Graph y+1=-1/12*(x-3)^2

Math
y+1=-112⋅(x-3)2
Solve for y.
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Simplify -112⋅(x-3)2.
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Rewrite (x-3)2 as (x-3)(x-3).
y+1=-112⋅((x-3)(x-3))
Expand (x-3)(x-3) using the FOIL Method.
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Apply the distributive property.
y+1=-112⋅(x(x-3)-3(x-3))
Apply the distributive property.
y+1=-112⋅(x⋅x+x⋅-3-3(x-3))
Apply the distributive property.
y+1=-112⋅(x⋅x+x⋅-3-3x-3⋅-3)
y+1=-112⋅(x⋅x+x⋅-3-3x-3⋅-3)
Simplify and combine like terms.
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Simplify each term.
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Multiply x by x.
y+1=-112⋅(x2+x⋅-3-3x-3⋅-3)
Move -3 to the left of x.
y+1=-112⋅(x2-3⋅x-3x-3⋅-3)
Multiply -3 by -3.
y+1=-112⋅(x2-3x-3x+9)
y+1=-112⋅(x2-3x-3x+9)
Subtract 3x from -3x.
y+1=-112⋅(x2-6x+9)
y+1=-112⋅(x2-6x+9)
Apply the distributive property.
y+1=-112×2-112(-6x)-112⋅9
Simplify.
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Combine x2 and 112.
y+1=-x212-112(-6x)-112⋅9
Cancel the common factor of 6.
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Move the leading negative in -112 into the numerator.
y+1=-x212+-112(-6x)-112⋅9
Factor 6 out of 12.
y+1=-x212+-16(2)(-6x)-112⋅9
Factor 6 out of -6x.
y+1=-x212+-16(2)(6(-x))-112⋅9
Cancel the common factor.
y+1=-x212+-16⋅2(6(-x))-112⋅9
Rewrite the expression.
y+1=-x212+-12(-x)-112⋅9
y+1=-x212+-12(-x)-112⋅9
Combine -12 and x.
y+1=-x212–x2-112⋅9
Cancel the common factor of 3.
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Move the leading negative in -112 into the numerator.
y+1=-x212–x2+-112⋅9
Factor 3 out of 12.
y+1=-x212–x2+-13(4)⋅9
Factor 3 out of 9.
y+1=-x212–x2+-13⋅4⋅(3⋅3)
Cancel the common factor.
y+1=-x212–x2+-13⋅4⋅(3⋅3)
Rewrite the expression.
y+1=-x212–x2+-14⋅3
y+1=-x212–x2+-14⋅3
Combine -14 and 3.
y+1=-x212–x2+-1⋅34
Multiply -1 by 3.
y+1=-x212–x2+-34
y+1=-x212–x2+-34
Simplify each term.
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Move the negative in front of the fraction.
y+1=-x212–x2+-34
Multiply –x2.
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Multiply -1 by -1.
y+1=-x212+1×2+-34
Multiply x2 by 1.
y+1=-x212+x2+-34
y+1=-x212+x2+-34
Move the negative in front of the fraction.
y+1=-x212+x2-34
y+1=-x212+x2-34
y+1=-x212+x2-34
Move all terms not containing y to the right side of the equation.
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Subtract 1 from both sides of the equation.
y=-x212+x2-34-1
To write -1 as a fraction with a common denominator, multiply by 44.
y=-x212+x2-34-1⋅44
Combine -1 and 44.
y=-x212+x2-34+-1⋅44
Combine the numerators over the common denominator.
y=-x212+x2+-3-1⋅44
Simplify the numerator.
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Multiply -1 by 4.
y=-x212+x2+-3-44
Subtract 4 from -3.
y=-x212+x2+-74
y=-x212+x2+-74
Move the negative in front of the fraction.
y=-x212+x2-74
y=-x212+x2-74
y=-x212+x2-74
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 -x212+x2-74.
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Use the form ax2+bx+c, to find the values of a, b, and c.
a=-112,b=12,c=-74
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(-112)
Simplify the right side.
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Multiply the numerator by the reciprocal of the denominator.
d=12⋅12⋅(-1(112))
Cancel the common factor of 1 and -1.
