What is the correct standard form of the equation of the parabola?
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To determine the standard form of the equation of a parabola, we need to consider its vertex form, which is given by:
$$y = a(x-h)^2 + k$$
where (h, k) represents the vertex of the parabola.
If the given equation of the parabola is not in vertex form, we can manipulate it to obtain the vertex form.
Please provide the given equation of the parabola so that we can work through the steps together.
you need to use the focus and the directrix. Using the values for the focus on one side and the directrix on the other, you are going to now use the Pythagorean Theorem to help you solve this question. You need to be sure you know what values are the x’s and which ones are the y’s.
My apologies for the confusion. You are correct that using the focus and directrix, we can determine the standard form of the equation of the parabola.
Let's consider a parabola with its focus at the point (f, 0) and the directrix as the line y = -d.
Using the Pythagorean Theorem, we can say that the distance from any point (x, y) on the parabola to the focus is equal to the distance from that point to the directrix.
The distance between a point (x, y) and the focus (f, 0) is given by:
Distance to the focus = √((x-f)^2 + (y-0)^2)
The distance between a point (x, y) and the directrix (y = -d) is given by:
Distance to the directrix = |y + d|
Since these two distances are equal, we can set up the equation:
√((x-f)^2 + y^2) = |y + d|
Squaring both sides to eliminate the square root:
(x-f)^2 + y^2 = (y + d)^2
Expanding the right side:
(x-f)^2 + y^2 = y^2 + 2dy + d^2
Simplifying and rearranging the equation:
(x-f)^2 = 2dy + d^2
Now, let's convert this equation to the standard form y^2 = 4ax.
Subtracting y^2 from both sides:
(x-f)^2 - y^2 = 2dy + d^2 - y^2
Rearranging terms:
(x-f)^2 - (y^2 + 2dy + d^2) = 0
Expanding and simplifying:
(x^2 - 2fx + f^2) - (y^2 + 2dy + d^2) = 0
x^2 - 2fx + f^2 - y^2 - 2dy - d^2 = 0
Combining like terms:
(x^2 - 2fx + f^2) - (y^2 + 2dy + d^2) + f^2 - d^2 = 0
(x^2 - 2fx + f^2) - (y^2 + 2dy + d^2) = -(f^2 - d^2)
Finally, we can rewrite the equation in standard form:
(x - f)^2 - (y + d)^2 = -(f^2 - d^2)
Note: In standard form, we usually require the coefficient of the x^2 term to be positive. If the coefficient is negative, we can multiply the entire equation by -1 to obtain the positive coefficient.
the parabola is (-3,2)
Given that the parabola has a focus at (-3, 2), we can proceed with finding the standard form of its equation using the given information.
We know that the vertex of the parabola is halfway between the focus and the directrix. Since the directrix is a horizontal line, the vertex will have the same y-coordinate as the focus.
Therefore, the vertex of the parabola is V = (-3, 2).
The distance from the vertex to the focus is defined as the focal length, which we will denote by p. The focal length is also equal to the distance from the vertex to the directrix.
Since the focus has the y-coordinate 2, the directrix will be a horizontal line passing through the point (x, -p). However, we are not given the value of p.
To determine p, we need additional information, such as another point on the parabola or the equation of the parabola in another form (such as the directrix form or the focus-directrix form). Without more information, we cannot determine the standard form of the equation of the parabola with only the focus given.
To find the standard form of the equation of a parabola, you need to start with the general form of the equation, which is:
y = a(x - h)^2 + k
In this equation, (h, k) represents the vertex of the parabola.
To convert the general form to standard form, you need to complete the square. Here are the steps:
1. Start with the general form: y = a(x - h)^2 + k
2. Expand the square: y = a(x^2 - 2hx + h^2) + k
3. Distribute the 'a' through the parentheses: y = ax^2 - 2ahx + ah^2 + k
4. Rearrange the terms: y = ax^2 - 2ahx + (ah^2 + k)
5. To complete the square, add and subtract the square of half the coefficient of 'x' within the parentheses: y = ax^2 - 2ahx + (ah^2 + k) + (-1) * (ah)^2
6. Simplify inside the parentheses: y = ax^2 - 2ahx + (ah^2 - (ah)^2 + k)
7. Combine like terms within the parentheses: y = ax^2 - 2ahx + (ah^2 - a^2h^2 + k)
8. Factor out an 'a' from the last two terms within the parentheses: y = ax^2 - 2ahx + a(h^2 - ah^2/k)
9. Simplify further: y = ax^2 - 2ahx + a(1 - ah/k) * h^2
10. Now, we can see that (1 - ah/k) is a constant. Let's call it 'p'. Rewrite the equation as: y = ax^2 - 2ahx + aph^2
11. Finally, bring the 'a' term outside of the parentheses: y = a(x^2 - 2hx + ph^2)
So, the standard form of the equation of the parabola is: y = a(x^2 - 2hx + ph^2)
To find the correct standard form of the equation of a parabola, we need to consider a few factors. A parabola is a curve formed by points that are equidistant from a fixed point (called the focus) and a fixed line (called the directrix). The standard form of the equation of a parabola is given by:
(x - h)^2 = 4p(y - k)
where (h, k) is the vertex of the parabola and p is the distance between the vertex and the focus/directrix.
To determine the standard form of the equation, we need to know the vertex coordinates, and either the distance between the vertex and the focus (p), or the distance between the vertex and the directrix.
Once you have these values, substitute them into the equation and simplify if necessary.
If you provide the specific information regarding the vertex, focus, or directrix, I can help you find the correct standard form of the equation of the parabola.