Consider the functions y = x ², y = x ² + 4, and y = x ² - 4. The difference between y = x ² and y = x ² + 4, is the fact that the second function has been shifted 4 units up. The difference between y = x ² and y = x ² - 4 is the fact that the second function has been shifted down four units.

FACT: y = f (x) + k. If k > 0, then f (x) is shifted up k units, and if k < 0, then f (x) is shifted down k units.

y = x ²

y = x ² + 4

y = x ² - 4

Consider the functions y = x ², y = (x - 3) ², and y = (x + 3) ². The difference between y = x ² and y = (x - 3) ² is the fact that y = (x - 3) ² is shifted 3 units to the right. The difference between y = x ² and y = (x + 3) ² is the fact that the second one has been shifted 3 units to the left.

FACT: y = f (x - h). If h > 0, then the graph is shifted h units to right. If h < 0, then the graph is shifted h units to the left.

y = x ²

y = (x - 3) ²

y = (x + 3) ²

DEFINITION:

A circle is the set of points in a plane whose distance from a given fixed point in the plane is constant. The fixed point is the center of the circle, and the constant distance is the radius.

STANDARD EQUATION OF A CIRCLE

(x - h) ² + (y - k) ² = a ² where the center is (h, k) and the radius is a.

EXAMPLE 1:

Write the equation of a circle with center (2, -1) and has radius of 3.

SOLUTION: (x - 2) ² + (y - (-1)) ² = 3 ² ® (x - 2) ² + (y + 1) ² = 9

EXAMPLE 2:

Determine the center, radius, and intercepts of the circle x ² + y ² - 16x + 4y + 25 = 0.

SOLUTION: To be able the find the center and radius, I must complete the square.

To find the x-intercepts, let y = 0 and solve for x.

To find the y-intercepts, let x = 0 and solve for y.

We cannot take the square root of a negative number; therefore there are not y-intercepts.

The points that lie inside the circle (x - h) ² + (y - k) ² = a ² are the points less than a units from the center (h, k). These points satisfy the inequality (x - h) ² + (y - k) ² < a ², and they are called the interior of the circle. The circle's exterior consists of the points that lie more than a units from the center (h, k) and satisfy the inequality (x - h) ² + (y - k) ² > a ².

FACT:

y = ax ² is a parabola whose axis of symmetry is the y-axis and has vertex at the origin. (See figure 1) If a > 0, then the parabola will open upwards. If a < 0, then the parabola will open downwards.

figure 1

We can determine the vertex, axis of symmetry, and which way the parabola will open by completing the square to transform y = ax ² + bx + c into the form y = a (x - h) ² + k where the vertex is (h, k). The axis of symmetry is x = h, and a behaves like above.

EXAMPLE 3:

Determine the vertex, axis of symmetry, intercepts, and the direction it opens for the parabola y = -3x ² - 12x + 6.

SOLUTION: First we must complete the square.

The vertex is (-2, 18) and the axis of symmetry is x = -2. This parabola opens downward.

Now to find the intercepts. When x = 0, y = 6. When y = 0,