Rewrite the equation in vertex form.

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.

Cancel the common factor of and .

Factor out of .

Cancel the common factors.

Cancel the common factor.

Rewrite the expression.

Cancel the common factor of and .

Factor out of .

Move the negative one from the denominator of .

Simplify the expression.

Rewrite as .

Multiply by .

Find the value of using the formula .

Simplify each term.

Raising to any positive power yields .

Multiply by .

Divide by .

Multiply by .

Add and .

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 negative, the parabola opens down.

Opens Down

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.

Multiply by .

Move the negative in front of the fraction.

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 Down

Vertex:

Focus:

Axis of Symmetry:

Directrix:

Direction: Opens Down

Vertex:

Focus:

Axis of Symmetry:

Directrix:

Replace the variable with in the expression.

Simplify the result.

Simplify each term.

Raise to the power of .

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.

Multiply by by adding the exponents.

Multiply by .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Raise to the power of .

Add and .

The final answer is .

The value at is .

Replace the variable with in the expression.

Simplify the result.

Simplify each term.

One to any power is one.

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 .

Multiply by .

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 Down

Vertex:

Focus:

Axis of Symmetry:

Directrix:

Graph f(x)=-2x^2+4