To determine the equation of the quadratic function [tex]\( f(x) \)[/tex] that matches the given table of values, let's analyze each potential function and verify if it aligns with the data points.
Given data points are:
[tex]\[
\begin{tabular}{|c|l|}
\hline
$x$ & $f(x)$ \\
\hline
-8 & 13 \\
\hline
-7 & 6 \\
\hline
-6 & 1 \\
\hline
-5 & -2 \\
\hline
-4 & -3 \\
\hline
-3 & -2 \\
\hline
-2 & 1 \\
\hline
-1 & 6 \\
\hline
0 & 13 \\
\hline
\end{tabular}
\][/tex]
Potential equations for [tex]\( f(x) \)[/tex] are:
1. [tex]\( f(x) = (x + 5)^2 - 2 \)[/tex]
2. [tex]\( f(x) = (x + 4)^2 - 3 \)[/tex]
3. [tex]\( f(x) = (x - 4)^2 - 3 \)[/tex]
4. [tex]\( f(x) = (x - 5)^2 - 2 \)[/tex]
We will test these equations one by one against the given data points:
1. Testing [tex]\( f(x) = (x + 5)^2 - 2 \)[/tex]:
- For [tex]\( x = -8 \)[/tex], [tex]\( f(-8) = ((-8) + 5)^2 - 2 = (-3)^2 - 2 = 9 - 2 = 7 \)[/tex]. This does not match [tex]\( f(-8) = 13 \)[/tex].
Since the first function does not match the data point for [tex]\( x = -8 \)[/tex], we can eliminate it.
2. Testing [tex]\( f(x) = (x + 4)^2 - 3 \)[/tex]:
- For [tex]\( x = -8 \)[/tex], [tex]\( f(-8) = ((-8) + 4)^2 - 3 = (-4)^2 - 3 = 16 - 3 = 13 \)[/tex]. This matches [tex]\( f(-8) = 13 \)[/tex].
- For [tex]\( x = -7 \)[/tex], [tex]\( f(-7) = ((-7) + 4)^2 - 3 = (-3)^2 - 3 = 9 - 3 = 6 \)[/tex]. This matches [tex]\( f(-7) = 6 \)[/tex].
- For [tex]\( x = -6 \)[/tex], [tex]\( f(-6) = ((-6) + 4)^2 - 3 = (-2)^2 - 3 = 4 - 3 = 1 \)[/tex]. This matches [tex]\( f(-6) = 1 \)[/tex].
- For [tex]\( x = -5 \)[/tex], [tex]\( f(-5) = ((-5) + 4)^2 - 3 = (-1)^2 - 3 = 1 - 3 = -2 \)[/tex]. This matches [tex]\( f(-5) = -2 \)[/tex].
- For [tex]\( x = -4 \)[/tex], [tex]\( f(-4) = ((-4) + 4)^2 - 3 = (0)^2 - 3 = 0 - 3 = -3 \)[/tex]. This matches [tex]\( f(-4) = -3 \)[/tex].
- For [tex]\( x = -3 \)[/tex], [tex]\( f(-3) = ((-3) + 4)^2 - 3 = (1)^2 - 3 = 1 - 3 = -2 \)[/tex]. This matches [tex]\( f(-3) = -2 \)[/tex].
- For [tex]\( x = -2 \)[/tex], [tex]\( f(-2) = ((-2) + 4)^2 - 3 = (2)^2 - 3 = 4 - 3 = 1 \)[/tex]. This matches [tex]\( f(-2) = 1 \)[/tex].
- For [tex]\( x = -1 \)[/tex], [tex]\( f(-1) = ((-1) + 4)^2 - 3 = (3)^2 - 3 = 9 - 3 = 6 \)[/tex]. This matches [tex]\( f(-1) = 6 \)[/tex].
- For [tex]\( x = 0 \)[/tex], [tex]\( f(0) = (0 + 4)^2 - 3 = (4)^2 - 3 = 16 - 3 = 13 \)[/tex]. This matches [tex]\( f(0) = 13 \)[/tex].
The second function [tex]\( f(x) = (x + 4)^2 - 3 \)[/tex] perfectly matches all given data points.
Conclusion:
The equation of [tex]\( f(x) \)[/tex] that aligns with the given data points is:
[tex]\[
f(x) = (x + 4)^2 - 3.
\][/tex]
