y=6×2-12x

Rewrite the equation in vertex form.

Complete the square for 6×2-12x.

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

a=6,b=-12,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=-122(6)

Simplify the right side.

Cancel the common factor of 12 and 2.

Factor 2 out of 12.

d=-2⋅62⋅6

Cancel the common factors.

Factor 2 out of 2⋅6.

d=-2⋅62(6)

Cancel the common factor.

d=-2⋅62⋅6

Rewrite the expression.

d=-66

d=-66

d=-66

Cancel the common factor of 6.

Cancel the common factor.

d=-66

Divide 1 by 1.

d=-1⋅1

d=-1⋅1

Multiply -1 by 1.

d=-1

d=-1

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

Simplify each term.

Raise -12 to the power of 2.

e=0-1444⋅6

Multiply 4 by 6.

e=0-14424

Divide 144 by 24.

e=0-1⋅6

Multiply -1 by 6.

e=0-6

e=0-6

Subtract 6 from 0.

e=-6

e=-6

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

6(x-1)2-6

6(x-1)2-6

Set y equal to the new right side.

y=6(x-1)2-6

y=6(x-1)2-6

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

a=6

h=1

k=-6

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

Opens Up

Find the vertex (h,k).

(1,-6)

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⋅6

Multiply 4 by 6.

124

124

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.

(1,-14324)

(1,-14324)

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

x=1

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=-14524

y=-14524

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

Direction: Opens Up

Vertex: (1,-6)

Focus: (1,-14324)

Axis of Symmetry: x=1

Directrix: y=-14524

Direction: Opens Up

Vertex: (1,-6)

Focus: (1,-14324)

Axis of Symmetry: x=1

Directrix: y=-14524

Replace the variable x with 0 in the expression.

f(0)=6(0)2-12⋅0

Simplify the result.

Simplify each term.

Raising 0 to any positive power yields 0.

f(0)=6⋅0-12⋅0

Multiply 6 by 0.

f(0)=0-12⋅0

Multiply -12 by 0.

f(0)=0+0

f(0)=0+0

Add 0 and 0.

f(0)=0

The final answer is 0.

0

0

The y value at x=0 is 0.

y=0

Replace the variable x with -1 in the expression.

f(-1)=6(-1)2-12⋅-1

Simplify the result.

Simplify each term.

Raise -1 to the power of 2.

f(-1)=6⋅1-12⋅-1

Multiply 6 by 1.

f(-1)=6-12⋅-1

Multiply -12 by -1.

f(-1)=6+12

f(-1)=6+12

Add 6 and 12.

f(-1)=18

The final answer is 18.

18

18

The y value at x=-1 is 18.

y=18

Replace the variable x with 2 in the expression.

f(2)=6(2)2-12⋅2

Simplify the result.

Simplify each term.

Raise 2 to the power of 2.

f(2)=6⋅4-12⋅2

Multiply 6 by 4.

f(2)=24-12⋅2

Multiply -12 by 2.

f(2)=24-24

f(2)=24-24

Subtract 24 from 24.

f(2)=0

The final answer is 0.

0

0

The y value at x=2 is 0.

y=0

Replace the variable x with 3 in the expression.

f(3)=6(3)2-12⋅3

Simplify the result.

Simplify each term.

Raise 3 to the power of 2.

f(3)=6⋅9-12⋅3

Multiply 6 by 9.

f(3)=54-12⋅3

Multiply -12 by 3.

f(3)=54-36

f(3)=54-36

Subtract 36 from 54.

f(3)=18

The final answer is 18.

18

18

The y value at x=3 is 18.

y=18

Graph the parabola using its properties and the selected points.

xy-118001-620318

xy-118001-620318

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: (1,-6)

Focus: (1,-14324)

Axis of Symmetry: x=1

Directrix: y=-14524

xy-118001-620318

Graph y=6x^2-12x