3×2+12x-18

Rewrite the equation in vertex form.

Complete the square for 3×2+12x-18.

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

a=3,b=12,c=-18

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=122(3)

Simplify the right side.

Cancel the common factor of 12 and 2.

Factor 2 out of 12.

d=2⋅62⋅3

Cancel the common factors.

Factor 2 out of 2⋅3.

d=2⋅62(3)

Cancel the common factor.

d=2⋅62⋅3

Rewrite the expression.

d=63

d=63

d=63

Cancel the common factor of 6 and 3.

Factor 3 out of 6.

d=3⋅23

Cancel the common factors.

Factor 3 out of 3.

d=3⋅23(1)

Cancel the common factor.

d=3⋅23⋅1

Rewrite the expression.

d=21

Divide 2 by 1.

d=2

d=2

d=2

d=2

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

Simplify each term.

Raise 12 to the power of 2.

e=-18-1444⋅3

Multiply 4 by 3.

e=-18-14412

Divide 144 by 12.

e=-18-1⋅12

Multiply -1 by 12.

e=-18-12

e=-18-12

Subtract 12 from -18.

e=-30

e=-30

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

3(x+2)2-30

3(x+2)2-30

Set y equal to the new right side.

y=3(x+2)2-30

y=3(x+2)2-30

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

a=3

h=-2

k=-30

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

Opens Up

Find the vertex (h,k).

(-2,-30)

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⋅3

Multiply 4 by 3.

112

112

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,-35912)

(-2,-35912)

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=-36112

y=-36112

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

Direction: Opens Up

Vertex: (-2,-30)

Focus: (-2,-35912)

Axis of Symmetry: x=-2

Directrix: y=-36112

Direction: Opens Up

Vertex: (-2,-30)

Focus: (-2,-35912)

Axis of Symmetry: x=-2

Directrix: y=-36112

Replace the variable x with -3 in the expression.

f(-3)=3(-3)2+12(-3)-18

Simplify the result.

Simplify each term.

Raise -3 to the power of 2.

f(-3)=3⋅9+12(-3)-18

Multiply 3 by 9.

f(-3)=27+12(-3)-18

Multiply 12 by -3.

f(-3)=27-36-18

f(-3)=27-36-18

Simplify by subtracting numbers.

Subtract 36 from 27.

f(-3)=-9-18

Subtract 18 from -9.

f(-3)=-27

f(-3)=-27

The final answer is -27.

-27

-27

The y value at x=-3 is -27.

y=-27

Replace the variable x with -4 in the expression.

f(-4)=3(-4)2+12(-4)-18

Simplify the result.

Simplify each term.

Raise -4 to the power of 2.

f(-4)=3⋅16+12(-4)-18

Multiply 3 by 16.

f(-4)=48+12(-4)-18

Multiply 12 by -4.

f(-4)=48-48-18

f(-4)=48-48-18

Simplify by subtracting numbers.

Subtract 48 from 48.

f(-4)=0-18

Subtract 18 from 0.

f(-4)=-18

f(-4)=-18

The final answer is -18.

-18

-18

The y value at x=-4 is -18.

y=-18

Replace the variable x with -1 in the expression.

f(-1)=3(-1)2+12(-1)-18

Simplify the result.

Simplify each term.

Raise -1 to the power of 2.

f(-1)=3⋅1+12(-1)-18

Multiply 3 by 1.

f(-1)=3+12(-1)-18

Multiply 12 by -1.

f(-1)=3-12-18

f(-1)=3-12-18

Simplify by subtracting numbers.

Subtract 12 from 3.

f(-1)=-9-18

Subtract 18 from -9.

f(-1)=-27

f(-1)=-27

The final answer is -27.

-27

-27

The y value at x=-1 is -27.

y=-27

Replace the variable x with 0 in the expression.

f(0)=3(0)2+12(0)-18

Simplify the result.

Simplify each term.

Raising 0 to any positive power yields 0.

f(0)=3⋅0+12(0)-18

Multiply 3 by 0.

f(0)=0+12(0)-18

Multiply 12 by 0.

f(0)=0+0-18

f(0)=0+0-18

Simplify by adding zeros.

Add 0 and 0.

f(0)=0-18

Subtract 18 from 0.

f(0)=-18

f(0)=-18

The final answer is -18.

-18

-18

The y value at x=0 is -18.

y=-18

Graph the parabola using its properties and the selected points.

xy-4-18-3-27-2-30-1-270-18

xy-4-18-3-27-2-30-1-270-18

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: (-2,-30)

Focus: (-2,-35912)

Axis of Symmetry: x=-2

Directrix: y=-36112

xy-4-18-3-27-2-30-1-270-18

Graph 3x^2+12x-18