parabola help!!!?

Question

how do you name the vertex, axis of sym, focus, directrix and direction of opening???
maybe you can give me an example on how to do it.....
such as-
#1 4(y+2)=3(x-1)^2

#2 y/5=(x+2)^2

#3 x= 2y^2-8y+7

Answer 1

Let's look at the principles of a parabola and then apply them to your problems.

Given the following equation of a parabola:

4p(y - k) = (x - h)²

The parabola opens vertically because x is the squared term.

If p is positive it opens upward, and if p is negative it opens downward.

The vertex is (h,k).

Since the parabola opens vertically, the axis of symmetry is a vertical line. The line of symmetry goes thru the vertex. Its equation is x = h.

The focus is inside the parabola a distance of p from the vertex along the axis of symmetry. It is located at (h,k+p).

The directrix is outside the parabola. It is a distance of p from the vertex in the direction opposite that of the focus. It is perpendicular to the axis of symmetry. The equation of the directrix is:

y = k - p
__________________________

Now let's look at your first question and apply the principles above.

#1
4(y + 2) = 3(x - 1)²
(4/3)(y + 2) = (x - 1)²

The parabola opens vertically because x is the squared term.

4p = 4/3
p = 1/3

Since p is positive it opens upward.

The vertex (h,k) is (1,-2).

Since the parabola opens vertically, the axis of symmetry is a vertical line. The line of symmetry goes thru the vertex. Its equation is x = 1.

The focus is inside the parabola a distance of 1/3 from the vertex along the axis of symmetry.
It is located at (1,-2+1/3) = (1,-5/3).

The directrix is outside the parabola. It is a distance of 1/3 from the vertex in the direction opposite that of the focus. It is perpendicular to the axis of symmetry. The equation of the directrix is:

y = -2 - 1/3 = -7/3

The other two are done the same way.