# Precalculus Examples

Step 1

Write as an equation.

Step 2

Step 2.1

Complete the square for .

Step 2.1.1

Use the form , to find the values of , , and .

Step 2.1.2

Consider the vertex form of a parabola.

Step 2.1.3

Find the value of using the formula .

Step 2.1.3.1

Substitute the values of and into the formula .

Step 2.1.3.2

Simplify the right side.

Step 2.1.3.2.1

Multiply by .

Step 2.1.3.2.2

Dividing two negative values results in a positive value.

Step 2.1.4

Find the value of using the formula .

Step 2.1.4.1

Substitute the values of , and into the formula .

Step 2.1.4.2

Simplify the right side.

Step 2.1.4.2.1

Simplify each term.

Step 2.1.4.2.1.1

Raise to the power of .

Step 2.1.4.2.1.2

Multiply by .

Step 2.1.4.2.1.3

Move the negative in front of the fraction.

Step 2.1.4.2.1.4

Multiply .

Step 2.1.4.2.1.4.1

Multiply by .

Step 2.1.4.2.1.4.2

Multiply by .

Step 2.1.4.2.2

To write as a fraction with a common denominator, multiply by .

Step 2.1.4.2.3

Combine and .

Step 2.1.4.2.4

Combine the numerators over the common denominator.

Step 2.1.4.2.5

Simplify the numerator.

Step 2.1.4.2.5.1

Multiply by .

Step 2.1.4.2.5.2

Add and .

Step 2.1.5

Substitute the values of , , and into the vertex form .

Step 2.2

Set equal to the new right side.

Step 3

Use the vertex form, , to determine the values of , , and .

Step 4

Since the value of is negative, the parabola opens down.

Opens Down

Step 5

Find the vertex .

Step 6

Step 6.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

Step 6.2

Substitute the value of into the formula.

Step 6.3

Cancel the common factor of and .

Step 6.3.1

Rewrite as .

Step 6.3.2

Move the negative in front of the fraction.

Step 7

Step 7.1

The focus of a parabola can be found by adding to the y-coordinate if the parabola opens up or down.

Step 7.2

Substitute the known values of , , and into the formula and simplify.

Step 8

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

Step 9

Step 9.1

The directrix of a parabola is the horizontal line found by subtracting from the y-coordinate of the vertex if the parabola opens up or down.

Step 9.2

Substitute the known values of and into the formula and simplify.

Step 10

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Down

Vertex:

Focus:

Axis of Symmetry:

Directrix:

Step 11