f(x)=x2+7x+9

Rewrite the equation in vertex form.

Complete the square for x2+7x+9.

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

a=1,b=7,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=72(1)

Multiply 2 by 1.

d=72

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

Simplify each term.

Raise 7 to the power of 2.

e=9-494⋅1

Multiply 4 by 1.

e=9-494

e=9-494

To write 9 as a fraction with a common denominator, multiply by 44.

e=9⋅44-494

Combine 9 and 44.

e=9⋅44-494

Combine the numerators over the common denominator.

e=9⋅4-494

Simplify the numerator.

Multiply 9 by 4.

e=36-494

Subtract 49 from 36.

e=-134

e=-134

Move the negative in front of the fraction.

e=-134

e=-134

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

(x+72)2-134

(x+72)2-134

Set y equal to the new right side.

y=(x+72)2-134

y=(x+72)2-134

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

a=1

h=-72

k=-134

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

Opens Up

Find the vertex (h,k).

(-72,-134)

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.

(-72,-3)

(-72,-3)

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

x=-72

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=-72

y=-72

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

Direction: Opens Up

Vertex: (-72,-134)

Focus: (-72,-3)

Axis of Symmetry: x=-72

Directrix: y=-72

Direction: Opens Up

Vertex: (-72,-134)

Focus: (-72,-3)

Axis of Symmetry: x=-72

Directrix: y=-72

Replace the variable x with -5 in the expression.

f(-5)=(-5)2+7(-5)+9

Simplify the result.

Simplify each term.

Raise -5 to the power of 2.

f(-5)=25+7(-5)+9

Multiply 7 by -5.

f(-5)=25-35+9

f(-5)=25-35+9

Simplify by adding and subtracting.

Subtract 35 from 25.

f(-5)=-10+9

Add -10 and 9.

f(-5)=-1

f(-5)=-1

The final answer is -1.

-1

-1

The y value at x=-5 is -1.

y=-1

Replace the variable x with -6 in the expression.

f(-6)=(-6)2+7(-6)+9

Simplify the result.

Simplify each term.

Raise -6 to the power of 2.

f(-6)=36+7(-6)+9

Multiply 7 by -6.

f(-6)=36-42+9

f(-6)=36-42+9

Simplify by adding and subtracting.

Subtract 42 from 36.

f(-6)=-6+9

Add -6 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+7(-3)+9

Simplify the result.

Simplify each term.

Raise -3 to the power of 2.

f(-3)=9+7(-3)+9

Multiply 7 by -3.

f(-3)=9-21+9

f(-3)=9-21+9

Simplify by adding and subtracting.

Subtract 21 from 9.

f(-3)=-12+9

Add -12 and 9.

f(-3)=-3

f(-3)=-3

The final answer is -3.

-3

-3

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

y=-3

Replace the variable x with -2 in the expression.

f(-2)=(-2)2+7(-2)+9

Simplify the result.

Simplify each term.

Raise -2 to the power of 2.

f(-2)=4+7(-2)+9

Multiply 7 by -2.

f(-2)=4-14+9

f(-2)=4-14+9

Simplify by adding and subtracting.

Subtract 14 from 4.

f(-2)=-10+9

Add -10 and 9.

f(-2)=-1

f(-2)=-1

The final answer is -1.

-1

-1

The y value at x=-2 is -1.

y=-1

Graph the parabola using its properties and the selected points.

xy-63-5-1-72-134-3-3-2-1

xy-63-5-1-72-134-3-3-2-1

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: (-72,-134)

Focus: (-72,-3)

Axis of Symmetry: x=-72

Directrix: y=-72

xy-63-5-1-72-134-3-3-2-1

Graph f(x)=x^2+7x+9