Graph f(x)=3x^2-24x-8

Math
f(x)=3×2-24x-8
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 3×2-24x-8.
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Use the form ax2+bx+c, to find the values of a, b, and c.
a=3,b=-24,c=-8
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=-242(3)
Simplify the right side.
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Cancel the common factor of 24 and 2.
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Factor 2 out of 24.
d=-2⋅122⋅3
Cancel the common factors.
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Factor 2 out of 2⋅3.
d=-2⋅122(3)
Cancel the common factor.
d=-2⋅122⋅3
Rewrite the expression.
d=-123
d=-123
d=-123
Cancel the common factor of 12 and 3.
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Factor 3 out of 12.
d=-3⋅43
Cancel the common factors.
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Factor 3 out of 3.
d=-3⋅43(1)
Cancel the common factor.
d=-3⋅43⋅1
Rewrite the expression.
d=-41
Divide 4 by 1.
d=-1⋅4
d=-1⋅4
d=-1⋅4
Multiply -1 by 4.
d=-4
d=-4
Find the value of e using the formula e=c-b24a.
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Simplify each term.
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Raise -24 to the power of 2.
e=-8-5764⋅3
Multiply 4 by 3.
e=-8-57612
Divide 576 by 12.
e=-8-1⋅48
Multiply -1 by 48.
e=-8-48
e=-8-48
Subtract 48 from -8.
e=-56
e=-56
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
3(x-4)2-56
3(x-4)2-56
Set y equal to the new right side.
y=3(x-4)2-56
y=3(x-4)2-56
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=3
h=4
k=-56
Since the value of a is positive, the parabola opens up.
Opens Up
Find the vertex (h,k).
(4,-56)
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⋅3
Multiply 4 by 3.
112
112
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.
(4,-67112)
(4,-67112)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=4
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=-67312
y=-67312
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex: (4,-56)
Focus: (4,-67112)
Axis of Symmetry: x=4
Directrix: y=-67312
Direction: Opens Up
Vertex: (4,-56)
Focus: (4,-67112)
Axis of Symmetry: x=4
Directrix: y=-67312
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 3 in the expression.
f(3)=3(3)2-24⋅3-8
Simplify the result.
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Simplify each term.
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Multiply 3 by (3)2 by adding the exponents.
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Multiply 3 by (3)2.
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Raise 3 to the power of 1.
f(3)=3(3)2-24⋅3-8
Use the power rule aman=am+n to combine exponents.
f(3)=31+2-24⋅3-8
f(3)=31+2-24⋅3-8
Add 1 and 2.
f(3)=33-24⋅3-8
f(3)=33-24⋅3-8
Raise 3 to the power of 3.
f(3)=27-24⋅3-8
Multiply -24 by 3.
f(3)=27-72-8
f(3)=27-72-8
Simplify by subtracting numbers.
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Subtract 72 from 27.
f(3)=-45-8
Subtract 8 from -45.
f(3)=-53
f(3)=-53
The final answer is -53.
-53
-53
The y value at x=3 is -53.
y=-53
Replace the variable x with 2 in the expression.
f(2)=3(2)2-24⋅2-8
Simplify the result.
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Simplify each term.
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Raise 2 to the power of 2.
f(2)=3⋅4-24⋅2-8
Multiply 3 by 4.
f(2)=12-24⋅2-8
Multiply -24 by 2.
f(2)=12-48-8
f(2)=12-48-8
Simplify by subtracting numbers.
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Subtract 48 from 12.
f(2)=-36-8
Subtract 8 from -36.
f(2)=-44
f(2)=-44
The final answer is -44.
-44
-44
The y value at x=2 is -44.
y=-44
Replace the variable x with 5 in the expression.
f(5)=3(5)2-24⋅5-8
Simplify the result.
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Simplify each term.
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Raise 5 to the power of 2.
f(5)=3⋅25-24⋅5-8
Multiply 3 by 25.
f(5)=75-24⋅5-8
Multiply -24 by 5.
f(5)=75-120-8
f(5)=75-120-8
Simplify by subtracting numbers.
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Subtract 120 from 75.
f(5)=-45-8
Subtract 8 from -45.
f(5)=-53
f(5)=-53
The final answer is -53.
-53
-53
The y value at x=5 is -53.
y=-53
Replace the variable x with 6 in the expression.
f(6)=3(6)2-24⋅6-8
Simplify the result.
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Simplify each term.
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Raise 6 to the power of 2.
f(6)=3⋅36-24⋅6-8
Multiply 3 by 36.
f(6)=108-24⋅6-8
Multiply -24 by 6.
f(6)=108-144-8
f(6)=108-144-8
Simplify by subtracting numbers.
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Subtract 144 from 108.
f(6)=-36-8
Subtract 8 from -36.
f(6)=-44
f(6)=-44
The final answer is -44.
-44
-44
The y value at x=6 is -44.
y=-44
Graph the parabola using its properties and the selected points.
xy2-443-534-565-536-44
xy2-443-534-565-536-44
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex: (4,-56)
Focus: (4,-67112)
Axis of Symmetry: x=4
Directrix: y=-67312
xy2-443-534-565-536-44
Graph f(x)=3x^2-24x-8

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