Algebra Examples

Find the Axis of Symmetry x=-1/2y^2-4y-7
Step 1
Rewrite the equation in vertex form.
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Step 1.1
Combine and .
Step 1.2
Complete the square for .
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Step 1.2.1
Use the form , to find the values of , , and .
Step 1.2.2
Consider the vertex form of a parabola.
Step 1.2.3
Find the value of using the formula .
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Step 1.2.3.1
Substitute the values of and into the formula .
Step 1.2.3.2
Simplify the right side.
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Step 1.2.3.2.1
Cancel the common factor of and .
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Step 1.2.3.2.1.1
Factor out of .
Step 1.2.3.2.1.2
Cancel the common factors.
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Step 1.2.3.2.1.2.1
Cancel the common factor.
Step 1.2.3.2.1.2.2
Rewrite the expression.
Step 1.2.3.2.2
Multiply the numerator by the reciprocal of the denominator.
Step 1.2.3.2.3
Multiply .
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Step 1.2.3.2.3.1
Multiply by .
Step 1.2.3.2.3.2
Multiply by .
Step 1.2.4
Find the value of using the formula .
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Step 1.2.4.1
Substitute the values of , and into the formula .
Step 1.2.4.2
Simplify the right side.
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Step 1.2.4.2.1
Simplify each term.
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Step 1.2.4.2.1.1
Cancel the common factor of and .
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Step 1.2.4.2.1.1.1
Rewrite as .
Step 1.2.4.2.1.1.2
Apply the product rule to .
Step 1.2.4.2.1.1.3
Raise to the power of .
Step 1.2.4.2.1.1.4
Multiply by .
Step 1.2.4.2.1.1.5
Factor out of .
Step 1.2.4.2.1.1.6
Cancel the common factors.
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Step 1.2.4.2.1.1.6.1
Cancel the common factor.
Step 1.2.4.2.1.1.6.2
Rewrite the expression.
Step 1.2.4.2.1.2
Multiply the numerator by the reciprocal of the denominator.
Step 1.2.4.2.1.3
Multiply .
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Step 1.2.4.2.1.3.1
Multiply by .
Step 1.2.4.2.1.3.2
Multiply by .
Step 1.2.4.2.1.3.3
Multiply by .
Step 1.2.4.2.2
Add and .
Step 1.2.5
Substitute the values of , , and into the vertex form .
Step 1.3
Set equal to the new right side.
Step 2
Use the vertex form, , to determine the values of , , and .
Step 3
Since the value of is negative, the parabola opens left.
Opens Left
Step 4
Find the vertex .
Step 5
Find , the distance from the vertex to the focus.
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Step 5.1
Find the distance from the vertex to a focus of the parabola by using the following formula.
Step 5.2
Substitute the value of into the formula.
Step 5.3
Simplify.
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Step 5.3.1
Cancel the common factor of and .
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Step 5.3.1.1
Rewrite as .
Step 5.3.1.2
Move the negative in front of the fraction.
Step 5.3.2
Combine and .
Step 5.3.3
Divide by .
Step 6
Find the focus.
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Step 6.1
The focus of a parabola can be found by adding to the x-coordinate if the parabola opens left or right.
Step 6.2
Substitute the known values of , , and into the formula and simplify.
Step 7
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
Step 8