f(x)=x2+4x-y

Rewrite the equation in vertex form.

Isolate y to the left side of the equation.

Move all terms containing y to the left side of the equation.

Add y to both sides of the equation.

y+y=x2+4x

Add y and y.

2y=x2+4x

2y=x2+4x

Divide each term by 2 and simplify.

Divide each term in 2y=x2+4x by 2.

2y2=x22+4×2

Cancel the common factor of 2.

Cancel the common factor.

2y2=x22+4×2

Divide y by 1.

y=x22+4×2

y=x22+4×2

Cancel the common factor of 4 and 2.

Factor 2 out of 4x.

y=x22+2(2x)2

Cancel the common factors.

Factor 2 out of 2.

y=x22+2(2x)2(1)

Cancel the common factor.

y=x22+2(2x)2⋅1

Rewrite the expression.

y=x22+2×1

Divide 2x by 1.

y=x22+2x

y=x22+2x

y=x22+2x

y=x22+2x

y=x22+2x

Complete the square for x22+2x.

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

a=12,b=2,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=22(12)

Simplify the right side.

Cancel the common factor of 2.

Cancel the common factor.

d=22(12)

Rewrite the expression.

d=112

d=112

Multiply the numerator by the reciprocal of the denominator.

d=1⋅2

Multiply 2 by 1.

d=2

d=2

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

Simplify each term.

Raise 2 to the power of 2.

e=0-44(12)

Combine 4 and 12.

e=0-442

Divide 4 by 2.

e=0-42

Divide 4 by 2.

e=0-1⋅2

Multiply -1 by 2.

e=0-2

e=0-2

Subtract 2 from 0.

e=-2

e=-2

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

12⋅(x+2)2-2

12⋅(x+2)2-2

Set y equal to the new right side.

y=12⋅(x+2)2-2

y=12⋅(x+2)2-2

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

a=12

h=-2

k=-2

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

Opens Up

Find the vertex (h,k).

(-2,-2)

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⋅12

Simplify.

Combine 4 and 12.

142

Divide 4 by 2.

12

12

12

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.

(-2,-32)

(-2,-32)

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

x=-2

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=-52

y=-52

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

Direction: Opens Up

Vertex: (-2,-2)

Focus: (-2,-32)

Axis of Symmetry: x=-2

Directrix: y=-52

Direction: Opens Up

Vertex: (-2,-2)

Focus: (-2,-32)

Axis of Symmetry: x=-2

Directrix: y=-52

Replace the variable x with -4 in the expression.

f(-4)=(-4)22+2(-4)

Simplify the result.

Simplify each term.

Raise -4 to the power of 2.

f(-4)=162+2(-4)

Divide 16 by 2.

f(-4)=8+2(-4)

Multiply 2 by -4.

f(-4)=8-8

f(-4)=8-8

Subtract 8 from 8.

f(-4)=0

The final answer is 0.

0

0

The y value at x=-4 is 0.

y=0

Replace the variable x with -3 in the expression.

f(-3)=(-3)22+2(-3)

Simplify the result.

Simplify each term.

Raise -3 to the power of 2.

f(-3)=92+2(-3)

Multiply 2 by -3.

f(-3)=92-6

f(-3)=92-6

To write -6 as a fraction with a common denominator, multiply by 22.

f(-3)=92-6⋅22

Combine -6 and 22.

f(-3)=92+-6⋅22

Combine the numerators over the common denominator.

f(-3)=9-6⋅22

Simplify the numerator.

Multiply -6 by 2.

f(-3)=9-122

Subtract 12 from 9.

f(-3)=-32

f(-3)=-32

Move the negative in front of the fraction.

f(-3)=-32

The final answer is -32.

-32

-32

The y value at x=-3 is -32.

y=-32

Replace the variable x with 0 in the expression.

f(0)=(0)22+2(0)

Simplify the result.

Simplify each term.

Raising 0 to any positive power yields 0.

f(0)=02+2(0)

Divide 0 by 2.

f(0)=0+2(0)

Multiply 2 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)=(-1)22+2(-1)

Simplify the result.

Simplify each term.

Raise -1 to the power of 2.

f(-1)=12+2(-1)

Multiply 2 by -1.

f(-1)=12-2

f(-1)=12-2

To write -2 as a fraction with a common denominator, multiply by 22.

f(-1)=12-2⋅22

Combine -2 and 22.

f(-1)=12+-2⋅22

Combine the numerators over the common denominator.

f(-1)=1-2⋅22

Simplify the numerator.

Multiply -2 by 2.

f(-1)=1-42

Subtract 4 from 1.

f(-1)=-32

f(-1)=-32

Move the negative in front of the fraction.

f(-1)=-32

The final answer is -32.

-32

-32

The y value at x=-1 is -32.

y=-32

Graph the parabola using its properties and the selected points.

xy-40-3-32-2-2-1-3200

xy-40-3-32-2-2-1-3200

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: (-2,-2)

Focus: (-2,-32)

Axis of Symmetry: x=-2

Directrix: y=-52

xy-40-3-32-2-2-1-3200

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