Algebra Examples

f(x)=-x2-5x-5
Step 1
Write f(x)=-x2-5x-5 as an equation.
y=-x2-5x-5
Step 2
Rewrite the equation in vertex form.
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Step 2.1
Complete the square for -x2-5x-5.
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Step 2.1.1
Use the form ax2+bx+c, to find the values of a, b, and c.
a=-1
b=-5
c=-5
Step 2.1.2
Consider the vertex form of a parabola.
a(x+d)2+e
Step 2.1.3
Find the value of d using the formula d=b2a.
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Step 2.1.3.1
Substitute the values of a and b into the formula d=b2a.
d=-52-1
Step 2.1.3.2
Simplify the right side.
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Step 2.1.3.2.1
Multiply 2 by -1.
d=-5-2
Step 2.1.3.2.2
Dividing two negative values results in a positive value.
d=52
d=52
d=52
Step 2.1.4
Find the value of e using the formula e=c-b24a.
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Step 2.1.4.1
Substitute the values of c, b and a into the formula e=c-b24a.
e=-5-(-5)24-1
Step 2.1.4.2
Simplify the right side.
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Step 2.1.4.2.1
Simplify each term.
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Step 2.1.4.2.1.1
Raise -5 to the power of 2.
e=-5-254-1
Step 2.1.4.2.1.2
Multiply 4 by -1.
e=-5-25-4
Step 2.1.4.2.1.3
Move the negative in front of the fraction.
e=-5--254
Step 2.1.4.2.1.4
Multiply --254.
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Step 2.1.4.2.1.4.1
Multiply -1 by -1.
e=-5+1(254)
Step 2.1.4.2.1.4.2
Multiply 254 by 1.
e=-5+254
e=-5+254
e=-5+254
Step 2.1.4.2.2
To write -5 as a fraction with a common denominator, multiply by 44.
e=-544+254
Step 2.1.4.2.3
Combine -5 and 44.
e=-544+254
Step 2.1.4.2.4
Combine the numerators over the common denominator.
e=-54+254
Step 2.1.4.2.5
Simplify the numerator.
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Step 2.1.4.2.5.1
Multiply -5 by 4.
e=-20+254
Step 2.1.4.2.5.2
Add -20 and 25.
e=54
e=54
e=54
e=54
Step 2.1.5
Substitute the values of a, d, and e into the vertex form -(x+52)2+54.
-(x+52)2+54
-(x+52)2+54
Step 2.2
Set y equal to the new right side.
y=-(x+52)2+54
y=-(x+52)2+54
Step 3
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=-1
h=-52
k=54
Step 4
Since the value of a is negative, the parabola opens down.
Opens Down
Step 5
Find the vertex (h,k).
(-52,54)
Step 6
Find p, the distance from the vertex to the focus.
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Step 6.1
Find the distance from the vertex to a focus of the parabola by using the following formula.
14a
Step 6.2
Substitute the value of a into the formula.
14-1
Step 6.3
Cancel the common factor of 1 and -1.
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Step 6.3.1
Rewrite 1 as -1(-1).
-1(-1)4-1
Step 6.3.2
Move the negative in front of the fraction.
-14
-14
-14
Step 7
Find the focus.
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Step 7.1
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)
Step 7.2
Substitute the known values of h, p, and k into the formula and simplify.
(-52,1)
(-52,1)
Step 8
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=-52
Step 9
Find the directrix.
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Step 9.1
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
Step 9.2
Substitute the known values of p and k into the formula and simplify.
y=32
y=32
Step 10
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Down
Vertex: (-52,54)
Focus: (-52,1)
Axis of Symmetry: x=-52
Directrix: y=32
Step 11
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 [x2  12  π  xdx ]