f(x)=x2+8x+9

Rewrite the equation in vertex form.

Complete the square for x2+8x+9.

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

a=1,b=8,c=9

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(1)

Cancel the common factor of 8 and 2.

Factor 2 out of 8.

d=2⋅42⋅1

Cancel the common factors.

Factor 2 out of 2⋅1.

d=2⋅42(1)

Cancel the common factor.

d=2⋅42⋅1

Rewrite the expression.

d=41

Divide 4 by 1.

d=4

d=4

d=4

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

Simplify each term.

Raise 8 to the power of 2.

e=9-644⋅1

Multiply 4 by 1.

e=9-644

Divide 64 by 4.

e=9-1⋅16

Multiply -1 by 16.

e=9-16

e=9-16

Subtract 16 from 9.

e=-7

e=-7

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

(x+4)2-7

(x+4)2-7

Set y equal to the new right side.

y=(x+4)2-7

y=(x+4)2-7

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

a=1

h=-4

k=-7

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

Opens Up

Find the vertex (h,k).

(-4,-7)

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.

(-4,-274)

(-4,-274)

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

x=-4

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=-294

y=-294

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

Direction: Opens Up

Vertex: (-4,-7)

Focus: (-4,-274)

Axis of Symmetry: x=-4

Directrix: y=-294

Direction: Opens Up

Vertex: (-4,-7)

Focus: (-4,-274)

Axis of Symmetry: x=-4

Directrix: y=-294

Replace the variable x with -5 in the expression.

f(-5)=(-5)2+8(-5)+9

Simplify the result.

Simplify each term.

Raise -5 to the power of 2.

f(-5)=25+8(-5)+9

Multiply 8 by -5.

f(-5)=25-40+9

f(-5)=25-40+9

Simplify by adding and subtracting.

Subtract 40 from 25.

f(-5)=-15+9

Add -15 and 9.

f(-5)=-6

f(-5)=-6

The final answer is -6.

-6

-6

The y value at x=-5 is -6.

y=-6

Replace the variable x with -6 in the expression.

f(-6)=(-6)2+8(-6)+9

Simplify the result.

Simplify each term.

Raise -6 to the power of 2.

f(-6)=36+8(-6)+9

Multiply 8 by -6.

f(-6)=36-48+9

f(-6)=36-48+9

Simplify by adding and subtracting.

Subtract 48 from 36.

f(-6)=-12+9

Add -12 and 9.

f(-6)=-3

f(-6)=-3

The final answer is -3.

-3

-3

The y value at x=-6 is -3.

y=-3

Replace the variable x with -3 in the expression.

f(-3)=(-3)2+8(-3)+9

Simplify the result.

Simplify each term.

Raise -3 to the power of 2.

f(-3)=9+8(-3)+9

Multiply 8 by -3.

f(-3)=9-24+9

f(-3)=9-24+9

Simplify by adding and subtracting.

Subtract 24 from 9.

f(-3)=-15+9

Add -15 and 9.

f(-3)=-6

f(-3)=-6

The final answer is -6.

-6

-6

The y value at x=-3 is -6.

y=-6

Replace the variable x with -2 in the expression.

f(-2)=(-2)2+8(-2)+9

Simplify the result.

Simplify each term.

Raise -2 to the power of 2.

f(-2)=4+8(-2)+9

Multiply 8 by -2.

f(-2)=4-16+9

f(-2)=4-16+9

Simplify by adding and subtracting.

Subtract 16 from 4.

f(-2)=-12+9

Add -12 and 9.

f(-2)=-3

f(-2)=-3

The final answer is -3.

-3

-3

The y value at x=-2 is -3.

y=-3

Graph the parabola using its properties and the selected points.

xy-6-3-5-6-4-7-3-6-2-3

xy-6-3-5-6-4-7-3-6-2-3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: (-4,-7)

Focus: (-4,-274)

Axis of Symmetry: x=-4

Directrix: y=-294

xy-6-3-5-6-4-7-3-6-2-3

Graph f(x)=x^2+8x+9