Question
vertex is (3,-9)
(write your response as an ordered pair).
2) The lesser zero is (0,0)
(write your response as an ordered pair)
3) The other zero is (6,0)
The y-intercept is (0,0)
what is the line of symmetry?
(write your response as an ordered pair).
2) The lesser zero is (0,0)
(write your response as an ordered pair)
3) The other zero is (6,0)
The y-intercept is (0,0)
what is the line of symmetry?
Answers
Answer
turn it into a f(x) equation.
Answered by
GPT-4o mini
To find the line of symmetry for a parabola given its vertex and zeros, you can use the vertex and the x-coordinates of the zeros.
The given vertex is at (3, -9), and the zeros of the parabola are at (0, 0) and (6, 0).
The line of symmetry of a parabola is located halfway between the two zeros. You can calculate it using the formula for the x-coordinate of the line of symmetry:
\[
x = \frac{x_1 + x_2}{2}
\]
where \( x_1 \) and \( x_2 \) are the x-coordinates of the zeros.
For the zeros (0, 0) and (6, 0):
\[
x = \frac{0 + 6}{2} = \frac{6}{2} = 3
\]
Thus, the line of symmetry is the vertical line where \( x = 3 \).
So, the line of symmetry can be expressed as:
\[
x = 3
\]
If you need it in the form of an ordered pair, the line of symmetry does not have endpoints as it extends vertically. However, you can write it as:
\[
(3, y) \quad \text{for all } y \in \mathbb{R}
\]
But typically, we simply state that the line of symmetry is at \( x = 3 \).
The given vertex is at (3, -9), and the zeros of the parabola are at (0, 0) and (6, 0).
The line of symmetry of a parabola is located halfway between the two zeros. You can calculate it using the formula for the x-coordinate of the line of symmetry:
\[
x = \frac{x_1 + x_2}{2}
\]
where \( x_1 \) and \( x_2 \) are the x-coordinates of the zeros.
For the zeros (0, 0) and (6, 0):
\[
x = \frac{0 + 6}{2} = \frac{6}{2} = 3
\]
Thus, the line of symmetry is the vertical line where \( x = 3 \).
So, the line of symmetry can be expressed as:
\[
x = 3
\]
If you need it in the form of an ordered pair, the line of symmetry does not have endpoints as it extends vertically. However, you can write it as:
\[
(3, y) \quad \text{for all } y \in \mathbb{R}
\]
But typically, we simply state that the line of symmetry is at \( x = 3 \).
Answered by
GPT-4o mini
To turn the given information about the vertex and zeros into a function \( f(x) \), we can use the vertex form of a quadratic function, which is:
\[
f(x) = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex. In this case, the vertex is at \((3, -9)\). So, we have:
\[
f(x) = a(x - 3)^2 - 9
\]
Next, we need to determine the value of \( a \) using one of the zeros. The zeros are given as \((0, 0)\) and \((6, 0)\). Let's use \((0, 0)\):
Substituting \( x = 0 \) and \( f(0) = 0 \) into the equation:
\[
0 = a(0 - 3)^2 - 9
\]
This simplifies to:
\[
0 = a(9) - 9
\]
\[
9 = 9a
\]
\[
a = 1
\]
Now that we have determined \( a = 1 \), we can substitute it back into the vertex form equation:
\[
f(x) = 1(x - 3)^2 - 9
\]
This can be simplified to:
\[
f(x) = (x - 3)^2 - 9
\]
Next, it is useful to expand this into standard form:
\[
f(x) = (x^2 - 6x + 9) - 9
\]
\[
f(x) = x^2 - 6x + 0
\]
\[
f(x) = x^2 - 6x
\]
Thus, the function \( f(x) \) is:
\[
\boxed{f(x) = x^2 - 6x}
\]
\[
f(x) = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex. In this case, the vertex is at \((3, -9)\). So, we have:
\[
f(x) = a(x - 3)^2 - 9
\]
Next, we need to determine the value of \( a \) using one of the zeros. The zeros are given as \((0, 0)\) and \((6, 0)\). Let's use \((0, 0)\):
Substituting \( x = 0 \) and \( f(0) = 0 \) into the equation:
\[
0 = a(0 - 3)^2 - 9
\]
This simplifies to:
\[
0 = a(9) - 9
\]
\[
9 = 9a
\]
\[
a = 1
\]
Now that we have determined \( a = 1 \), we can substitute it back into the vertex form equation:
\[
f(x) = 1(x - 3)^2 - 9
\]
This can be simplified to:
\[
f(x) = (x - 3)^2 - 9
\]
Next, it is useful to expand this into standard form:
\[
f(x) = (x^2 - 6x + 9) - 9
\]
\[
f(x) = x^2 - 6x + 0
\]
\[
f(x) = x^2 - 6x
\]
Thus, the function \( f(x) \) is:
\[
\boxed{f(x) = x^2 - 6x}
\]