Explain what each of the following represents, and how equations (a) and (b) are equivalent.
(a) y = a(x - h)2 + k, a ≠ 0
(b) (x - h)2 = 4p(y - k), p ≠ 0
(c) (y - k)2 = 4p(x - h), p ≠ 0
can you check my work and explain how equations (a) and (b) are equivalent.
(b)This equation 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.
(c) This 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.
(a) and (b) are equivalent, since you can write them as
(y-k) = a(x-h)^2
(y-k) = 4p(x-h)^2
You have (b) and (c) right.
I thought we already went over this...
Well, equations (a) and (b) are actually equivalent because they represent the same type of parabola, just written in different forms. Equation (a) is in the standard form of a parabolic equation, where (h, k) represents the vertex. The variable "a" determines the shape and direction of the parabola.
On the other hand, equation (b) is in the standard form for a parabolic equation with vertical axis. Here, (h, k) also represents the vertex, but "p" represents a different property. It determines the distance between the vertex and the focus, as well as the directrix.
So, equations (a) and (b) are equivalent because they both represent parabolas with the same vertex (h, k), but with different properties: "a" in equation (a) and "p" in equation (b) determine different characteristics of the parabolas, namely the shape and direction in equation (a), and the focus and directrix in equation (b).
Your understanding of the representations and interpretations of equations (b) and (c) is correct. Equation (b) represents a parabola with a vertical axis, where the vertex is at (h, k), the focus is at (h, k+p), and the directrix is given by the equation y = k-p. Equation (c) represents a parabola with a horizontal axis, where the vertex is at (h, k), the focus is at (h+p, k), and the directrix is given by the equation x = h-p.
Now, let's discuss how equations (a) and (b) are equivalent. To show this, we'll need to expand equation (a) and rearrange it to match the form of equation (b):
(a) y = a(x - h)^2 + k
Expanding the equation:
(a) y = a(x^2 - 2hx + h^2) + k
(a) y = ax^2 - 2ahx + ah^2 + k
Comparing this with equation (b) (x - h)^2 = 4p(y - k), we can see that it matches if we let:
4p = a
-2ah = 0 (which implies a = 0 or h = 0, but since a ≠ 0, h = 0)
ah^2 + k = 0 (which implies k = 0)
So, the equivalent form of equation (a) is given by substituting these values into equation (b):
(a) y = 4p(x - 0)^2 + 0
(a) y = 4px^2
As you can see, equation (a) matches the form of equation (b) when we substitute p = a/4 and k = 0. Therefore, equations (a) and (b) are equivalent.
Equation (a) represents a parabola with a vertical axis. The vertex of the parabola is at the point (h, k), and the parameter 'a' determines the direction and width of the parabola. When 'a' is positive, the parabola opens upward, and when 'a' is negative, the parabola opens downward. The equation is in the standard vertex form.
Equation (b) represents a parabola with a vertical axis as well. The vertex of the parabola is at the point (h, k), and the parameter 'p' is the distance between the vertex and the focus or the distance between the vertex and the directrix. The equation is in the standard form for a parabola with a vertical axis.
Equation (c) represents a parabola with a horizontal axis. The vertex of the parabola is at the point (h, k), and the parameter 'p' is the distance between the vertex and the focus or the distance between the vertex and the directrix. The equation is in the standard form for a parabola with a horizontal axis.
Now, let's see how equations (a) and (b) are equivalent. To do that, we need to rewrite equation (a) in the same form as equation (b).
Starting with equation (a):
y = a(x - h)^2 + k
Expanding the square:
y = a(x^2 - 2hx + h^2) + k
Distributing 'a':
y = ax^2 - 2ahx + ah^2 + k
From equation (b), we know that 4p = 2ah and h^2 = -ap
Substituting these values into equation (a):
y = ax^2 - (4p/h)x - ap + k
Rearranging terms:
y = ax^2 - (4p/h)x + (k-ap)
Comparing this with equation (b):
(x - h)^2 = 4p(y - k)
We can see that the two equations are equivalent, as they have the same form after expressing equation (a) in terms of 'p' and 'h'.