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Rewrite 1 as -1(-1).
d=12⋅-1⋅-12⋅(-1(112))
Move the negative in front of the fraction.
d=12⋅(-12(112))
d=12⋅(-12(112))
Combine 2 and 112.
d=12⋅(-1212)
Cancel the common factor of 2 and 12.
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Factor 2 out of 2.
d=12⋅(-12(1)12)
Cancel the common factors.
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Factor 2 out of 12.
d=12⋅(-12⋅12⋅6)
Cancel the common factor.
d=12⋅(-12⋅12⋅6)
Rewrite the expression.
d=12⋅(-116)
d=12⋅(-116)
d=12⋅(-116)
Multiply.
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Multiply the numerator by the reciprocal of the denominator.
d=12⋅(-(1⋅6))
Multiply 6 by 1.
d=12⋅(-1⋅6)
Multiply -1 by 6.
d=12⋅-6
d=12⋅-6
Cancel the common factor of 2.
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Factor 2 out of -6.
d=12⋅(2(-3))
Cancel the common factor.
d=12⋅(2⋅-3)
Rewrite the expression.
d=-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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Simplify the numerator.
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Apply the product rule to 12.
e=-74-12224⋅(-1(112))
One to any power is one.
e=-74-1224⋅(-1(112))
Raise 2 to the power of 2.
e=-74-144⋅(-1(112))
e=-74-144⋅(-1(112))
Simplify the denominator.
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Multiply 4 by -1.
e=-74-14-4(112)
Combine -4 and 112.
e=-74-14-412
e=-74-14-412
Reduce the expression by cancelling the common factors.
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Cancel the common factor of -4 and 12.
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Factor 4 out of -4.
e=-74-144(-1)12
Cancel the common factors.
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Factor 4 out of 12.
e=-74-144⋅-14⋅3
Cancel the common factor.
e=-74-144⋅-14⋅3
Rewrite the expression.
e=-74-14-13
e=-74-14-13
e=-74-14-13
Move the negative in front of the fraction.
e=-74-14-13
e=-74-14-13
Multiply the numerator by the reciprocal of the denominator.
e=-74-(14⋅(-1⋅3))
Multiply -1 by 3.
e=-74-(14⋅-3)
Combine 14 and -3.
e=-74–34
Move the negative in front of the fraction.
e=-74+34
Multiply –34.
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Multiply -1 by -1.
e=-74+1(34)
Multiply 34 by 1.
e=-74+34
e=-74+34
e=-74+34
Combine the numerators over the common denominator.
e=-7+34
Add -7 and 3.
e=-44
Divide -4 by 4.
e=-1
e=-1
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
-112⋅(x-3)2-1
-112⋅(x-3)2-1
Set y equal to the new right side.
y=-112⋅(x-3)2-1
y=-112⋅(x-3)2-1
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=-112
h=3
k=-1
Since the value of a is negative, the parabola opens down.
Opens Down
Find the vertex (h,k).
(3,-1)
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⋅(-112)
Simplify.
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Cancel the common factor of 1 and -1.
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Rewrite 1 as -1(-1).
-1(-1)4⋅(-112)
Move the negative in front of the fraction.
-14(112)
-14(112)
Combine 4 and 112.
-1412
Cancel the common factor of 4 and 12.
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Factor 4 out of 4.
-14(1)12
Cancel the common factors.
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Factor 4 out of 12.
-14⋅14⋅3
Cancel the common factor.
-14⋅14⋅3
Rewrite the expression.
-113
-113
-113
Multiply the numerator by the reciprocal of the denominator.
-(1⋅3)
Multiply 3 by 1.
-1⋅3
Multiply -1 by 3.
-3
-3
-3
Find the focus.
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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.
(3,-4)
(3,-4)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=3
Find the directrix.
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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=2
y=2
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Down
Vertex: (3,-1)
Focus: (3,-4)
Axis of Symmetry: x=3
Directrix: y=2
Direction: Opens Down
Vertex: (3,-1)
Focus: (3,-4)
Axis of Symmetry: x=3
Directrix: y=2
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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Replace the variable x with 2 in the expression.
f(2)=-(2)212+22-74
Simplify the result.
