Graph 3x^2+15x-6

Math
3×2+15x-6
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+15x-6.
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Use the form ax2+bx+c, to find the values of a, b, and c.
a=3,b=15,c=-6
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=152(3)
Cancel the common factor of 15 and 3.
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Factor 3 out of 15.
d=3⋅52⋅3
Cancel the common factors.
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Factor 3 out of 2⋅3.
d=3⋅53⋅2
Cancel the common factor.
d=3⋅53⋅2
Rewrite the expression.
d=52
d=52
d=52
Find the value of e using the formula e=c-b24a.
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Simplify each term.
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Raise 15 to the power of 2.
e=-6-2254⋅3
Multiply 4 by 3.
e=-6-22512
Cancel the common factor of 225 and 12.
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Factor 3 out of 225.
e=-6-3(75)12
Cancel the common factors.
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Factor 3 out of 12.
e=-6-3⋅753⋅4
Cancel the common factor.
e=-6-3⋅753⋅4
Rewrite the expression.
e=-6-754
e=-6-754
e=-6-754
e=-6-754
To write -6 as a fraction with a common denominator, multiply by 44.
e=-6⋅44-754
Combine -6 and 44.
e=-6⋅44-754
Combine the numerators over the common denominator.
e=-6⋅4-754
Simplify the numerator.
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Multiply -6 by 4.
e=-24-754
Subtract 75 from -24.
e=-994
e=-994
Move the negative in front of the fraction.
e=-994
e=-994
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
3(x+52)2-994
3(x+52)2-994
Set y equal to the new right side.
y=3(x+52)2-994
y=3(x+52)2-994
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=3
h=-52
k=-994
Since the value of a is positive, the parabola opens up.
Opens Up
Find the vertex (h,k).
(-52,-994)
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.
(-52,-743)
(-52,-743)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=-52
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=-1496
y=-1496
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex: (-52,-994)
Focus: (-52,-743)
Axis of Symmetry: x=-52
Directrix: y=-1496
Direction: Opens Up
Vertex: (-52,-994)
Focus: (-52,-743)
Axis of Symmetry: x=-52
Directrix: y=-1496
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+15(-3)-6
Simplify the result.
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Simplify each term.
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Raise -3 to the power of 2.
f(-3)=3⋅9+15(-3)-6
Multiply 3 by 9.
f(-3)=27+15(-3)-6
Multiply 15 by -3.
f(-3)=27-45-6
f(-3)=27-45-6
Simplify by subtracting numbers.
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Subtract 45 from 27.
f(-3)=-18-6
Subtract 6 from -18.
f(-3)=-24
f(-3)=-24
The final answer is -24.
-24
-24
The y value at x=-3 is -24.
y=-24
Replace the variable x with -4 in the expression.
f(-4)=3(-4)2+15(-4)-6
Simplify the result.
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Simplify each term.
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Raise -4 to the power of 2.
f(-4)=3⋅16+15(-4)-6
Multiply 3 by 16.
f(-4)=48+15(-4)-6
Multiply 15 by -4.
f(-4)=48-60-6
f(-4)=48-60-6
Simplify by subtracting numbers.
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Subtract 60 from 48.
f(-4)=-12-6
Subtract 6 from -12.
f(-4)=-18
f(-4)=-18
The final answer is -18.
-18
-18
The y value at x=-4 is -18.
y=-18
Replace the variable x with -1 in the expression.
f(-1)=3(-1)2+15(-1)-6
Simplify the result.
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Simplify each term.
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Raise -1 to the power of 2.
f(-1)=3⋅1+15(-1)-6
Multiply 3 by 1.
f(-1)=3+15(-1)-6
Multiply 15 by -1.
f(-1)=3-15-6
f(-1)=3-15-6
Simplify by subtracting numbers.
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Subtract 15 from 3.
f(-1)=-12-6
Subtract 6 from -12.
f(-1)=-18
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 0 in the expression.
f(0)=3(0)2+15(0)-6
Simplify the result.
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Simplify each term.
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Raising 0 to any positive power yields 0.
f(0)=3⋅0+15(0)-6
Multiply 3 by 0.
f(0)=0+15(0)-6
Multiply 15 by 0.
f(0)=0+0-6
f(0)=0+0-6
Simplify by adding zeros.
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Add 0 and 0.
f(0)=0-6
Subtract 6 from 0.
f(0)=-6
f(0)=-6
The final answer is -6.
-6
-6
The y value at x=0 is -6.
y=-6
Graph the parabola using its properties and the selected points.
xy-4-18-3-24-52-994-1-180-6
xy-4-18-3-24-52-994-1-180-6
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex: (-52,-994)
Focus: (-52,-743)
Axis of Symmetry: x=-52
Directrix: y=-1496
xy-4-18-3-24-52-994-1-180-6
Graph 3x^2+15x-6

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