Problem Overview:
The task involves evaluating the function \( f(x) = -x^2 - 6x - 1 \) for specific values of \( x \). The given values of \( x \) are: \(-6, -5, -4, -3, -2, -1, 0\). The solution provided in the image demonstrates how to substitute each value of \( x \) into the function and simplify to find \( f(x) \).
Solution Explanation:
#### Step-by-Step Evaluation of \( f(x) \):
1.
Substitute \( x = -6 \):
\[
f(x) = -x^2 - 6x - 1
\]
Substitute \( x = -6 \):
\[
f(-6) = -(-6)^2 - 6(-6) - 1
\]
Simplify:
\[
f(-6) = -(36) - (-36) - 1 = -36 + 36 - 1 = -1
\]
So, \( f(-6) = -1 \).
2.
Substitute \( x = -5 \):
\[
f(x) = -x^2 - 6x - 1
\]
Substitute \( x = -5 \):
\[
f(-5) = -(-5)^2 - 6(-5) - 1
\]
Simplify:
\[
f(-5) = -(25) - (-30) - 1 = -25 + 30 - 1 = 4
\]
So, \( f(-5) = 4 \).
3.
Substitute \( x = -4 \):
\[
f(x) = -x^2 - 6x - 1
\]
Substitute \( x = -4 \):
\[
f(-4) = -(-4)^2 - 6(-4) - 1
\]
Simplify:
\[
f(-4) = -(16) - (-24) - 1 = -16 + 24 - 1 = 7
\]
So, \( f(-4) = 7 \).
4.
Substitute \( x = -3 \):
\[
f(x) = -x^2 - 6x - 1
\]
Substitute \( x = -3 \):
\[
f(-3) = -(-3)^2 - 6(-3) - 1
\]
Simplify:
\[
f(-3) = -(9) - (-18) - 1 = -9 + 18 - 1 = 8
\]
So, \( f(-3) = 8 \).
5.
Substitute \( x = -2 \):
\[
f(x) = -x^2 - 6x - 1
\]
Substitute \( x = -2 \):
\[
f(-2) = -(-2)^2 - 6(-2) - 1
\]
Simplify:
\[
f(-2) = -(4) - (-12) - 1 = -4 + 12 - 1 = 7
\]
So, \( f(-2) = 7 \).
6.
Substitute \( x = -1 \):
\[
f(x) = -x^2 - 6x - 1
\]
Substitute \( x = -1 \):
\[
f(-1) = -(-1)^2 - 6(-1) - 1
\]
Simplify:
\[
f(-1) = -(1) - (-6) - 1 = -1 + 6 - 1 = 4
\]
So, \( f(-1) = 4 \).
7.
Substitute \( x = 0 \):
\[
f(x) = -x^2 - 6x - 1
\]
Substitute \( x = 0 \):
\[
f(0) = -(0)^2 - 6(0) - 1
\]
Simplify:
\[
f(0) = -0 - 0 - 1 = -1
\]
So, \( f(0) = -1 \).
Final Results:
The values of \( f(x) \) for the given \( x \)-values are summarized in the table:
| \( x \) | \( f(x) \) |
|---------|------------|
| \(-6\) | \(-1\) |
| \(-5\) | \(4\) |
| \(-4\) | \(7\) |
| \(-3\) | \(8\) |
| \(-2\) | \(7\) |
| \(-1\) | \(4\) |
| \(0\) | \(-1\) |
Final Answer:
\[
\boxed{-1, 4, 7, 8, 7, 4, -1}
\]
Parent Tip: Review the logic above to help your child master the concept of quadratic functions worksheet with answers.
