The hyperbola in the question's graph is .

This could be discerned because it subtracts from and not the other way around, putting the zeros on the axis. This narrows the answers down to and .

Test a point to tell which inequality is being used in the graph.

The point is within the shaded region, for example.

, which is greater than 16, so the answer is .

The portion results in the graph being shifted 3 units to the left, while the results in the graph being shifted six units down. Vertical shifts are the same sign as the number outside the parentheses, while horizontal shifts are the OPPOSITE direction as the sign inside the parentheses, associated with .

To find the x-intercepts of the equation, we set the numerator equal to zero.

Remember that the "shifts" involved with circular functions are sort of like those found in parabolas. When you shift a parabola left or right, you have to think "oppositely". A right shift requires you to subtract from the x-component, and a left one requires you to add. Hence, this circle has no horizontal shift, but does shift 6 upward for the vertical component.

You can also remember the general formula for a circle with center at and a radius of .

Comparing this to the given equation, we can determine the center point.

The center point is at (0,6) and the circle has a radius of 5.

The equation of a circle is , in which (h, k) is the center of the circle. To derive the center of a circle from its equation, identify the constants immediately following x and y, and flip their signs. In the given equation, x is followed by -1 and y is followed by -6, so the coordinates of the center must be (1, 6).

First determine the center of the circle. The "x-3" portion of the circle equation tells us that the x coordinate is equal to 3. The "y-3" portion of the circle equation tells us that the y coordinate is equal to 3 as well. Therefore, the center of the circle is at (3,3).

To find the distance between (3,3) and (0,0), it is necessary to use the Pythagorean Theorem . Where "a" and "b" are equal to 3

(to visualize, you may draw the two points on a graph, and create a triangle. The line connecting the two points is the hypotenuse, aka "c." )

The equation given represents a circle.

represents the center, and is the radius.

The center is at:

Set up an equation to solve the radius.

The radius is:

The answer is:

A parabola is one example of a quadratic function, regardless of whether it points upwards or downwards.

The red line represents a quadratic function and will have a formula similar to .

The blue line represents a linear function and will have a formula similar to .

The green line represents an exponential function and will have a formula similar to .

The purple line represents an absolute value function and will have a formula similar to .

For any parabola, the general equation is

, and the x-coordinate of its vertex is given by

.

For the given problem, the x-coordinate is

.

To find the y-coordinate, plug into the original equation:

Therefore the vertex is at .

The portion results in the graph being shifted 3 units to the left, while the results in the graph being shifted six units down. Vertical shifts are the same sign as the number outside the parentheses, while horizontal shifts are the OPPOSITE direction as the sign inside the parentheses, associated with .