Graph y=6x^2-12x

Math
y=6×2-12x
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 6×2-12x.
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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.
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Cancel the common factor of 12 and 2.
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Factor 2 out of 12.
d=-2⋅62⋅6
Cancel the common factors.
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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.
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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.
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Simplify each term.
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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.
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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⋅6
Multiply 4 by 6.
124
124
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.
(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.
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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=-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
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 0 in the expression.
f(0)=6(0)2-12⋅0
Simplify the result.
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Simplify each term.
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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.
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Simplify each term.
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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.
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Simplify each term.
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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.
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Simplify each term.
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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

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