To determine the equation for the given quadratic function \( f(x) \) that matches the table of values, we need to compare each of the given possible equations with the values in the table. Here is a detailed step-by-step solution:
1. Identifying the Points from the Table:
The table presents the following points \((x, f(x))\):
[tex]\[
(-8, 13), (-7, 6), (-6, 1), (-5, -2), (-4, -3), (-3, -2), (-2, 1), (-1, 6), (0, 13)
\][/tex]
2. Possible Equations:
We have three potential equations for \( f(x) \):
- \( f(x) = (x + 5)^2 - 2 \)
- \( f(x) = (x + 4)^2 - 3 \)
- \( f(x) = (x - 4)^2 - 3 \)
3. Testing the First Equation:
Let's test \( f(x) = (x + 5)^2 - 2 \):
- For \( x = -8 \): \( f(-8) = ((-8) + 5)^2 - 2 = (-3)^2 - 2 = 9 - 2 = 7 \) (not 13)
Since it does not match the first point, this equation is incorrect.
4. Testing the Second Equation:
Let's test \( f(x) = (x + 4)^2 - 3 \):
- For \( x = -8 \): \( f(-8) = ((-8) + 4)^2 - 3 = (-4)^2 - 3 = 16 - 3 = 13 \) (matches the table)
- For \( x = -7 \): \( f(-7) = ((-7) + 4)^2 - 3 = (-3)^2 - 3 = 9 - 3 = 6 \) (matches the table)
- For \( x = -6 \): \( f(-6) = ((-6) + 4)^2 - 3 = (-2)^2 - 3 = 4 - 3 = 1 \) (matches the table)
- For \( x = -5 \): \( f(-5) = ((-5) + 4)^2 - 3 = (-1)^2 - 3 = 1 - 3 = -2 \) (matches the table)
- For \( x = -4 \): \( f(-4) = ((-4) + 4)^2 - 3 = (0)^2 - 3 = 0 - 3 = -3 \) (matches the table)
- For \( x = -3 \): \( f(-3) = ((-3) + 4)^2 - 3 = (1)^2 - 3 = 1 - 3 = -2 \) (matches the table)
- For \( x = -2 \): \( f(-2) = ((-2) + 4)^2 - 3 = (2)^2 - 3 = 4 - 3 = 1 \) (matches the table)
- For \( x = -1 \): \( f(-1) = ((-1) + 4)^2 - 3 = (3)^2 - 3 = 9 - 3 = 6 \) (matches the table)
- For \( x = 0 \): \( f(0) = ((0) + 4)^2 - 3 = (4)^2 - 3 = 16 - 3 = 13 \) (matches the table)
Since all points match, this equation is correct.
5. Testing the Third Equation:
Let's test \( f(x) = (x - 4)^2 - 3 \):
- For \( x = -8 \): \( f(-8) = ((-8) - 4)^2 - 3 = (-12)^2 - 3 = 144 - 3 = 141 \) (not 13)
Since it does not match the first point, this equation is incorrect.
Given the checks above, the equation of \( f(x) \) that correctly matches all the values in the table is:
[tex]\[
f(x) = (x + 4)^2 - 3
\][/tex]
