f(x)=-4×2-8x

Rewrite the equation in vertex form.

Complete the square for -4×2-8x.

Use the form ax2+bx+c, to find the values of a, b, and c.

a=-4,b=-8,c=0

Consider the vertex form of a parabola.

a(x+d)2+e

Substitute the values of a and b into the formula d=b2a.

d=-82(-4)

Simplify the right side.

Cancel the common factor of 8 and 2.

Factor 2 out of 8.

d=-2⋅42⋅-4

Cancel the common factors.

Factor 2 out of 2⋅-4.

d=-2⋅42(-4)

Cancel the common factor.

d=-2⋅42⋅-4

Rewrite the expression.

d=-4-4

d=-4-4

d=-4-4

Cancel the common factor of 4 and -4.

Factor 4 out of 4.

d=-4(1)-4

Move the negative one from the denominator of 1-1.

d=-(-1⋅1)

d=-(-1⋅1)

Multiply.

Multiply -1 by 1.

d=1

Multiply -1 by -1.

d=1

d=1

d=1

Find the value of e using the formula e=c-b24a.

Simplify each term.

Raise -8 to the power of 2.

e=0-644⋅-4

Multiply 4 by -4.

e=0-64-16

Divide 64 by -16.

e=0+4

Multiply -1 by -4.

e=0+4

e=0+4

Add 0 and 4.

e=4

e=4

Substitute the values of a, d, and e into the vertex form a(x+d)2+e.

-4(x+1)2+4

-4(x+1)2+4

Set y equal to the new right side.

y=-4(x+1)2+4

y=-4(x+1)2+4

Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.

a=-4

h=-1

k=4

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

Opens Down

Find the vertex (h,k).

(-1,4)

Find p, 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.

14a

Substitute the value of a into the formula.

14⋅-4

Simplify.

Multiply 4 by -4.

1-16

Move the negative in front of the fraction.

-116

-116

-116

Find the focus.

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

(h,k+p)

Substitute the known values of h, p, and k into the formula and simplify.

(-1,6316)

(-1,6316)

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

x=-1

Find the directrix.

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

y=k-p

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

y=6516

y=6516

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

Direction: Opens Down

Vertex: (-1,4)

Focus: (-1,6316)

Axis of Symmetry: x=-1

Directrix: y=6516

Direction: Opens Down

Vertex: (-1,4)

Focus: (-1,6316)

Axis of Symmetry: x=-1

Directrix: y=6516

Replace the variable x with -2 in the expression.

f(-2)=-4(-2)2-8⋅-2

Simplify the result.

Simplify each term.

Raise -2 to the power of 2.

f(-2)=-4⋅4-8⋅-2

Multiply -4 by 4.

f(-2)=-16-8⋅-2

Multiply -8 by -2.

f(-2)=-16+16

f(-2)=-16+16

Add -16 and 16.

f(-2)=0

The final answer is 0.

0

0

The y value at x=-2 is 0.

y=0

Replace the variable x with -3 in the expression.

f(-3)=-4(-3)2-8⋅-3

Simplify the result.

Simplify each term.

Raise -3 to the power of 2.

f(-3)=-4⋅9-8⋅-3

Multiply -4 by 9.

f(-3)=-36-8⋅-3

Multiply -8 by -3.

f(-3)=-36+24

f(-3)=-36+24

Add -36 and 24.

f(-3)=-12

The final answer is -12.

-12

-12

The y value at x=-3 is -12.

y=-12

Replace the variable x with 0 in the expression.

f(0)=-4(0)2-8⋅0

Simplify the result.

Simplify each term.

Raising 0 to any positive power yields 0.

f(0)=-4⋅0-8⋅0

Multiply -4 by 0.

f(0)=0-8⋅0

Multiply -8 by 0.

f(0)=0+0

f(0)=0+0

Add 0 and 0.

f(0)=0

The final answer is 0.

0

0

The y value at x=0 is 0.

y=0

Replace the variable x with 1 in the expression.

f(1)=-4(1)2-8⋅1

Simplify the result.

Simplify each term.

One to any power is one.

f(1)=-4⋅1-8⋅1

Multiply -4 by 1.

f(1)=-4-8⋅1

Multiply -8 by 1.

f(1)=-4-8

f(1)=-4-8

Subtract 8 from -4.

f(1)=-12

The final answer is -12.

-12

-12

The y value at x=1 is -12.

y=-12

Graph the parabola using its properties and the selected points.

xy-3-12-20-14001-12

xy-3-12-20-14001-12

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: (-1,4)

Focus: (-1,6316)

Axis of Symmetry: x=-1

Directrix: y=6516

xy-3-12-20-14001-12

Graph f(x)=-4x^2-8x