Fibd dilation factor of a transformed parabola, when vertex 3, 30 and x intercepts 0,0 and 6,0

Give answer in y=a(x-h)^2 +k

Using the intercepts, you have

y = a(x-0)(x-6) = a(x^2 - 6x)
Since the vertex is midway between the roots, you know that it is at x=3
Since the vertex of x^2-6x is at (3,-9) and you want it to be at (3,30), you can see that a = -30/9 = -10/3
so now you need to put that into vertex form
x^2-6x+9 = (x-3)^2
So, now you need
y = -10/3 (x^2-6x+9-9) = -10/3 (x-3)^2 + 30

You could, of course, come at it in other ways, but this is the way I handled it.

To find the dilation factor of a transformed parabola, you need to compare the distance between the vertex and the x-intercepts of the original parabola (in this case, the parent function y = x^2) to the distance between the vertex and the x-intercepts of the transformed parabola.

Let's start by analyzing the given information:

1. Original vertex: (0, 0)
2. Transformed vertex: (3, 30)
3. Original x-intercepts: (0, 0) and (6, 0)

The transformation of the parabola involves a horizontal shift of 3 units to the right and a vertical shift of 30 units upward. Before we talk about the dilation factor, let's first understand the effects of the vertical shift.

The vertical shift does not affect the shape of the parabola, so it does not alter the dilation factor. It only moves the entire parabola upward or downward. Therefore, we can ignore the vertical shift for now and focus on the horizontal shift.

Since the vertex of the transformed parabola is (3, 30), and the original x-intercepts are (0, 0) and (6, 0), we can calculate the distances.

Original distance between vertex and x-intercept:
d_original = 6 units (6 - 0)

Transformed distance between vertex and x-intercept:
d_transformed = 3 units (3 - 0)

Now, calculate the dilation factor (k) by dividing the transformed distance by the original distance:

k = d_transformed / d_original
= 3 / 6
= 1/2

Therefore, the dilation factor of the transformed parabola is 1/2.

In summary, to find the dilation factor of a transformed parabola, you need to compare the distance between the vertex and the x-intercepts of the original parabola to the distance between the vertex and the x-intercepts of the transformed parabola. Divide the transformed distance by the original distance to get the dilation factor.