Graph y=x^2-4x+10

Math
y=x2-4x+10
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 x2-4x+10.
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Use the form ax2+bx+c, to find the values of a, b, and c.
a=1,b=-4,c=10
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=-42(1)
Simplify the right side.
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Cancel the common factor of 4 and 2.
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Factor 2 out of 4.
d=-2⋅22⋅1
Cancel the common factors.
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Factor 2 out of 2⋅1.
d=-2⋅22(1)
Cancel the common factor.
d=-2⋅22⋅1
Rewrite the expression.
d=-21
Divide 2 by 1.
d=-1⋅2
d=-1⋅2
d=-1⋅2
Multiply -1 by 2.
d=-2
d=-2
Find the value of e using the formula e=c-b24a.
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Simplify each term.
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Cancel the common factor of (-4)2 and 4.
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Rewrite -4 as -1(4).
e=10-(-1⋅4)24(1)
Apply the product rule to -1(4).
e=10-(-1)2⋅424(1)
Raise -1 to the power of 2.
e=10-1⋅424(1)
Multiply 42 by 1.
e=10-424(1)
Factor 4 out of 42.
e=10-4⋅44⋅1
Cancel the common factors.
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Factor 4 out of 4⋅1.
e=10-4⋅44(1)
Cancel the common factor.
e=10-4⋅44⋅1
Rewrite the expression.
e=10-41
Divide 4 by 1.
e=10-1⋅4
e=10-1⋅4
e=10-1⋅4
Multiply -1 by 4.
e=10-4
e=10-4
Subtract 4 from 10.
e=6
e=6
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
(x-2)2+6
(x-2)2+6
Set y equal to the new right side.
y=(x-2)2+6
y=(x-2)2+6
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=1
h=2
k=6
Since the value of a is positive, the parabola opens up.
Opens Up
Find the vertex (h,k).
(2,6)
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⋅1
Cancel the common factor of 1.
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Cancel the common factor.
14⋅1
Rewrite the expression.
14
14
14
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.
(2,254)
(2,254)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=2
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=234
y=234
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex: (2,6)
Focus: (2,254)
Axis of Symmetry: x=2
Directrix: y=234
Direction: Opens Up
Vertex: (2,6)
Focus: (2,254)
Axis of Symmetry: x=2
Directrix: y=234
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)=(1)2-4⋅1+10
Simplify the result.
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Simplify each term.
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One to any power is one.
f(1)=1-4⋅1+10
Multiply -4 by 1.
f(1)=1-4+10
f(1)=1-4+10
Simplify by adding and subtracting.
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Subtract 4 from 1.
f(1)=-3+10
Add -3 and 10.
f(1)=7
f(1)=7
The final answer is 7.
7
7
The y value at x=1 is 7.
y=7
Replace the variable x with 0 in the expression.
f(0)=(0)2-4⋅0+10
Simplify the result.
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Simplify each term.
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Raising 0 to any positive power yields 0.
f(0)=0-4⋅0+10
Multiply -4 by 0.
f(0)=0+0+10
f(0)=0+0+10
Simplify by adding zeros.
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Add 0 and 0.
f(0)=0+10
Add 0 and 10.
f(0)=10
f(0)=10
The final answer is 10.
10
10
The y value at x=0 is 10.
y=10
Replace the variable x with 3 in the expression.
f(3)=(3)2-4⋅3+10
Simplify the result.
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Simplify each term.
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Raise 3 to the power of 2.
f(3)=9-4⋅3+10
Multiply -4 by 3.
f(3)=9-12+10
f(3)=9-12+10
Simplify by adding and subtracting.
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Subtract 12 from 9.
f(3)=-3+10
Add -3 and 10.
f(3)=7
f(3)=7
The final answer is 7.
7
7
The y value at x=3 is 7.
y=7
Replace the variable x with 4 in the expression.
f(4)=(4)2-4⋅4+10
Simplify the result.
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Simplify each term.
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Raise 4 to the power of 2.
f(4)=16-4⋅4+10
Multiply -4 by 4.
f(4)=16-16+10
f(4)=16-16+10
Simplify by subtracting numbers.
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Subtract 16 from 16.
f(4)=0+10
Add 0 and 10.
f(4)=10
f(4)=10
The final answer is 10.
10
10
The y value at x=4 is 10.
y=10
Graph the parabola using its properties and the selected points.
xy010172637410
xy010172637410
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex: (2,6)
Focus: (2,254)
Axis of Symmetry: x=2
Directrix: y=234
xy010172637410
Graph y=x^2-4x+10

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