A circle is defined by an equation in the format \(\displaystyle (x-h)^2+(y-k)^2=r^2\).

The center is indicated by the point \(\displaystyle (h,k)\) and the radius \(\displaystyle r\).

In the equation \(\displaystyle (x-2)^2+(y)^2=36=6^2\), the center is \(\displaystyle (2,0)\) and the radius is \(\displaystyle 6\).

An ellipse has an equation that can be written in the format\(\displaystyle \frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1\). The center is indicated by \(\displaystyle (h,k)\), or in this case \(\displaystyle (0,0)\).

The simplest way to know what kind of conic section an equation represents is by checking the coefficients in front of each variable. The equation must be in general form while you do this check. Luckily, this equation is already in general form, so it's easy to see. The general equation for a conic section is the following:

\(\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0\)

Assuming the term \(\displaystyle Bxy\) is 0 (which it usually is):

• If A equals C, the equation is a circle.
• If A and C have the same sign (but are not equal to each other), the equation is an ellipse.
• If either A or C equals 0, the equation is a parabola.
• If A and C are different signs (i.e. one is negative and one is positive), the equation is a hyperbola.

First, we need to make sure the conic section equation is in a form we recognize. Luckily, this equation is already in standard form:

\(\displaystyle \frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1\)

The first step is to determine the type of conic section this equation represents. Because there are two squared variables (\(\displaystyle x^{2}\) and \(\displaystyle y^{2}\)), this equation cannot be a parabola.  Because the coefficients in front of the squared variables are different signs (i.e. one is negative and the other is positive), this equation must be a hyperbola, not an ellipse.

In a hyperbola, the squared term with a positive coefficient represents the direction in which the hyperbola opens. In other words, if the \(\displaystyle x^{2}\) term is positive, the hyperbola opens horizontally. If the \(\displaystyle y^{2}\) term is positive, the hyperbola opens vertically. Therefore, this is a horizontal hyperbola.

The center is always found at \(\displaystyle (h, k)\), which in this case is \(\displaystyle (3, -2)\).

That leaves only the asymptotes. For a hyperbola, the slopes of the asymptotes can be found by dividing \(\displaystyle b\) by \(\displaystyle a\) (remember to always put the vertical value, \(\displaystyle b\), above the horizontal value, \(\displaystyle a\)). Remember that these slopes always come in pairs, with one being positive and the other being negative.

In this case, \(\displaystyle b\) is 3 and \(\displaystyle a\) is 2, so we get slopes of \(\displaystyle \frac{3}{2}\) and \(\displaystyle -\frac{3}{2}\).

For any parabola of the form  \(\displaystyle ax^2+bx+c\) ,  the \(\displaystyle x\)-coordinate of its vertex is 

\(\displaystyle x = \frac{-b}{2a}\)

So here, we have

\(\displaystyle x=\frac{-(-6))}{2(3)}\)

= \(\displaystyle 1\)

We plug this back into the original equation to find \(\displaystyle y\):

\(\displaystyle y = 3(1)^2-6(1)+1\)

= \(\displaystyle -2\)

Since this is an upwards-opening parabola, its minimum value will occur at the vertex.  The \(\displaystyle x\)-coordinate for the vertex of any parabola of the form

\(\displaystyle y = ax^2 + bx+c\)

is at

\(\displaystyle x = \frac{-b}{2a}\)

So here, 

\(\displaystyle x = \frac{-b}{2a}\)

\(\displaystyle = \frac{-(16)}{2(2)}\)

\(\displaystyle = -4\)

We plug this value back into the equation of the parabola, to find the value of the function at this \(\displaystyle x\). 

\(\displaystyle y = 2(-4)^2 + 16(4) - 7\)

\(\displaystyle =-39\)

Thus the minimal value of the expression is \(\displaystyle -39\).