f(x)=7×2+40x-12

Rewrite the equation in vertex form.

Complete the square for 7×2+40x-12.

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

a=7,b=40,c=-12

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=402(7)

Cancel the common factor of 40 and 2.

Factor 2 out of 40.

d=2⋅202⋅7

Cancel the common factors.

Factor 2 out of 2⋅7.

d=2⋅202(7)

Cancel the common factor.

d=2⋅202⋅7

Rewrite the expression.

d=207

d=207

d=207

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

Simplify each term.

Raise 40 to the power of 2.

e=-12-16004⋅7

Multiply 4 by 7.

e=-12-160028

Cancel the common factor of 1600 and 28.

Factor 4 out of 1600.

e=-12-4(400)28

Cancel the common factors.

Factor 4 out of 28.

e=-12-4⋅4004⋅7

Cancel the common factor.

e=-12-4⋅4004⋅7

Rewrite the expression.

e=-12-4007

e=-12-4007

e=-12-4007

e=-12-4007

To write -12 as a fraction with a common denominator, multiply by 77.

e=-12⋅77-4007

Combine -12 and 77.

e=-12⋅77-4007

Combine the numerators over the common denominator.

e=-12⋅7-4007

Simplify the numerator.

Multiply -12 by 7.

e=-84-4007

Subtract 400 from -84.

e=-4847

e=-4847

Move the negative in front of the fraction.

e=-4847

e=-4847

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

7(x+207)2-4847

7(x+207)2-4847

Set y equal to the new right side.

y=7(x+207)2-4847

y=7(x+207)2-4847

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

a=7

h=-207

k=-4847

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

Opens Up

Find the vertex (h,k).

(-207,-4847)

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⋅7

Multiply 4 by 7.

128

128

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.

(-207,-193528)

(-207,-193528)

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

x=-207

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=-193728

y=-193728

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

Direction: Opens Up

Vertex: (-207,-4847)

Focus: (-207,-193528)

Axis of Symmetry: x=-207

Directrix: y=-193728

Direction: Opens Up

Vertex: (-207,-4847)

Focus: (-207,-193528)

Axis of Symmetry: x=-207

Directrix: y=-193728

Replace the variable x with -4 in the expression.

f(-4)=7(-4)2+40(-4)-12

Simplify the result.

Simplify each term.

Raise -4 to the power of 2.

f(-4)=7⋅16+40(-4)-12

Multiply 7 by 16.

f(-4)=112+40(-4)-12

Multiply 40 by -4.

f(-4)=112-160-12

f(-4)=112-160-12

Simplify by subtracting numbers.

Subtract 160 from 112.

f(-4)=-48-12

Subtract 12 from -48.

f(-4)=-60

f(-4)=-60

The final answer is -60.

-60

-60

The y value at x=-4 is -60.

y=-60

Replace the variable x with -2 in the expression.

f(-2)=7(-2)2+40(-2)-12

Simplify the result.

Simplify each term.

Raise -2 to the power of 2.

f(-2)=7⋅4+40(-2)-12

Multiply 7 by 4.

f(-2)=28+40(-2)-12

Multiply 40 by -2.

f(-2)=28-80-12

f(-2)=28-80-12

Simplify by subtracting numbers.

Subtract 80 from 28.

f(-2)=-52-12

Subtract 12 from -52.

f(-2)=-64

f(-2)=-64

The final answer is -64.

-64

-64

The y value at x=-2 is -64.

y=-64

Replace the variable x with -1 in the expression.

f(-1)=7(-1)2+40(-1)-12

Simplify the result.

Simplify each term.

Raise -1 to the power of 2.

f(-1)=7⋅1+40(-1)-12

Multiply 7 by 1.

f(-1)=7+40(-1)-12

Multiply 40 by -1.

f(-1)=7-40-12

f(-1)=7-40-12

Simplify by subtracting numbers.

Subtract 40 from 7.

f(-1)=-33-12

Subtract 12 from -33.

f(-1)=-45

f(-1)=-45

The final answer is -45.

-45

-45

The y value at x=-1 is -45.

y=-45

Graph the parabola using its properties and the selected points.

xy-4-60-207-4847-2-64-1-45

xy-4-60-207-4847-2-64-1-45

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: (-207,-4847)

Focus: (-207,-193528)

Axis of Symmetry: x=-207

Directrix: y=-193728

xy-4-60-207-4847-2-64-1-45

Graph f(x)=7x^2+40x-12