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Algebra Examples
f(x)=-3x2+12x-6f(x)=−3x2+12x−6
Step 1
Write f(x)=-3x2+12x-6f(x)=−3x2+12x−6 as an equation.
y=-3x2+12x-6y=−3x2+12x−6
Step 2
Step 2.1
Complete the square for -3x2+12x-6−3x2+12x−6.
Step 2.1.1
Use the form ax2+bx+cax2+bx+c, to find the values of aa, bb, and cc.
a=-3a=−3
b=12b=12
c=-6c=−6
Step 2.1.2
Consider the vertex form of a parabola.
a(x+d)2+ea(x+d)2+e
Step 2.1.3
Find the value of dd using the formula d=b2ad=b2a.
Step 2.1.3.1
Substitute the values of aa and bb into the formula d=b2ad=b2a.
d=122⋅-3d=122⋅−3
Step 2.1.3.2
Simplify the right side.
Step 2.1.3.2.1
Cancel the common factor of 1212 and 22.
Step 2.1.3.2.1.1
Factor 22 out of 1212.
d=2⋅62⋅-3d=2⋅62⋅−3
Step 2.1.3.2.1.2
Cancel the common factors.
Step 2.1.3.2.1.2.1
Factor 22 out of 2⋅-32⋅−3.
d=2⋅62(-3)d=2⋅62(−3)
Step 2.1.3.2.1.2.2
Cancel the common factor.
d=2⋅62⋅-3
Step 2.1.3.2.1.2.3
Rewrite the expression.
d=6-3
d=6-3
d=6-3
Step 2.1.3.2.2
Cancel the common factor of 6 and -3.
Step 2.1.3.2.2.1
Factor 3 out of 6.
d=3(2)-3
Step 2.1.3.2.2.2
Move the negative one from the denominator of 2-1.
d=-1⋅2
d=-1⋅2
Step 2.1.3.2.3
Multiply -1 by 2.
d=-2
d=-2
d=-2
Step 2.1.4
Find the value of e using the formula e=c-b24a.
Step 2.1.4.1
Substitute the values of c, b and a into the formula e=c-b24a.
e=-6-1224⋅-3
Step 2.1.4.2
Simplify the right side.
Step 2.1.4.2.1
Simplify each term.
Step 2.1.4.2.1.1
Raise 12 to the power of 2.
e=-6-1444⋅-3
Step 2.1.4.2.1.2
Multiply 4 by -3.
e=-6-144-12
Step 2.1.4.2.1.3
Divide 144 by -12.
e=-6--12
Step 2.1.4.2.1.4
Multiply -1 by -12.
e=-6+12
e=-6+12
Step 2.1.4.2.2
Add -6 and 12.
e=6
e=6
e=6
Step 2.1.5
Substitute the values of a, d, and e into the vertex form -3(x-2)2+6.
-3(x-2)2+6
-3(x-2)2+6
Step 2.2
Set y equal to the new right side.
y=-3(x-2)2+6
y=-3(x-2)2+6
Step 3
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=-3
h=2
k=6
Step 4
Since the value of a is negative, the parabola opens down.
Opens Down
Step 5
Find the vertex (h,k).
(2,6)
Step 6
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⋅-3
Step 6.3
Simplify.
Step 6.3.1
Multiply 4 by -3.
1-12
Step 6.3.2
Move the negative in front of the fraction.
-112
-112
-112
Step 7
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.
(2,7112)
(2,7112)
Step 8
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=2
Step 9