-2y2+12y-x-20=0

Add 2y2 to both sides of the equation.

12y-x-20=2y2

Subtract 12y from both sides of the equation.

-x-20=2y2-12y

Add 20 to both sides of the equation.

-x=2y2-12y+20

-x=2y2-12y+20

Multiply each term in -x=2y2-12y+20 by -1.

(-x)⋅-1=2y2⋅-1+(-12y)⋅-1+20⋅-1

Multiply (-x)⋅-1.

Multiply -1 by -1.

1x=2y2⋅-1+(-12y)⋅-1+20⋅-1

Multiply x by 1.

x=2y2⋅-1+(-12y)⋅-1+20⋅-1

x=2y2⋅-1+(-12y)⋅-1+20⋅-1

Simplify each term.

Multiply -1 by 2.

x=-2y2+(-12y)⋅-1+20⋅-1

Multiply -1 by -12.

x=-2y2+12y+20⋅-1

Multiply 20 by -1.

x=-2y2+12y-20

x=-2y2+12y-20

x=-2y2+12y-20

Rewrite the equation in vertex form.

Complete the square for -2y2+12y-20.

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

a=-2,b=12,c=-20

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(-2)

Simplify the right side.

Cancel the common factor of 12 and 2.

Factor 2 out of 12.

d=2⋅62⋅-2

Cancel the common factors.

Factor 2 out of 2⋅-2.

d=2⋅62(-2)

Cancel the common factor.

d=2⋅62⋅-2

Rewrite the expression.

d=6-2

d=6-2

d=6-2

Cancel the common factor of 6 and -2.

Factor 2 out of 6.

d=2(3)-2

Move the negative one from the denominator of 3-1.

d=-1⋅3

d=-1⋅3

Multiply -1 by 3.

d=-3

d=-3

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

Simplify each term.

Raise 12 to the power of 2.

e=-20-1444⋅-2

Multiply 4 by -2.

e=-20-144-8

Divide 144 by -8.

e=-20+18

Multiply -1 by -18.

e=-20+18

e=-20+18

Add -20 and 18.

e=-2

e=-2

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

-2(y-3)2-2

-2(y-3)2-2

Set x equal to the new right side.

x=-2(y-3)2-2

x=-2(y-3)2-2

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

a=-2

h=-2

k=3

Since the value of a is negative, the parabola opens left.

Opens Left

Find the vertex (h,k).

(-2,3)

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⋅-2

Simplify.

Multiply 4 by -2.

1-8

Move the negative in front of the fraction.

-18

-18

-18

Find the focus.

The focus of a parabola can be found by adding p to the x-coordinate h if the parabola opens left or right.

(h+p,k)

Substitute the known values of h, p, and k into the formula and simplify.

(-178,3)

(-178,3)

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

y=3

Find the directrix.

The directrix of a parabola is the vertical line found by subtracting p from the x-coordinate h of the vertex if the parabola opens left or right.

x=h-p

Substitute the known values of p and h into the formula and simplify.

x=-158

x=-158

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

Direction: Opens Left

Vertex: (-2,3)

Focus: (-178,3)

Axis of Symmetry: y=3

Directrix: x=-158

Direction: Opens Left

Vertex: (-2,3)

Focus: (-178,3)

Axis of Symmetry: y=3

Directrix: x=-158

Substitute the x value -4 into f(x)=6+2(-x-2)2. In this case, the point is (-4,4).

Replace the variable x with -4 in the expression.

f(-4)=6+2(-(-4)-2)2

Simplify the result.

Simplify the numerator.

Multiply -1 by -4.

f(-4)=6+2(4-2)2

Subtract 2 from 4.

f(-4)=6+2⋅22

Multiply 2 by 2.

f(-4)=6+42

Rewrite 4 as 22.

f(-4)=6+222

Pull terms out from under the radical, assuming positive real numbers.

f(-4)=6+22

Add 6 and 2.

f(-4)=82

f(-4)=82

Divide 8 by 2.

f(-4)=4

The final answer is 4.

y=4

y=4

Convert 4 to decimal.

=4

=4

Substitute the x value -4 into f(x)=6-2(-x-2)2. In this case, the point is (-4,2).

Replace the variable x with -4 in the expression.

f(-4)=6-2(-(-4)-2)2

Simplify the result.

Simplify the numerator.

Multiply -1 by -4.

f(-4)=6-2(4-2)2

Subtract 2 from 4.

f(-4)=6-2⋅22

Multiply 2 by 2.

f(-4)=6-42

Rewrite 4 as 22.

f(-4)=6-222

Pull terms out from under the radical, assuming positive real numbers.

f(-4)=6-1⋅22

Multiply -1 by 2.

f(-4)=6-22

Subtract 2 from 6.

f(-4)=42

f(-4)=42

Divide 4 by 2.

f(-4)=2

The final answer is 2.

y=2

y=2

Convert 2 to decimal.

=2

=2

Substitute the x value -3 into f(x)=6+2(-x-2)2. In this case, the point is (-3,3.70710678).

Replace the variable x with -3 in the expression.

f(-3)=6+2(-(-3)-2)2

Simplify the result.

Simplify the numerator.

Multiply -1 by -3.

f(-3)=6+2(3-2)2

Subtract 2 from 3.

f(-3)=6+2⋅12

Multiply 2 by 1.

f(-3)=6+22

f(-3)=6+22

The final answer is 6+22.

y=6+22

y=6+22

Convert 6+22 to decimal.

=3.70710678

=3.70710678

Substitute the x value -3 into f(x)=6-2(-x-2)2. In this case, the point is (-3,2.29289321).

Replace the variable x with -3 in the expression.

f(-3)=6-2(-(-3)-2)2

Simplify the result.

Simplify the numerator.

Multiply -1 by -3.

f(-3)=6-2(3-2)2

Subtract 2 from 3.

f(-3)=6-2⋅12

Multiply 2 by 1.

f(-3)=6-22

f(-3)=6-22

The final answer is 6-22.

y=6-22

y=6-22

Convert 6-22 to decimal.

=2.29289321

=2.29289321

Graph the parabola using its properties and the selected points.

xy-44-42-33.71-32.29-23

xy-44-42-33.71-32.29-23

Graph the parabola using its properties and the selected points.

Direction: Opens Left

Vertex: (-2,3)

Focus: (-178,3)

Axis of Symmetry: y=3

Directrix: x=-158

xy-44-42-33.71-32.29-23

Graph -2y^2+12y-x-20=0