Graph f(x)=x^2+10x-2

Math
f(x)=x2+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 x2+10x-2.
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Use the form ax2+bx+c, to find the values of a, b, and c.
a=1,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(1)
Cancel the common factor of 10 and 2.
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Factor 2 out of 10.
d=2⋅52⋅1
Cancel the common factors.
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Factor 2 out of 2⋅1.
d=2⋅52(1)
Cancel the common factor.
d=2⋅52⋅1
Rewrite the expression.
d=51
Divide 5 by 1.
d=5
d=5
d=5
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⋅1
Multiply 4 by 1.
e=-2-1004
Divide 100 by 4.
e=-2-1⋅25
Multiply -1 by 25.
e=-2-25
e=-2-25
Subtract 25 from -2.
e=-27
e=-27
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
(x+5)2-27
(x+5)2-27
Set y equal to the new right side.
y=(x+5)2-27
y=(x+5)2-27
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=1
h=-5
k=-27
Since the value of a is positive, the parabola opens up.
Opens Up
Find the vertex (h,k).
(-5,-27)
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.
(-5,-1074)
(-5,-1074)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=-5
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=-1094
y=-1094
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex: (-5,-27)
Focus: (-5,-1074)
Axis of Symmetry: x=-5
Directrix: y=-1094
Direction: Opens Up
Vertex: (-5,-27)
Focus: (-5,-1074)
Axis of Symmetry: x=-5
Directrix: y=-1094
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 -6 in the expression.
f(-6)=(-6)2+10(-6)-2
Simplify the result.
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Simplify each term.
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Raise -6 to the power of 2.
f(-6)=36+10(-6)-2
Multiply 10 by -6.
f(-6)=36-60-2
f(-6)=36-60-2
Simplify by subtracting numbers.
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Subtract 60 from 36.
f(-6)=-24-2
Subtract 2 from -24.
f(-6)=-26
f(-6)=-26
The final answer is -26.
-26
-26
The y value at x=-6 is -26.
y=-26
Replace the variable x with -7 in the expression.
f(-7)=(-7)2+10(-7)-2
Simplify the result.
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Simplify each term.
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Raise -7 to the power of 2.
f(-7)=49+10(-7)-2
Multiply 10 by -7.
f(-7)=49-70-2
f(-7)=49-70-2
Simplify by subtracting numbers.
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Subtract 70 from 49.
f(-7)=-21-2
Subtract 2 from -21.
f(-7)=-23
f(-7)=-23
The final answer is -23.
-23
-23
The y value at x=-7 is -23.
y=-23
Replace the variable x with -4 in the expression.
f(-4)=(-4)2+10(-4)-2
Simplify the result.
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Simplify each term.
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Raise -4 to the power of 2.
f(-4)=16+10(-4)-2
Multiply 10 by -4.
f(-4)=16-40-2
f(-4)=16-40-2
Simplify by subtracting numbers.
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Subtract 40 from 16.
f(-4)=-24-2
Subtract 2 from -24.
f(-4)=-26
f(-4)=-26
The final answer is -26.
-26
-26
The y value at x=-4 is -26.
y=-26
Replace the variable x with -3 in the expression.
f(-3)=(-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)=9+10(-3)-2
Multiply 10 by -3.
f(-3)=9-30-2
f(-3)=9-30-2
Simplify by subtracting numbers.
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Subtract 30 from 9.
f(-3)=-21-2
Subtract 2 from -21.
f(-3)=-23
f(-3)=-23
The final answer is -23.
-23
-23
The y value at x=-3 is -23.
y=-23
Graph the parabola using its properties and the selected points.
xy-7-23-6-26-5-27-4-26-3-23
xy-7-23-6-26-5-27-4-26-3-23
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex: (-5,-27)
Focus: (-5,-1074)
Axis of Symmetry: x=-5
Directrix: y=-1094
xy-7-23-6-26-5-27-4-26-3-23
Graph f(x)=x^2+10x-2

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