I have a question and I can not figure it out. Any answers to the following along with any explanation would be great!
Here is the equation:
x=.2y^2-2y-3
Now I know that is an equation for a parabola. Now, I need the following:
Equation in standard form
Vertices
Co-vertices
Center
Foci
Directric
Major Axis
Minor Axis
Some may not be needed based on it being a parabola. Any help would be great!
Here is the equation:
x=.2y^2-2y-3
x+3 = .2 y^2 -2 y
multiply both sides by 5 to get 1 as coefficient of y^2
5 x + 15 = y^2 - 20 y
add 100 to both sides to get perfect square on the right
5 x + 115 = y^2 -20 y +100 = (y-10)^2
so
(y-10)^2 = 5 ( x + 23)
vertex at (-23,10) etc
Wow thanks for that. Could you help me with all the other things I listed above. This stuff is confusing to me.
Here is the equation:
x=.2y^2-2y-3
x+3 = .2 y^2 -2 y
multiply both sides by 5 to get 1 as coefficient of y^2
5 x + 15 = y^2 - 10 y
add 25 to both sides to get perfect square on the right
5 x + 40 = y^2 - 10 y + 25 = (y-5)^2
so
(y-5)^2 = 5 ( x + 8)
vertex at (-8,5) etc
Actually, I belive you made a mistake there. Multiplying both sides by five would get my -10y and therefor, I would add 25 to both sides right?
(y-5)^2 = 5 ( x + 8)
y gets big + as x gets big + so opens to the right
well, all that is left is a focus and directrix. parabolas do not have major and minor axes like ellipses.
if in the form
(y-k)^2 = 4 a (x-h)
so here a = 5/4
then
vertex at (h,k)
vertex to focus = a so at x= -8+5/4 = -6.75 so (-6.75 , 5)
vertex to directrix = a so at x = -9.25
When you say vertex, you mean the center so what about vertices and co-vertices. And there is only 1 foci?
Another thing, would a=1.25? Because 4*1.25 equals 5
yes, 1.25 = 5/4
when I say vertex I mean the leftmost point of the parabola. It has no center like an ellipse and it only has one vertex and one focus.
Look in your text book for labeled pictures of:
parabola
ellipse
hyperbola
Sorry but I am trying to understand your reasoning for the directrix. How did you get -9.25. I would think it would be -1.25.
To find the equation of the parabola in standard form, we need to complete the square.
1. Equation in standard form:
Let's start by rewriting the equation as follows:
x = 0.2y^2 - 2y - 3
Now, we want to complete the square to get the equation in the form (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p is the distance from the vertex to the focus.
To complete the square, we need to focus on the y terms. We can rewrite the equation as:
x = 0.2(y^2 - 10y) - 3
Next, we need to take half of the coefficient of the y term (-10/2 = -5) and square it (-5^2 = 25). We add and subtract this value inside the parentheses to maintain the equation's balance:
x = 0.2(y^2 - 10y + 25 - 25) - 3
Now we can rewrite it as:
x = 0.2((y - 5)^2 - 25) - 3
x = 0.2(y - 5)^2 - 5 - 3
x = 0.2(y - 5)^2 - 8
Therefore, the equation in standard form is:
x = 0.2(y - 5)^2 - 8
2. Vertex:
Comparing the equation in standard form, we can see that the vertex is at (h, k) = (0, 5).
3. Co-vertices:
Since the equation given is for a parabola, it does not have co-vertices. Co-vertices are properties of ellipses and hyperbolas.
4. Center:
Again, since it is a parabola, it does not have a center. Centers are properties of circles and ellipses.
5. Foci:
To find the foci of a parabola, we need to use the formula:
Foci = (h, k + p), (h, k - p)
From the equation in standard form, we can see that p = -8/0.2 = -40. Thus, the foci are:
Foci = (0, 5 - 40), (0, 5 + 40)
Foci = (0, -35), (0, 45)
6. Directrix:
To find the directrix, we use the formula: y = k - p
In this case, the directrix is:
y = 5 - (-40) = 45
7. Major Axis:
In a parabola, the major axis refers to the x-axis, so it is a line segment parallel to the x-axis, passing through the vertex.
8. Minor Axis:
Since it is a parabola, there is no minor axis. The minor axis is specific to ellipses.