Graph y=3x^2+15x+18

Math
y=3×2+15x+18
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+18.
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Use the form ax2+bx+c, to find the values of a, b, and c.
a=3,b=15,c=18
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=18-2254⋅3
Multiply 4 by 3.
e=18-22512
Cancel the common factor of 225 and 12.
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Factor 3 out of 225.
e=18-3(75)12
Cancel the common factors.
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Factor 3 out of 12.
e=18-3⋅753⋅4
Cancel the common factor.
e=18-3⋅753⋅4
Rewrite the expression.
e=18-754
e=18-754
e=18-754
e=18-754
To write 18 as a fraction with a common denominator, multiply by 44.
e=18⋅44-754
Combine 18 and 44.
e=18⋅44-754
Combine the numerators over the common denominator.
e=18⋅4-754
Simplify the numerator.
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Multiply 18 by 4.
e=72-754
Subtract 75 from 72.
e=-34
e=-34
Move the negative in front of the fraction.
e=-34
e=-34
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
3(x+52)2-34
3(x+52)2-34
Set y equal to the new right side.
y=3(x+52)2-34
y=3(x+52)2-34
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=3
h=-52
k=-34
Since the value of a is positive, the parabola opens up.
Opens Up
Find the vertex (h,k).
(-52,-34)
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,-23)
(-52,-23)
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=-56
y=-56
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex: (-52,-34)
Focus: (-52,-23)
Axis of Symmetry: x=-52
Directrix: y=-56
Direction: Opens Up
Vertex: (-52,-34)
Focus: (-52,-23)
Axis of Symmetry: x=-52
Directrix: y=-56
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)+18
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)+18
Multiply 3 by 9.
f(-3)=27+15(-3)+18
Multiply 15 by -3.
f(-3)=27-45+18
f(-3)=27-45+18
Simplify by adding and subtracting.
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Subtract 45 from 27.
f(-3)=-18+18
Add -18 and 18.
f(-3)=0
f(-3)=0
The final answer is 0.
0
0
The y value at x=-3 is 0.
y=0
Replace the variable x with -4 in the expression.
f(-4)=3(-4)2+15(-4)+18
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)+18
Multiply 3 by 16.
f(-4)=48+15(-4)+18
Multiply 15 by -4.
f(-4)=48-60+18
f(-4)=48-60+18
Simplify by adding and subtracting.
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Subtract 60 from 48.
f(-4)=-12+18
Add -12 and 18.
f(-4)=6
f(-4)=6
The final answer is 6.
6
6
The y value at x=-4 is 6.
y=6
Replace the variable x with -1 in the expression.
f(-1)=3(-1)2+15(-1)+18
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)+18
Multiply 3 by 1.
f(-1)=3+15(-1)+18
Multiply 15 by -1.
f(-1)=3-15+18
f(-1)=3-15+18
Simplify by adding and subtracting.
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Subtract 15 from 3.
f(-1)=-12+18
Add -12 and 18.
f(-1)=6
f(-1)=6
The final answer is 6.
6
6
The y value at x=-1 is 6.
y=6
Replace the variable x with 0 in the expression.
f(0)=3(0)2+15(0)+18
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)+18
Multiply 3 by 0.
f(0)=0+15(0)+18
Multiply 15 by 0.
f(0)=0+0+18
f(0)=0+0+18
Simplify by adding zeros.
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Add 0 and 0.
f(0)=0+18
Add 0 and 18.
f(0)=18
f(0)=18
The final answer is 18.
18
18
The y value at x=0 is 18.
y=18
Graph the parabola using its properties and the selected points.
xy-46-30-52-34-16018
xy-46-30-52-34-16018
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex: (-52,-34)
Focus: (-52,-23)
Axis of Symmetry: x=-52
Directrix: y=-56
xy-46-30-52-34-16018
Graph y=3x^2+15x+18

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