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Simplify each term.
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Raise 2 to the power of 2.
f(2)=-412+22-74
Cancel the common factor of 4 and 12.
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Factor 4 out of 4.
f(2)=-4(1)12+22-74
Cancel the common factors.
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Factor 4 out of 12.
f(2)=-4⋅14⋅3+22-74
Cancel the common factor.
f(2)=-4⋅14⋅3+22-74
Rewrite the expression.
f(2)=-13+22-74
f(2)=-13+22-74
f(2)=-13+22-74
Divide 2 by 2.
f(2)=-13+1-74
f(2)=-13+1-74
Simplify the expression.
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Write 1 as a fraction with a common denominator.
f(2)=-13+33-74
Combine the numerators over the common denominator.
f(2)=-1+33-74
f(2)=-1+33-74
Add -1 and 3.
f(2)=23-74
To write 23 as a fraction with a common denominator, multiply by 44.
f(2)=23⋅44-74
To write -74 as a fraction with a common denominator, multiply by 33.
f(2)=23⋅44-74⋅33
Write each expression with a common denominator of 12, by multiplying each by an appropriate factor of 1.
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Multiply 23 and 44.
f(2)=2⋅43⋅4-74⋅33
Multiply 3 by 4.
f(2)=2⋅412-74⋅33
Multiply 74 and 33.
f(2)=2⋅412-7⋅34⋅3
Multiply 4 by 3.
f(2)=2⋅412-7⋅312
f(2)=2⋅412-7⋅312
Combine the numerators over the common denominator.
f(2)=2⋅4-7⋅312
Simplify the numerator.
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Multiply 2 by 4.
f(2)=8-7⋅312
Multiply -7 by 3.
f(2)=8-2112
Subtract 21 from 8.
f(2)=-1312
f(2)=-1312
Move the negative in front of the fraction.
f(2)=-1312
The final answer is -1312.
-1312
-1312
The y value at x=2 is -1312.
y=-1312
Replace the variable x with 1 in the expression.
f(1)=-(1)212+12-74
Simplify the result.
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One to any power is one.
f(1)=-112+12-74
To write 12 as a fraction with a common denominator, multiply by 66.
f(1)=-112+12⋅66-74
Write each expression with a common denominator of 12, by multiplying each by an appropriate factor of 1.
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Multiply 12 and 66.
f(1)=-112+62⋅6-74
Multiply 2 by 6.
f(1)=-112+612-74
f(1)=-112+612-74
Combine the numerators over the common denominator.
f(1)=-1+612-74
Add -1 and 6.
f(1)=512-74
To write -74 as a fraction with a common denominator, multiply by 33.
f(1)=512-74⋅33
Write each expression with a common denominator of 12, by multiplying each by an appropriate factor of 1.
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Multiply 74 and 33.
f(1)=512-7⋅34⋅3
Multiply 4 by 3.
f(1)=512-7⋅312
f(1)=512-7⋅312
Combine the numerators over the common denominator.
f(1)=5-7⋅312
Simplify the numerator.
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Multiply -7 by 3.
f(1)=5-2112
Subtract 21 from 5.
f(1)=-1612
f(1)=-1612
Cancel the common factor of -16 and 12.
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Factor 4 out of -16.
f(1)=4(-4)12
Cancel the common factors.
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Factor 4 out of 12.
f(1)=4⋅-44⋅3
Cancel the common factor.
f(1)=4⋅-44⋅3
Rewrite the expression.
f(1)=-43
f(1)=-43
f(1)=-43
Move the negative in front of the fraction.
f(1)=-43
The final answer is -43.
-43
-43
The y value at x=1 is -43.
y=-43
Replace the variable x with 4 in the expression.
f(4)=-(4)212+42-74
Simplify the result.
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Simplify each term.
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Raise 4 to the power of 2.
f(4)=-1612+42-74
Cancel the common factor of 16 and 12.
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Factor 4 out of 16.
f(4)=-4(4)12+42-74
Cancel the common factors.
