Question

the vertex of the graph is (3, -9)
the lesser zero is (0,0)
the other zero is (6,0)
the line of symmetry is x =3
The y intercept is (0,0)

Complete the table

x values, -1,0,1,3,5
what are the f(x) values

Answer 1

To complete the table with the \( f(x) \) values based on the given information about the quadratic function, we can first derive the function itself.

Given:
- Vertex: \( (3, -9) \)
- Zeros (x-intercepts): \( (0, 0) \) and \( (6, 0) \)
- Line of symmetry: \( x = 3 \)
- The function is a quadratic in the form \( f(x) = a(x - r_1)(x - r_2) \), where \( r_1 \) and \( r_2 \) are the zeros.

The zeros are \( r_1 = 0 \) and \( r_2 = 6 \), so the function can be written as:
\[
f(x) = a(x - 0)(x - 6) = ax(x - 6)
\]
which simplifies to:
\[
f(x) = a(x^2 - 6x)
\]

To find the value of \( a \), we use the vertex. The vertex occurs at \( x = 3 \), and we know that \( f(3) = -9 \).
Inserting \( x = 3 \) into the function gives:

\[
f(3) = a(3^2 - 6 \cdot 3) = a(9 - 18) = a(-9)
\]
Setting this equal to -9 (the y-value of the vertex), we get:
\[
a(-9) = -9 \quad \Rightarrow \quad a = 1
\]

Thus, the equation of the quadratic function is:
\[
f(x) = x^2 - 6x
\]

Now, we can calculate \( f(x) \) for the required \( x \) values: -1, 0, 1, 3, and 5.

Calculating each:
- For \( x = -1 \):
\[
f(-1) = (-1)^2 - 6(-1) = 1 + 6 = 7
\]

- For \( x = 0 \):
\[
f(0) = 0^2 - 6(0) = 0
\]

- For \( x = 1 \):
\[
f(1) = 1^2 - 6(1) = 1 - 6 = -5
\]

- For \( x = 3 \):
\[
f(3) = 3^2 - 6(3) = 9 - 18 = -9
\]

- For \( x = 5 \):
\[
f(5) = 5^2 - 6(5) = 25 - 30 = -5
\]

Now, we can summarize the \( f(x) \) values:

| \( x \) | \( f(x) \) |
|---------|------------|
| -1 | 7 |
| 0 | 0 |
| 1 | -5 |
| 3 | -9 |
| 5 | -5 |

Feel free to ask if you need further assistance!