Graph y=5x^2-10x+2

Math
y=5×2-10x+2
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 5×2-10x+2.
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Use the form ax2+bx+c, to find the values of a, b, and c.
a=5,b=-10,c=2
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=-102(5)
Simplify the right side.
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Cancel the common factor of 10 and 2.
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Factor 2 out of 10.
d=-2⋅52⋅5
Cancel the common factors.
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Factor 2 out of 2⋅5.
d=-2⋅52(5)
Cancel the common factor.
d=-2⋅52⋅5
Rewrite the expression.
d=-55
d=-55
d=-55
Cancel the common factor of 5.
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Cancel the common factor.
d=-55
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 -10 to the power of 2.
e=2-1004⋅5
Multiply 4 by 5.
e=2-10020
Divide 100 by 20.
e=2-1⋅5
Multiply -1 by 5.
e=2-5
e=2-5
Subtract 5 from 2.
e=-3
e=-3
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
5(x-1)2-3
5(x-1)2-3
Set y equal to the new right side.
y=5(x-1)2-3
y=5(x-1)2-3
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=5
h=1
k=-3
Since the value of a is positive, the parabola opens up.
Opens Up
Find the vertex (h,k).
(1,-3)
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⋅5
Multiply 4 by 5.
120
120
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,-5920)
(1,-5920)
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=-6120
y=-6120
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex: (1,-3)
Focus: (1,-5920)
Axis of Symmetry: x=1
Directrix: y=-6120
Direction: Opens Up
Vertex: (1,-3)
Focus: (1,-5920)
Axis of Symmetry: x=1
Directrix: y=-6120
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)=5(0)2-10⋅0+2
Simplify the result.
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Simplify each term.
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Raising 0 to any positive power yields 0.
f(0)=5⋅0-10⋅0+2
Multiply 5 by 0.
f(0)=0-10⋅0+2
Multiply -10 by 0.
f(0)=0+0+2
f(0)=0+0+2
Simplify by adding zeros.
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Add 0 and 0.
f(0)=0+2
Add 0 and 2.
f(0)=2
f(0)=2
The final answer is 2.
2
2
The y value at x=0 is 2.
y=2
Replace the variable x with -1 in the expression.
f(-1)=5(-1)2-10⋅-1+2
Simplify the result.
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Simplify each term.
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Raise -1 to the power of 2.
f(-1)=5⋅1-10⋅-1+2
Multiply 5 by 1.
f(-1)=5-10⋅-1+2
Multiply -10 by -1.
f(-1)=5+10+2
f(-1)=5+10+2
Simplify by adding numbers.
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Add 5 and 10.
f(-1)=15+2
Add 15 and 2.
f(-1)=17
f(-1)=17
The final answer is 17.
17
17
The y value at x=-1 is 17.
y=17
Replace the variable x with 2 in the expression.
f(2)=5(2)2-10⋅2+2
Simplify the result.
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Simplify each term.
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Raise 2 to the power of 2.
f(2)=5⋅4-10⋅2+2
Multiply 5 by 4.
f(2)=20-10⋅2+2
Multiply -10 by 2.
f(2)=20-20+2
f(2)=20-20+2
Simplify by subtracting numbers.
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Subtract 20 from 20.
f(2)=0+2
Add 0 and 2.
f(2)=2
f(2)=2
The final answer is 2.
2
2
The y value at x=2 is 2.
y=2
Replace the variable x with 3 in the expression.
f(3)=5(3)2-10⋅3+2
Simplify the result.
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Simplify each term.
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Raise 3 to the power of 2.
f(3)=5⋅9-10⋅3+2
Multiply 5 by 9.
f(3)=45-10⋅3+2
Multiply -10 by 3.
f(3)=45-30+2
f(3)=45-30+2
Simplify by adding and subtracting.
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Subtract 30 from 45.
f(3)=15+2
Add 15 and 2.
f(3)=17
f(3)=17
The final answer is 17.
17
17
The y value at x=3 is 17.
y=17
Graph the parabola using its properties and the selected points.
xy-117021-322317
xy-117021-322317
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex: (1,-3)
Focus: (1,-5920)
Axis of Symmetry: x=1
Directrix: y=-6120
xy-117021-322317
Graph y=5x^2-10x+2

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