The vertex can be thought of as the center of a parabola. Begin by finding the axis of symmetry with the following formula:

Where b and a come from the standard equation of a parabola:

So given our parabola

This gives us the x-coordinate of our vertex. find the y-coordinate by plugging in our x-coordinate.

So our vertex is:

The polynomial is already in  format.

To find the vertex, use the following equation:

Substitute the coefficients and solve for the vertex.

The vertex is at .

Rewrite  in standard parabolic form, .

Write the vertex formula and substitute the values.

The equation of the axis of symmetry is .

Substitute this value back into the original equation .

The vertex is at .

Find the axis of symmetry and the vertex of the parabola given by the following equation:

To find the axis of symmetry of a parabola in standard form, , use the following equation:

So...

This means that we have an axis of symmetry at . Or, to put it more plainly, at  we could draw a vertical line which would perfectly cut our parabola in half!

So, we are halfway there, now we need the coordinates of our vertex. We already know the x-coordinate, which is 7. To find the y-coordinate, simply plug 7 into the parabola's formula and solve!

This makes our vertex the point 

The vertex form for a parabola is given below:

To complete the square, take the coefficient next to the x term, divide by  and raise the number to the second power. In this case, . Then take value and add it to add inside the parenthesis and subtract on the outside. 

Now factor and simplify:

Fromt the values of  and , the vertex is at 

The vertex form for a parabola is given below:

To complete the square, take the value next to the x term, divide by 2 and raise the number to the second power. In this case,. Then take value and add it to add inside the parenthesis and subtract on the outside. 

Now factor and simplify:

Fromt the values of h and k, the vertex is at 

The vertex form for a parabola is given below:

To complete the square, take the value next to the x term, divide by  and raise the number to the second power. In this case, . Then take value and add it to add inside the parenthesis and subtract on the outside. Remember to distribute before subtracting to the outside.

Now factor and simplify:

From the values of  and , the vertex is at 

The vertex form for a parabola is given below:

Factor the equation and transform it into the vertex form.

To complete the square, take the value next to the x term, divide by  and raise the number to the second power. In this case, . Then take value and add it to add inside the parenthesis and subtract on the outside. 

Now factor and simplify:

From the values of  and , the vertex is at 

The vertex form for a parabola is given below:

To complete the square, take the value next to the x term, divide by 2 and raise the number to the second power. In this case, . Then take value and add it to add inside the parenthesis and subtract on the outside. 

Now factor and simplify:

Fromt the values of  and , the vertex is at 

The standard form for the equation of a circle with radius , and centered at point  is

.

Here, , so the equation is

.

Note: one way to think of this equation is to remember the Pythagorean Theorem.

If the center is at the origin then the equation is

.

This describes a right triangle for any x and y that satisfy this equation.  Here r is the hypotenues, but when all values of x and y are used it stays the same and the points map out a circle with radius r.

The rules of graph translation apply in the same way as with any function.  That is they move the origin in the opposite direction by a and/or b.