This is the equation for a right-opening parabola with a vertex at . The distance from the vertex to the focus can be found by solving , so . This places the focus at .

Because the focus is at the origin, and the parabola opens to the right, this equation is in the form .

This particular parabola has a polar equation .

This is the equation for a down-opening parabola. The vertex is at . We can figure out the location of the focus by solving . This means that , so the focus is at .

Because the focus is at the origin, and the parabola opens down, the polar form of the equation is .

This equation is

.

First, multiply out :

we can re-arrange this a little bit by subtracting 2 from both sides and putting next to :

Now we can make the substitutions and :

We can solve for r using the quadratic formula:

factor out 4 inside the square root

To convert, make the substitutions and

subtract from both sides

Now we can solve using the quadratic formula:

Make the substitutions and :

subtract from both sides

We can solve for r using the quadratic formula:

Which of the following could be the graph of f(x)?

Begin by realizing this must be a downward facing parabola with its vertex at (0,3)

We know this because of the negative sign in front of the 5, and by the constant term of 3 on the end.

This narrows our options down to 2. One is much narrower than the other, although it may seem counterintuitive, the narrower one is what we need. This is because for every increase in x, we get a corresponding increase of times 5 in y. This translates to a graph that will get to higher values of y faster than a basic parabola. So, we need the graph below. to further confirm, try to find f(1)

So, the point (1,-2) must be on the graph, which means we must have:

The coefficient of the squared term tells us whether the parabola faces up or down. Parabolas in general, as in the parent function, are in the shape of a U. In the equation given, the coefficient of the squared term is . Generally, if the coefficient of the squared term is positive, the parabola faces up. If the coefficient is negative, the parabola faces down. Since  is negative, our parabola must face down. 

The coefficient of the squared term tells us whether the parabola faces up or down. Parabolas in general, as in the parent function, are in the shape of a U. In the equation given, the coefficient of the squared term is . Generally, if the coefficient of the squared term is positive, the parabola faces up. If the coefficient is negative, the parabola faces down. Since  is positive, our parabola must face up. 

The first step of the problem is to find the axis of symmetry using the following formula:

Where a and b are determined from the format for the equation of a parabola:

We can see from the equation given in the problem that a=1 and b=-3, so we can plug these values into the formula to find the axis of symmetry of our parabola:

Keep in mind that the vertex of the parabola lies directly on the axis of symmetry. That is, the x-coordinate of the axis of symmetry will be the same as that of the vertex of the parabola. Now that we know the vertex is at the same x-coordinate as the axis of symmetry, we can simply plug this value into our function to find the y-coordinate of the vertex:

So the vertex occurs at the point:

Rewrite the equation in standard form .

The vertex formula is:

Determine the necessary coefficients.

Plug in these values to the vertex formula.

The axis of symmetry is .