Question

What is the equation of the parabola with vertex (3,4) and focus (6,4)

Answer 1

The equation of the parabola with vertex (3,4) and focus (6,4) is y = (1/12)(x - 3)^2 + 4.

Step-by-step explanation:

The equation of the parabola with vertex (3,4) and focus (6,4) can be found using the formula for the equation of a parabola in vertex form: y = a(x - h)^2 + k, where (h,k) is the vertex. Plugging in the values (3,4) for (h,k), we get the equation y = a(x - 3)^2 + 4. To find the value of 'a', we can use the distance formula between the vertex and the focus, which is given by: a = 1 / (4p), where 'p' is the distance between the vertex and the focus. In this case, 'p' is 3 units, so 'a' is 1/12.

Substituting the value of 'a' into the equation, we have y = (1/12)(x - 3)^2 + 4. This is the equation of the parabola.

Answer 2

This is a horizontal parabola that opens to the right. I know this because the y-coordinate is the same for these two points and the focus is to the right of the vertex.

The basic equation for such a vertical parabola looks like

4p(x-h) = (y-k)^2

Since the vertex is (3,4), we now have 4p(x-3) = (y-4)^2.

p in this case is a positive number that is the difference between the x-coordinates of the vertex and focus: this difference is 6-3, or 3.

Thus, 4p(x-3) = (y-4)^2 becomes 4(3)(x-3) = (y-4)^2, or

12(x-3) = (y-4)^2 (answer)