Trigonometry Examples

Find the Focus x^2=-4y
Step 1
Rewrite the equation in vertex form.
Tap for more steps...
Step 1.1
Isolate to the left side of the equation.
Tap for more steps...
Step 1.1.1
Rewrite the equation as .
Step 1.1.2
Divide each term in by and simplify.
Tap for more steps...
Step 1.1.2.1
Divide each term in by .
Step 1.1.2.2
Simplify the left side.
Tap for more steps...
Step 1.1.2.2.1
Cancel the common factor of .
Tap for more steps...
Step 1.1.2.2.1.1
Cancel the common factor.
Step 1.1.2.2.1.2
Divide by .
Step 1.1.2.3
Simplify the right side.
Tap for more steps...
Step 1.1.2.3.1
Move the negative in front of the fraction.
Step 1.2
Complete the square for .
Tap for more steps...
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 .
Tap for more steps...
Step 1.2.3.1
Substitute the values of and into the formula .
Step 1.2.3.2
Simplify the right side.
Tap for more steps...
Step 1.2.3.2.1
Cancel the common factor of and .
Tap for more steps...
Step 1.2.3.2.1.1
Factor out of .
Step 1.2.3.2.1.2
Cancel the common factors.
Tap for more steps...
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 .
Tap for more steps...
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 .
Tap for more steps...
Step 1.2.4.1
Substitute the values of , and into the formula .
Step 1.2.4.2
Simplify the right side.
Tap for more steps...
Step 1.2.4.2.1
Simplify each term.
Tap for more steps...
Step 1.2.4.2.1.1
Raising to any positive power yields .
Step 1.2.4.2.1.2
Simplify the denominator.
Tap for more steps...
Step 1.2.4.2.1.2.1
Multiply by .
Step 1.2.4.2.1.2.2
Combine and .
Step 1.2.4.2.1.3
Divide by .
Step 1.2.4.2.1.4
Divide by .
Step 1.2.4.2.1.5
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
Find the vertex .
Step 4
Find , the distance from the vertex to the focus.
Tap for more steps...
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
Simplify.
Tap for more steps...
Step 4.3.1
Cancel the common factor of and .
Tap for more steps...
Step 4.3.1.1
Rewrite as .
Step 4.3.1.2
Move the negative in front of the fraction.
Step 4.3.2
Combine and .
Step 4.3.3
Divide by .
Step 4.3.4
Cancel the common factor of .
Tap for more steps...
Step 4.3.4.1
Cancel the common factor.
Step 4.3.4.2
Rewrite the expression.
Step 4.3.5
Multiply by .
Step 5
Find the focus.
Tap for more steps...
Step 5.1
The focus of a parabola can be found by adding to the y-coordinate if the parabola opens up or down.
Step 5.2
Substitute the known values of , , and into the formula and simplify.
Step 6