Rewrite the equation in vertex form.

Simplify .

Apply the distributive property.

Simplify.

Combine and .

Cancel the common factor of .

Factor out of .

Factor out of .

Cancel the common factor.

Rewrite the expression.

Combine and .

Combine and .

Complete the square for .

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

Consider the vertex form of a parabola.

Substitute the values of and into the formula .

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

Combine and .

Cancel the common factor of and .

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Multiply the numerator by the reciprocal of the denominator.

Cancel the common factor of .

Factor out of .

Cancel the common factor.

Rewrite the expression.

Multiply by .

Find the value of using the formula .

Simplify each term.

Simplify the numerator.

Apply the product rule to .

Raise to the power of .

Apply the product rule to .

One to any power is one.

Raise to the power of .

Combine and .

Multiply by .

Divide by .

Divide by .

Combine the numerators over the common denominator.

Subtract from .

Cancel the common factor of and .

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

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

Set equal to the new right side.

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.

Simplify.

Combine and .

Simplify by dividing numbers.

Divide by .

Divide by .

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.

Raising to any positive power yields .

Divide by .

Divide by .

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.

Raise to the power of .

Move the negative in front of the fraction.

Multiply .

Multiply by .

Multiply by .

Combine fractions.

Combine fractions with similar denominators.

Simplify the expression.

Add and .

Divide by .

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

Combine and .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply by .

Add and .

The final answer is .

The value at is .

Replace the variable with in the expression.

Simplify the result.

Simplify each term.

Raise to the power of .

Divide by .

Divide by .

Multiply by .

Find the common denominator.

Write as a fraction with denominator .

Multiply by .

Multiply and .

Write as a fraction with denominator .

Multiply by .

Multiply and .

Combine fractions.

Combine fractions with similar denominators.

Multiply by .

Simplify the numerator.

Subtract from .

Add and .

The final answer is .

The value at is .

Replace the variable with in the expression.

Simplify the result.

Raise to the power of .

Combine fractions with similar denominators.

Simplify the expression.

Add and .

Divide by .

Move the negative in front of the fraction.

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

Combine and .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply by .

Subtract from .

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:

Graph f(t)=1/4*(t^2-2t+15)