Question
The vertex is (3,-9) the lesser zero is (0,0) the other zero is (6,0) the line of symmetry is x=3 and the y intercept is (0,0)
Complete the table x is -1,0,1,3,5 what would the f(x) of each of the x be
Complete the table x is -1,0,1,3,5 what would the f(x) of each of the x be
Answers
GPT-4o mini
Given the information you provided, we know that the vertex of the parabola is (3, -9), and the roots (or zeros) of the function are (0, 0) and (6, 0). The line of symmetry is \( x = 3 \), and the parabola opens upwards since the vertex is below the x-axis.
From the zeroes, we can express the quadratic function in factored form:
\[
f(x) = a(x - 0)(x - 6) = ax(x - 6) = a(x^2 - 6x)
\]
To find the value of \( a \), we can use the vertex form of the quadratic equation or simply plug in the vertex coordinates into the function.
Since the vertex is at (3, -9), we will find \( f(3) \) using our current expression for \( f(x) \):
\[
f(3) = a(3^2 - 6 \cdot 3) = a(9 - 18) = a(-9)
\]
Set this equal to the y-value of the vertex:
\[
a(-9) = -9
\]
Solving gives:
\[
a = 1
\]
Now we have:
\[
f(x) = x^2 - 6x
\]
Next, we will calculate \( f(x) \) for the given values \( x = -1, 0, 1, 3, 5 \).
1. For \( x = -1 \):
\[
f(-1) = (-1)^2 - 6(-1) = 1 + 6 = 7
\]
2. For \( x = 0 \):
\[
f(0) = 0^2 - 6(0) = 0
\]
3. For \( x = 1 \):
\[
f(1) = 1^2 - 6(1) = 1 - 6 = -5
\]
4. For \( x = 3 \):
\[
f(3) = 3^2 - 6(3) = 9 - 18 = -9
\]
5. For \( x = 5 \):
\[
f(5) = 5^2 - 6(5) = 25 - 30 = -5
\]
Now we can summarize the results in the table form:
\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-1 & 7 \\
0 & 0 \\
1 & -5 \\
3 & -9 \\
5 & -5 \\
\hline
\end{array}
\]
This table represents the values of \( f(x) \) for the provided \( x \) values.
From the zeroes, we can express the quadratic function in factored form:
\[
f(x) = a(x - 0)(x - 6) = ax(x - 6) = a(x^2 - 6x)
\]
To find the value of \( a \), we can use the vertex form of the quadratic equation or simply plug in the vertex coordinates into the function.
Since the vertex is at (3, -9), we will find \( f(3) \) using our current expression for \( f(x) \):
\[
f(3) = a(3^2 - 6 \cdot 3) = a(9 - 18) = a(-9)
\]
Set this equal to the y-value of the vertex:
\[
a(-9) = -9
\]
Solving gives:
\[
a = 1
\]
Now we have:
\[
f(x) = x^2 - 6x
\]
Next, we will calculate \( f(x) \) for the given values \( x = -1, 0, 1, 3, 5 \).
1. For \( x = -1 \):
\[
f(-1) = (-1)^2 - 6(-1) = 1 + 6 = 7
\]
2. For \( x = 0 \):
\[
f(0) = 0^2 - 6(0) = 0
\]
3. For \( x = 1 \):
\[
f(1) = 1^2 - 6(1) = 1 - 6 = -5
\]
4. For \( x = 3 \):
\[
f(3) = 3^2 - 6(3) = 9 - 18 = -9
\]
5. For \( x = 5 \):
\[
f(5) = 5^2 - 6(5) = 25 - 30 = -5
\]
Now we can summarize the results in the table form:
\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-1 & 7 \\
0 & 0 \\
1 & -5 \\
3 & -9 \\
5 & -5 \\
\hline
\end{array}
\]
This table represents the values of \( f(x) \) for the provided \( x \) values.