frac{1}{2} \right)^0 = 64 \cdot 1 = 64 \][/tex]
6. Graph [tex]\(f(x)\)[/tex]:
To graph the exponential function [tex]\( f(x) = 64 \cdot \left( \frac{1}{2} \right)^x \)[/tex], plot the given points and the computed y-intercept:
[tex]\[ \begin{align*} \text{At } x = 0, & \quad f(0) = 64 \\ \text{At } x = 2, & \quad f(2) = 16 \\ \text{At } x = 3, & \quad f(3) = 8 \\ \text{At } x = 4, & \quad f(4) = 4 \\ \text{At } x = 5, & \quad f(5) = 2 \end{align*} \][/tex]
The plot should display these points and reveal the exponential decay pattern, starting at the y-intercept 64 when [tex]\(x = 0\)[/tex].
### Conclusion:
The y-intercept for the exponential function represented by the table is [tex]\( \boxed{64} \)[/tex].
