Trigonometry Examples

Find the Directrix y=x^2+4x-2
Step 1
Rewrite the equation in vertex form.
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Step 1.1
Complete the square for .
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Step 1.1.1
Use the form , to find the values of , , and .
Step 1.1.2
Consider the vertex form of a parabola.
Step 1.1.3
Find the value of using the formula .
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Step 1.1.3.1
Substitute the values of and into the formula .
Step 1.1.3.2
Cancel the common factor of and .
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Step 1.1.3.2.1
Factor out of .
Step 1.1.3.2.2
Cancel the common factors.
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Step 1.1.3.2.2.1
Factor out of .
Step 1.1.3.2.2.2
Cancel the common factor.
Step 1.1.3.2.2.3
Rewrite the expression.
Step 1.1.3.2.2.4
Divide by .
Step 1.1.4
Find the value of using the formula .
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Step 1.1.4.1
Substitute the values of , and into the formula .
Step 1.1.4.2
Simplify the right side.
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Step 1.1.4.2.1
Simplify each term.
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Step 1.1.4.2.1.1
Cancel the common factor of and .
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Step 1.1.4.2.1.1.1
Factor out of .
Step 1.1.4.2.1.1.2
Cancel the common factors.
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Step 1.1.4.2.1.1.2.1
Factor out of .
Step 1.1.4.2.1.1.2.2
Cancel the common factor.
Step 1.1.4.2.1.1.2.3
Rewrite the expression.
Step 1.1.4.2.1.1.2.4
Divide by .
Step 1.1.4.2.1.2
Multiply by .
Step 1.1.4.2.2
Subtract from .
Step 1.1.5
Substitute the values of , , and into the vertex form .
Step 1.2
Set equal to the new right side.
Step 2
Use the vertex form, , to determine the values of , , and .
Step 3
Find the vertex .
Step 4
Find , the distance from the vertex to the focus.
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Step 4.1
Find the distance from the vertex to a focus of the parabola by using the following formula.
Step 4.2
Substitute the value of into the formula.
Step 4.3
Cancel the common factor of .
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Step 4.3.1
Cancel the common factor.
Step 4.3.2
Rewrite the expression.
Step 5
Find the directrix.
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Step 5.1
The directrix of a parabola is the horizontal line found by subtracting from the y-coordinate of the vertex if the parabola opens up or down.
Step 5.2
Substitute the known values of and into the formula and simplify.
Step 6