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

Math
-2y2+12y-x-20=0
Move all terms not containing x to the right side of the equation.
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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
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Multiply each term in -x=2y2-12y+20 by -1.
(-x)⋅-1=2y2⋅-1+(-12y)⋅-1+20⋅-1
Multiply (-x)⋅-1.
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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.
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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
Find the properties of the given parabola.
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Rewrite the equation in vertex form.
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Complete the square for -2y2+12y-20.
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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.
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Cancel the common factor of 12 and 2.
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Factor 2 out of 12.
d=2⋅62⋅-2
Cancel the common factors.
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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.
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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.
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Simplify each term.
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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.
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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.
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Multiply 4 by -2.
1-8
Move the negative in front of the fraction.
-18
-18
-18
Find the focus.
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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.
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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
Select a few x values, and plug them into the equation to find the corresponding y values. The x values should be selected around the vertex.
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Substitute the x value -4 into f(x)=6+2(-x-2)2. In this case, the point is (-4,4).
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Replace the variable x with -4 in the expression.
f(-4)=6+2(-(-4)-2)2
Simplify the result.
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Simplify the numerator.
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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).
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Replace the variable x with -4 in the expression.
f(-4)=6-2(-(-4)-2)2
Simplify the result.
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Simplify the numerator.
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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).
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Replace the variable x with -3 in the expression.
f(-3)=6+2(-(-3)-2)2
Simplify the result.
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Simplify the numerator.
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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).
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Replace the variable x with -3 in the expression.
f(-3)=6-2(-(-3)-2)2
Simplify the result.
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Simplify the numerator.
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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

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