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Factor 4 out of 12.
f(4)=-4⋅44⋅3+42-74
Cancel the common factor.
f(4)=-4⋅44⋅3+42-74
Rewrite the expression.
f(4)=-43+42-74
f(4)=-43+42-74
f(4)=-43+42-74
Divide 4 by 2.
f(4)=-43+2-74
f(4)=-43+2-74
To write 2 as a fraction with a common denominator, multiply by 33.
f(4)=-43+2⋅33-74
Combine 2 and 33.
f(4)=-43+2⋅33-74
Combine the numerators over the common denominator.
f(4)=-4+2⋅33-74
Simplify the numerator.
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Multiply 2 by 3.
f(4)=-4+63-74
Add -4 and 6.
f(4)=23-74
f(4)=23-74
To write 23 as a fraction with a common denominator, multiply by 44.
f(4)=23⋅44-74
To write -74 as a fraction with a common denominator, multiply by 33.
f(4)=23⋅44-74⋅33
Write each expression with a common denominator of 12, by multiplying each by an appropriate factor of 1.
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Multiply 23 and 44.
f(4)=2⋅43⋅4-74⋅33
Multiply 3 by 4.
f(4)=2⋅412-74⋅33
Multiply 74 and 33.
f(4)=2⋅412-7⋅34⋅3
Multiply 4 by 3.
f(4)=2⋅412-7⋅312
f(4)=2⋅412-7⋅312
Combine the numerators over the common denominator.
f(4)=2⋅4-7⋅312
Simplify the numerator.
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Multiply 2 by 4.
f(4)=8-7⋅312
Multiply -7 by 3.
f(4)=8-2112
Subtract 21 from 8.
f(4)=-1312
f(4)=-1312
Move the negative in front of the fraction.
f(4)=-1312
The final answer is -1312.
-1312
-1312
The y value at x=4 is -1312.
y=-1312
Replace the variable x with 5 in the expression.
f(5)=-(5)212+52-74
Simplify the result.
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Raise 5 to the power of 2.
f(5)=-2512+52-74
To write 52 as a fraction with a common denominator, multiply by 66.
f(5)=-2512+52⋅66-74
Write each expression with a common denominator of 12, by multiplying each by an appropriate factor of 1.
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Multiply 52 and 66.
f(5)=-2512+5⋅62⋅6-74
Multiply 2 by 6.
f(5)=-2512+5⋅612-74
f(5)=-2512+5⋅612-74
Combine the numerators over the common denominator.
f(5)=-25+5⋅612-74
Simplify the numerator.
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Multiply 5 by 6.
f(5)=-25+3012-74
Add -25 and 30.
f(5)=512-74
f(5)=512-74
To write -74 as a fraction with a common denominator, multiply by 33.
f(5)=512-74⋅33
Write each expression with a common denominator of 12, by multiplying each by an appropriate factor of 1.
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Multiply 74 and 33.
f(5)=512-7⋅34⋅3
Multiply 4 by 3.
f(5)=512-7⋅312
f(5)=512-7⋅312
Combine the numerators over the common denominator.
f(5)=5-7⋅312
Simplify the numerator.
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Multiply -7 by 3.
f(5)=5-2112
Subtract 21 from 5.
f(5)=-1612
f(5)=-1612
Cancel the common factor of -16 and 12.
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Factor 4 out of -16.
f(5)=4(-4)12
Cancel the common factors.
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Factor 4 out of 12.
f(5)=4⋅-44⋅3
Cancel the common factor.
f(5)=4⋅-44⋅3
Rewrite the expression.
f(5)=-43
f(5)=-43
f(5)=-43
Move the negative in front of the fraction.
f(5)=-43
The final answer is -43.
-43
-43
The y value at x=5 is -43.
y=-43
Graph the parabola using its properties and the selected points.
xy1-432-13123-14-13125-43
xy1-432-13123-14-13125-43
Graph the parabola using its properties and the selected points.
Direction: Opens Down
Vertex: (3,-1)
Focus: (3,-4)
Axis of Symmetry: x=3
Directrix: y=2
xy1-432-13123-14-13125-43
Graph y+1=-1/12*(x-3)^2

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