# Calculus Examples

Complete the square on the right side of the equation.

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

Consider the vertex form of a parabola.

Find the value of using the formula .

Multiply by .

Multiply by .

Find the value of using the formula .

Simplify each term.

Multiply by .

Raise to the power of .

Multiply by .

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

Write each expression with a common denominator of , by multiplying each by an appropriate factor of .

Combine.

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply by .

Subtract from .

Move the negative in front of the fraction.

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

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

Since the value of is positive, the parabola opens up.

Opens Up

Find the vertex .

Find , the distance from the vertex to the focus.

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

Substitute the value of into the formula.

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Rewrite the expression.

Find the focus.

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

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

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

Find the directrix.

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.

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

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

Direction: Opens Up

Vertex:

Focus:

Axis of Symmetry:

Directrix:

Direction: Opens Up

Vertex:

Focus:

Axis of Symmetry:

Directrix:

Replace the variable with in the expression.

Simplify the result.

Simplify each term.

Remove parentheses.

One to any power is one.

Multiply by .

Simplify by adding and subtracting.

Subtract from .

Add and .

The final answer is .

The value at is .

Replace the variable with in the expression.

Simplify the result.

Simplify each term.

Remove parentheses.

Raising to any positive power yields .

Multiply by .

Simplify by adding zeros.

Add and .

Add and .

The final answer is .

The value at is .

Replace the variable with in the expression.

Simplify the result.

Simplify each term.

Remove parentheses.

Raise to the power of .

Multiply by .

Simplify by adding and subtracting.

Subtract from .

Add and .

The final answer is .

The value at is .

Replace the variable with in the expression.

Simplify the result.

Simplify each term.

Remove parentheses.

Raise to the power of .

Multiply by .

Simplify by adding and subtracting.

Subtract from .

Add and .

The final answer is .

The value at is .

Graph the parabola using its properties and the selected points.

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex:

Focus:

Axis of Symmetry:

Directrix: