y=x2⋅6

Move 6 to the left of x2.

y=6×2

Rewrite the equation in vertex form.

Complete the square for 6×2.

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

a=6,b=0,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=02(6)

Simplify the right side.

Cancel the common factor of 0 and 2.

Factor 2 out of 0.

d=2(0)2(6)

Cancel the common factors.

Cancel the common factor.

d=2⋅02⋅6

Rewrite the expression.

d=06

d=06

d=06

Cancel the common factor of 0 and 6.

Factor 6 out of 0.

d=6(0)6

Cancel the common factors.

Factor 6 out of 6.

d=6⋅06⋅1

Cancel the common factor.

d=6⋅06⋅1

Rewrite the expression.

d=01

Divide 0 by 1.

d=0

d=0

d=0

d=0

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

Simplify each term.

Raising 0 to any positive power yields 0.

e=0-04⋅6

Multiply 4 by 6.

e=0-024

Divide 0 by 24.

e=0-0

Multiply -1 by 0.

e=0+0

e=0+0

Add 0 and 0.

e=0

e=0

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

6×2

6×2

Set y equal to the new right side.

y=6×2

y=6×2

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

a=6

h=0

k=0

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

Opens Up

Find the vertex (h,k).

(0,0)

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.

(0,124)

(0,124)

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

x=0

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=-124

y=-124

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

Direction: Opens Up

Vertex: (0,0)

Focus: (0,124)

Axis of Symmetry: x=0

Directrix: y=-124

Direction: Opens Up

Vertex: (0,0)

Focus: (0,124)

Axis of Symmetry: x=0

Directrix: y=-124

Replace the variable x with -1 in the expression.

f(-1)=6(-1)2

Simplify the result.

Raise -1 to the power of 2.

f(-1)=6⋅1

Multiply 6 by 1.

f(-1)=6

The final answer is 6.

6

6

The y value at x=-1 is 6.

y=6

Replace the variable x with -2 in the expression.

f(-2)=6(-2)2

Simplify the result.

Raise -2 to the power of 2.

f(-2)=6⋅4

Multiply 6 by 4.

f(-2)=24

The final answer is 24.

24

24

The y value at x=-2 is 24.

y=24

Replace the variable x with 1 in the expression.

f(1)=6(1)2

Simplify the result.

One to any power is one.

f(1)=6⋅1

Multiply 6 by 1.

f(1)=6

The final answer is 6.

6

6

The y value at x=1 is 6.

y=6

Replace the variable x with 2 in the expression.

f(2)=6(2)2

Simplify the result.

Raise 2 to the power of 2.

f(2)=6⋅4

Multiply 6 by 4.

f(2)=24

The final answer is 24.

24

24

The y value at x=2 is 24.

y=24

Graph the parabola using its properties and the selected points.

xy-224-160016224

xy-224-160016224

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: (0,0)

Focus: (0,124)

Axis of Symmetry: x=0

Directrix: y=-124

xy-224-160016224

Graph y=x^2*6