f(x)=x2+20x

Rewrite the equation in vertex form.

Complete the square for x2+20x.

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

a=1,b=20,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=202(1)

Cancel the common factor of 20 and 2.

Factor 2 out of 20.

d=2⋅102⋅1

Cancel the common factors.

Factor 2 out of 2⋅1.

d=2⋅102(1)

Cancel the common factor.

d=2⋅102⋅1

Rewrite the expression.

d=101

Divide 10 by 1.

d=10

d=10

d=10

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

Simplify each term.

Raise 20 to the power of 2.

e=0-4004⋅1

Multiply 4 by 1.

e=0-4004

Divide 400 by 4.

e=0-1⋅100

Multiply -1 by 100.

e=0-100

e=0-100

Subtract 100 from 0.

e=-100

e=-100

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

(x+10)2-100

(x+10)2-100

Set y equal to the new right side.

y=(x+10)2-100

y=(x+10)2-100

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

a=1

h=-10

k=-100

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

Opens Up

Find the vertex (h,k).

(-10,-100)

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⋅1

Cancel the common factor of 1.

Cancel the common factor.

14⋅1

Rewrite the expression.

14

14

14

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.

(-10,-3994)

(-10,-3994)

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

x=-10

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=-4014

y=-4014

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

Direction: Opens Up

Vertex: (-10,-100)

Focus: (-10,-3994)

Axis of Symmetry: x=-10

Directrix: y=-4014

Direction: Opens Up

Vertex: (-10,-100)

Focus: (-10,-3994)

Axis of Symmetry: x=-10

Directrix: y=-4014

Replace the variable x with -11 in the expression.

f(-11)=(-11)2+20(-11)

Simplify the result.

Simplify each term.

Raise -11 to the power of 2.

f(-11)=121+20(-11)

Multiply 20 by -11.

f(-11)=121-220

f(-11)=121-220

Subtract 220 from 121.

f(-11)=-99

The final answer is -99.

-99

-99

The y value at x=-11 is -99.

y=-99

Replace the variable x with -12 in the expression.

f(-12)=(-12)2+20(-12)

Simplify the result.

Simplify each term.

Raise -12 to the power of 2.

f(-12)=144+20(-12)

Multiply 20 by -12.

f(-12)=144-240

f(-12)=144-240

Subtract 240 from 144.

f(-12)=-96

The final answer is -96.

-96

-96

The y value at x=-12 is -96.

y=-96

Replace the variable x with -9 in the expression.

f(-9)=(-9)2+20(-9)

Simplify the result.

Simplify each term.

Raise -9 to the power of 2.

f(-9)=81+20(-9)

Multiply 20 by -9.

f(-9)=81-180

f(-9)=81-180

Subtract 180 from 81.

f(-9)=-99

The final answer is -99.

-99

-99

The y value at x=-9 is -99.

y=-99

Replace the variable x with -8 in the expression.

f(-8)=(-8)2+20(-8)

Simplify the result.

Simplify each term.

Raise -8 to the power of 2.

f(-8)=64+20(-8)

Multiply 20 by -8.

f(-8)=64-160

f(-8)=64-160

Subtract 160 from 64.

f(-8)=-96

The final answer is -96.

-96

-96

The y value at x=-8 is -96.

y=-96

Graph the parabola using its properties and the selected points.

xy-12-96-11-99-10-100-9-99-8-96

xy-12-96-11-99-10-100-9-99-8-96

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: (-10,-100)

Focus: (-10,-3994)

Axis of Symmetry: x=-10

Directrix: y=-4014

xy-12-96-11-99-10-100-9-99-8-96

Graph f(x)=x^2+20x