Question

what is the table of values for f(x)=-3/4(x+6)^2-7

Answer 1

To create a table of values for the function \( f(x) = -\frac{3}{4}(x + 6)^2 - 7 \), we can choose several values for \( x \), calculate the corresponding \( f(x) \) values, and then display them in a table format.

Let's evaluate \( f(x) \) at a few \( x \) values around the vertex:

1. **Vertex of the Quadratic Function**:
The function has a vertex, and since it's in the form of \( a(x-h)^2 + k \), the vertex is at \( (-6, -7) \).

Now let's calculate \( f(x) \) for \( x \) values from -10 to 0 (just as examples).

### Table of Values

\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-10 & -\frac{3}{4}(-10+6)^2 - 7 = -\frac{3}{4}(-4)^2 - 7 = -\frac{3}{4}(16) - 7 = -12 - 7 = -19 \\
-9 & -\frac{3}{4}(-9+6)^2 - 7 = -\frac{3}{4}(-3)^2 - 7 = -\frac{3}{4}(9) - 7 = -\frac{27}{4} - 7 = -\frac{27}{4} - \frac{28}{4} = -\frac{55}{4} = -13.75 \\
-8 & -\frac{3}{4}(-8+6)^2 - 7 = -\frac{3}{4}(-2)^2 - 7 = -\frac{3}{4}(4) - 7 = -3 - 7 = -10 \\
-7 & -\frac{3}{4}(-7+6)^2 - 7 = -\frac{3}{4}(-1)^2 - 7 = -\frac{3}{4}(1) - 7 = -\frac{3}{4} - 7 = -\frac{3}{4} - \frac{28}{4} = -\frac{31}{4} = -7.75 \\
-6 & -\frac{3}{4}(-6+6)^2 - 7 = -\frac{3}{4}(0)^2 - 7 = 0 - 7 = -7 \\
-5 & -\frac{3}{4}(-5+6)^2 - 7 = -\frac{3}{4}(1)^2 - 7 = -\frac{3}{4} - 7 = -\frac{3}{4} - \frac{28}{4} = -\frac{31}{4} = -7.75 \\
-4 & -\frac{3}{4}(-4+6)^2 - 7 = -\frac{3}{4}(2)^2 - 7 = -\frac{3}{4}(4) - 7 = -3 - 7 = -10 \\
-3 & -\frac{3}{4}(-3+6)^2 - 7 = -\frac{3}{4}(3)^2 - 7 = -\frac{3}{4}(9) - 7 = -\frac{27}{4} - 7 = -\frac{27}{4} - \frac{28}{4} = -\frac{55}{4} = -13.75 \\
-2 & -\frac{3}{4}(-2+6)^2 - 7 = -\frac{3}{4}(4)^2 - 7 = -\frac{3}{4}(16) - 7 = -12 - 7 = -19 \\
\hline
\end{array}
\]

### Summary of Values

\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-10 & -19 \\
-9 & -13.75 \\
-8 & -10 \\
-7 & -7.75 \\
-6 & -7 \\
-5 & -7.75 \\
-4 & -10 \\
-3 & -13.75 \\
-2 & -19 \\
\hline
\end{array}
\]

These values show that the function has a parabolic shape opening downwards, with the vertex at \( (-6, -7) \).