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Trigonometry Examples
Step 1
Step 1.1
Subtract from both sides of the equation.
Step 1.2
Subtract from both sides of the equation.
Step 1.3
Simplify .
Step 1.3.1
Rewrite as .
Step 1.3.2
Expand using the FOIL Method.
Step 1.3.2.1
Apply the distributive property.
Step 1.3.2.2
Apply the distributive property.
Step 1.3.2.3
Apply the distributive property.
Step 1.3.3
Simplify and combine like terms.
Step 1.3.3.1
Simplify each term.
Step 1.3.3.1.1
Multiply by .
Step 1.3.3.1.2
Move to the left of .
Step 1.3.3.1.3
Multiply by .
Step 1.3.3.2
Subtract from .
Step 1.3.4
Apply the distributive property.
Step 1.3.5
Simplify.
Step 1.3.5.1
Multiply by .
Step 1.3.5.2
Multiply by .
Step 1.4
Move all terms to the left side of the equation and simplify.
Step 1.4.1
Move all the expressions to the left side of the equation.
Step 1.4.1.1
Add to both sides of the equation.
Step 1.4.1.2
Subtract from both sides of the equation.
Step 1.4.1.3
Add to both sides of the equation.
Step 1.4.2
Simplify .
Step 1.4.2.1
Simplify each term.
Step 1.4.2.1.1
Rewrite as .
Step 1.4.2.1.2
Expand using the FOIL Method.
Step 1.4.2.1.2.1
Apply the distributive property.
Step 1.4.2.1.2.2
Apply the distributive property.
Step 1.4.2.1.2.3
Apply the distributive property.
Step 1.4.2.1.3
Simplify and combine like terms.
Step 1.4.2.1.3.1
Simplify each term.
Step 1.4.2.1.3.1.1
Multiply by .
Step 1.4.2.1.3.1.2
Multiply by .
Step 1.4.2.1.3.1.3
Multiply by .
Step 1.4.2.1.3.1.4
Multiply by .
Step 1.4.2.1.3.2
Add and .
Step 1.4.2.1.4
Rewrite as .
Step 1.4.2.1.5
Expand using the FOIL Method.
Step 1.4.2.1.5.1
Apply the distributive property.
Step 1.4.2.1.5.2
Apply the distributive property.
Step 1.4.2.1.5.3
Apply the distributive property.
Step 1.4.2.1.6
Simplify and combine like terms.
Step 1.4.2.1.6.1
Simplify each term.
Step 1.4.2.1.6.1.1
Multiply by .
Step 1.4.2.1.6.1.2
Move to the left of .
Step 1.4.2.1.6.1.3
Multiply by .
Step 1.4.2.1.6.2
Add and .
Step 1.4.2.1.7
Apply the distributive property.
Step 1.4.2.1.8
Simplify.
Step 1.4.2.1.8.1
Multiply by .
Step 1.4.2.1.8.2
Multiply by .
Step 1.4.2.2
Combine the opposite terms in .
Step 1.4.2.2.1
Subtract from .
Step 1.4.2.2.2
Add and .
Step 1.4.2.3
Subtract from .
Step 1.4.2.4
Subtract from .
Step 1.4.2.5
Add and .
Step 1.5
Move all terms not containing to the right side of the equation.
Step 1.5.1
Subtract from both sides of the equation.
Step 1.5.2
Add to both sides of the equation.
Step 1.5.3
Subtract from both sides of the equation.
Step 1.6
Divide each term in by and simplify.
Step 1.6.1
Divide each term in by .
Step 1.6.2
Simplify the left side.
Step 1.6.2.1
Cancel the common factor of .
Step 1.6.2.1.1
Cancel the common factor.
Step 1.6.2.1.2
Divide by .
Step 1.6.3
Simplify the right side.
Step 1.6.3.1
Simplify each term.
Step 1.6.3.1.1
Dividing two negative values results in a positive value.
Step 1.6.3.1.2
Cancel the common factor of and .
Step 1.6.3.1.2.1
Factor out of .
Step 1.6.3.1.2.2
Move the negative one from the denominator of .
Step 1.6.3.1.3
Rewrite as .
Step 1.6.3.1.4
Multiply by .
Step 1.6.3.1.5
Divide by .
Step 2
Step 2.1
Rewrite the equation in vertex form.
Step 2.1.1
Complete the square for .
Step 2.1.1.1
Use the form , to find the values of , , and .
Step 2.1.1.2
Consider the vertex form of a parabola.
Step 2.1.1.3
Find the value of using the formula .
Step 2.1.1.3.1
Substitute the values of and into the formula .
Step 2.1.1.3.2
Simplify the right side.
Step 2.1.1.3.2.1
Cancel the common factor of and .
Step 2.1.1.3.2.1.1
Factor out of .
Step 2.1.1.3.2.1.2
Move the negative one from the denominator of .
Step 2.1.1.3.2.2
Multiply by .
Step 2.1.1.4
Find the value of using the formula .
Step 2.1.1.4.1
Substitute the values of , and into the formula .
Step 2.1.1.4.2
Simplify the right side.
Step 2.1.1.4.2.1
Simplify each term.
Step 2.1.1.4.2.1.1
Raise to the power of .
Step 2.1.1.4.2.1.2
Multiply by .
Step 2.1.1.4.2.1.3
Divide by .
Step 2.1.1.4.2.1.4
Multiply by .
Step 2.1.1.4.2.2
Add and .
Step 2.1.1.5
Substitute the values of , , and into the vertex form .
Step 2.1.2
Set equal to the new right side.
Step 2.2
Use the vertex form, , to determine the values of , , and .
Step 2.3
Since the value of is negative, the parabola opens down.
Opens Down
Step 2.4
Find the vertex .
Step 2.5
Find , the distance from the vertex to the focus.
Step 2.5.1
Find the distance from the vertex to a focus of the parabola by using the following formula.
Step 2.5.2
Substitute the value of into the formula.
Step 2.5.3
Cancel the common factor of and .
Step 2.5.3.1
Rewrite as .
Step 2.5.3.2
Move the negative in front of the fraction.
Step 2.6
Find the focus.
Step 2.6.1
The focus of a parabola can be found by adding to the y-coordinate if the parabola opens up or down.
Step 2.6.2
Substitute the known values of , , and into the formula and simplify.
Step 2.7
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
Step 2.8
Find the directrix.
Step 2.8.1
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 2.8.2
Substitute the known values of and into the formula and simplify.
Step 2.9
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Down
Vertex:
Focus:
Axis of Symmetry:
Directrix:
Direction: Opens Down
Vertex:
Focus:
Axis of Symmetry:
Directrix:
Step 3