Graph y=3x^2-9x+5

Math
y=3×2-9x+5
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 3×2-9x+5.
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Use the form ax2+bx+c, to find the values of a, b, and c.
a=3,b=-9,c=5
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=-92(3)
Cancel the common factor of 9 and 3.
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Factor 3 out of 9.
d=-3⋅32⋅3
Cancel the common factors.
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Factor 3 out of 2⋅3.
d=-3⋅33⋅2
Cancel the common factor.
d=-3⋅33⋅2
Rewrite the expression.
d=-32
d=-32
d=-32
Find the value of e using the formula e=c-b24a.
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Simplify each term.
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Raise -9 to the power of 2.
e=5-814⋅3
Multiply 4 by 3.
e=5-8112
Cancel the common factor of 81 and 12.
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Factor 3 out of 81.
e=5-3(27)12
Cancel the common factors.
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Factor 3 out of 12.
e=5-3⋅273⋅4
Cancel the common factor.
e=5-3⋅273⋅4
Rewrite the expression.
e=5-274
e=5-274
e=5-274
e=5-274
To write 5 as a fraction with a common denominator, multiply by 44.
e=5⋅44-274
Combine 5 and 44.
e=5⋅44-274
Combine the numerators over the common denominator.
e=5⋅4-274
Simplify the numerator.
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Multiply 5 by 4.
e=20-274
Subtract 27 from 20.
e=-74
e=-74
Move the negative in front of the fraction.
e=-74
e=-74
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
3(x-32)2-74
3(x-32)2-74
Set y equal to the new right side.
y=3(x-32)2-74
y=3(x-32)2-74
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=3
h=32
k=-74
Since the value of a is positive, the parabola opens up.
Opens Up
Find the vertex (h,k).
(32,-74)
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.
(32,-53)
(32,-53)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=32
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=-116
y=-116
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex: (32,-74)
Focus: (32,-53)
Axis of Symmetry: x=32
Directrix: y=-116
Direction: Opens Up
Vertex: (32,-74)
Focus: (32,-53)
Axis of Symmetry: x=32
Directrix: y=-116
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 1 in the expression.
f(1)=3(1)2-9⋅1+5
Simplify the result.
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Simplify each term.
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One to any power is one.
f(1)=3⋅1-9⋅1+5
Multiply 3 by 1.
f(1)=3-9⋅1+5
Multiply -9 by 1.
f(1)=3-9+5
f(1)=3-9+5
Simplify by adding and subtracting.
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Subtract 9 from 3.
f(1)=-6+5
Add -6 and 5.
f(1)=-1
f(1)=-1
The final answer is -1.
-1
-1
The y value at x=1 is -1.
y=-1
Replace the variable x with 0 in the expression.
f(0)=3(0)2-9⋅0+5
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⋅0-9⋅0+5
Multiply 3 by 0.
f(0)=0-9⋅0+5
Multiply -9 by 0.
f(0)=0+0+5
f(0)=0+0+5
Simplify by adding zeros.
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Add 0 and 0.
f(0)=0+5
Add 0 and 5.
f(0)=5
f(0)=5
The final answer is 5.
5
5
The y value at x=0 is 5.
y=5
Replace the variable x with 3 in the expression.
f(3)=3(3)2-9⋅3+5
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)2-9⋅3+5
Use the power rule aman=am+n to combine exponents.
f(3)=31+2-9⋅3+5
f(3)=31+2-9⋅3+5
Add 1 and 2.
f(3)=33-9⋅3+5
f(3)=33-9⋅3+5
Raise 3 to the power of 3.
f(3)=27-9⋅3+5
Multiply -9 by 3.
f(3)=27-27+5
f(3)=27-27+5
Simplify by subtracting numbers.
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Subtract 27 from 27.
f(3)=0+5
Add 0 and 5.
f(3)=5
f(3)=5
The final answer is 5.
5
5
The y value at x=3 is 5.
y=5
Replace the variable x with 4 in the expression.
f(4)=3(4)2-9⋅4+5
Simplify the result.
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Simplify each term.
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Raise 4 to the power of 2.
f(4)=3⋅16-9⋅4+5
Multiply 3 by 16.
f(4)=48-9⋅4+5
Multiply -9 by 4.
f(4)=48-36+5
f(4)=48-36+5
Simplify by adding and subtracting.
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Subtract 36 from 48.
f(4)=12+5
Add 12 and 5.
f(4)=17
f(4)=17
The final answer is 17.
17
17
The y value at x=4 is 17.
y=17
Graph the parabola using its properties and the selected points.
xy051-132-7435417
xy051-132-7435417
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex: (32,-74)
Focus: (32,-53)
Axis of Symmetry: x=32
Directrix: y=-116
xy051-132-7435417
Graph y=3x^2-9x+5

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