Find an equation of the parabola with vertex at the origin.

Passes through the point (4,6); horizontal axis.

I know it has to be y^2=4px. But I don't know what to do with (4,6). Please show step by step. Thanks.

Oh, I see. Solve for P first then substitute it in. I guess I forgot that concept. Thanks!

Oh, I see you're trying to find the equation of a parabola! Let's go step by step, just like you asked.

First, since the parabola has a horizontal axis, we can use the equation "x^2 = 4py". But since we're given the vertex at the origin, (0, 0), the equation becomes "x^2 = 4py".

Now, we need to find the value of "p." We're told that the parabola passes through the point (4, 6). Since the parabola is symmetric, it will also pass through the point (-4, 6). These two points are equidistant from the vertex. The distance from the vertex to any point on the parabola is given by 4p.

So, let's find the distance from the origin to (4, 6). Using the distance formula, we have:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
d = sqrt((4 - 0)^2 + (6 - 0)^2)
d = sqrt(16 + 36)
d = sqrt(52)

Since the distance from the vertex to (4, 6) is 4p, we have:

4p = sqrt(52)

Now, let's solve for "p" by dividing both sides of the equation by 4:

p = sqrt(52) / 4

Finally, we substitute the value of "p" back into the equation x^2 = 4py:

x^2 = 4(sqrt(52) / 4) * y
x^2 = sqrt(52) * y

So, there you have it! The equation of the parabola with a vertex at the origin and passing through the point (4, 6) is x^2 = sqrt(52) * y.

I hope that helps! If you need any more assistance, feel free to ask.

To find the equation of a parabola with a vertex at the origin, we can use the general equation for a horizontally-oriented parabola, which is given by:

y^2 = 4px

where p is the distance between the vertex and the focus.

Step 1: Given that the vertex is at the origin, we know that the vertex coordinates are (0,0).

Step 2: The point (4,6) lies on the parabola. Since the parabola has a horizontal axis, the point lies on the right side of the vertex.

Step 3: To determine the value of p, we can use the distance formula between the vertex (0,0) and the given point (4,6):

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Substituting the coordinates, we have:
d = sqrt((4 - 0)^2 + (6 - 0)^2)
d = sqrt(16 + 36)
d = sqrt(52)

Step 4: The value of p is half of the distance between the vertex and the focus. Since the focus lies on the positive side of the vertex, we take the positive square root of 52 as p:

p = √52

Step 5: Now that we have the value of p, we can substitute it into the general equation for the horizontal parabola:

y^2 = 4px
y^2 = 4√52x
y^2 = 2√13x

Therefore, the equation of the parabola with a vertex at the origin that passes through the point (4,6) and has a horizontal axis is:
y^2 = 2√13x

To find the equation of a parabola with vertex at the origin, we start with the general equation of a parabola, which is:

y^2 = 4px

In this equation, p represents the distance between the vertex and the focus of the parabola. Since the parabola has a horizontal axis, p will be positive if the parabola opens to the right and negative if it opens to the left.

Since the vertex is at the origin (0,0), we know that the equation is of the form y^2 = 4px. We just need to find the value of p to complete the equation.

Next, we are given that the parabola passes through the point (4,6). Substituting these x and y coordinates into the equation, we get:

6^2 = 4p(4)

Simplifying this equation, we have:

36 = 16p

To find the value of p, divide both sides of the equation by 16:

p = 36/16
p = 9/4

Now that we have the value of p, we can substitute it back into the general equation of the parabola to get the final equation. So, the equation of the parabola with vertex at the origin and passes through the point (4,6), with a horizontal axis, is:

y^2 = 4(9/4)x

Simplifying further, we get:

y^2 = 9x

Thus, the equation of the parabola is y^2 = 9x.

Horizontal axis means of form:

(y-k)^2 = 4p(x-h)
if the vertex is at the origin (h,k) = (0,0)
y^2 = 4 p x
36 = 4 p (4)
p = (9/4)
so
y^2 = 9 x