can you check my work and explain the last question because I do not uderstand.
Explain what each of the following represents, and how y = a(x - h)2 + k, a ≠ 0 and (x - h)2 = 4p(y - k), p ≠ 0 are equivalent.
The equation (x - h)2 = 4p(y - k), p ≠ 0 is used when the parabola has a vertical axis. In this equation (h,k) represents the vertex, (h, k+p) represents the focus, and (y=k-p) represents the directrix. The axis is the line x=h. When p>0 the parabola opens upward and when p<0 it opens downward.
The equation (y - k)2 = 4p(x - h), p ≠ 0 equation is used when the parabola has a horizontal axis. In this equation (h,k) represents the vertex,( h+p,k)represents the focus, and (x=h-p) represents the directrix. The axis line is the line y=k. When p>0 the parabola opens to the right and when p<0 it opens to the left.
I don't see any questions. Just statements of the properties. If you understant
(x-h)^2 = 4p(y-k)
then I don't see what the problem is with
(y-k)^2 = 4p(x-k)
Go to wolframalpha.com and type in some functions and see what happens. For example,
http://www.wolframalpha.com/input/?i=parabola+(x-3)%5E2+%3D+8(y-2)
Sure! Let's walk through it.
The equations y = a(x - h)² + k and (x - h)² = 4p(y - k), a ≠ 0 and (x - h)² = 4p(y - k), p ≠ 0 represent the same parabola but in different forms.
First, let's look at the equation (x - h)² = 4p(y - k), p ≠ 0. This equation is used when the parabola has a vertical axis.
- The point (h, k) represents the vertex of the parabola, which is the lowest or highest point of the parabola depending on whether it opens upward or downward.
- The point (h, k+p) represents the focus of the parabola. It is a point which is closer to the vertex and is responsible for the shape and position of the parabola.
- The line y = k-p represents the directrix of the parabola. It is a horizontal line that is equidistant from every point on the parabola.
- The axis of the parabola is the vertical line x = h. This line passes through the vertex and divides the parabola into two symmetrical halves.
- If p > 0, the parabola opens upward. If p < 0, the parabola opens downward.
Now, let's look at the equation (y - k)² = 4p(x - h), p ≠ 0. This equation is used when the parabola has a horizontal axis.
- Again, the point (h, k) represents the vertex of the parabola, which is the leftmost or rightmost point of the parabola depending on whether it opens to the left or right.
- The point (h+p, k) represents the focus of the parabola. It is a point which is closer to the vertex and determines the shape and position of the parabola.
- The line x = h-p represents the directrix of the parabola. It is a vertical line that is equidistant from every point on the parabola.
- The axis of the parabola is the horizontal line y = k. This line passes through the vertex and divides the parabola into two symmetrical halves.
- If p > 0, the parabola opens to the right. If p < 0, the parabola opens to the left.
So, both equations represent the same parabola but in different forms depending on whether the axis is vertical or horizontal. The equations provide information about the vertex, focus, directrix, and axis, which are essential elements in representing and graphing a parabola.