# Parabola with Focus (-5,0) & Vertex (-5,-4): Find Equation

• TonyC
In summary, the equation for the parabola with focus at (-5,0) and vertex at (-5,-4) is x^2 + 10x + 16y + 89 = 0. The vertex can be found by solving for y and completing the square, while the focus can be determined using the equation for the focus of a given parabola.
TonyC
Fins an equation for the parabola with focus at (-5,0) and vertex at (-5,-4).

I have come up with:

x^2 + 10x + 16y + 89 = 0

How far off am I?

TonyC said:
Fins an equation for the parabola with focus at (-5,0) and vertex at (-5,-4).

I have come up with:

x^2 + 10x + 16y + 89 = 0

How far off am I?

Can't you check for yourself?

Is it a parabola? (That's easy- there is an x2 term but no y2 term.)

What is the vertex? (Solve for y, then complete the square.)

What is the focus? (You will need to know the equation for the focus of a given parabola.)

That is what I am having trouble determining. Can you show me step by step how to check myself?

## 1. What is the equation of a parabola with focus (-5,0) and vertex (-5,-4)?

The equation of a parabola with focus (h,k) and vertex (h,k-p), where p is the distance between the focus and vertex, is (x-h)^2=4p(y-k).

## 2. How do you find the direction of opening of a parabola with given focus and vertex?

The direction of opening of a parabola is determined by the sign of the coefficient of x^2 in the equation. If it is positive, the parabola opens upwards, and if it is negative, the parabola opens downwards.

## 3. Can the focus and vertex of a parabola be the same point?

No, the focus and vertex of a parabola cannot be the same point. The vertex represents the lowest or highest point on the parabola, while the focus is a point inside the parabola. Therefore, they must be distinct points.

## 4. How do you graph a parabola with focus (-5,0) and vertex (-5,-4)?

To graph a parabola with focus (h,k) and vertex (h,k-p), first plot the focus and vertex points on the coordinate plane. Then, use the distance p to plot two more points on either side of the vertex, making sure they are equidistant from the focus. Finally, draw a smooth curve connecting the four points to complete the parabola.

## 5. How can you determine the axis of symmetry of a parabola with focus (-5,0) and vertex (-5,-4)?

The axis of symmetry of a parabola is a vertical line that divides the parabola into two equal halves. For a parabola with focus (h,k) and vertex (h,k-p), the axis of symmetry is the line x=h. In this case, the axis of symmetry is the line x=-5.

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