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Trigonometry Examples
Step 1
Step 1.1
Remove parentheses.
Step 1.2
Simplify .
Step 1.2.1
Expand using the FOIL Method.
Step 1.2.1.1
Apply the distributive property.
Step 1.2.1.2
Apply the distributive property.
Step 1.2.1.3
Apply the distributive property.
Step 1.2.2
Simplify and combine like terms.
Step 1.2.2.1
Simplify each term.
Step 1.2.2.1.1
Multiply by .
Step 1.2.2.1.2
Move to the left of .
Step 1.2.2.1.3
Multiply by .
Step 1.2.2.2
Subtract from .
Step 1.2.3
Apply the distributive property.
Step 1.2.4
Simplify.
Step 1.2.4.1
Combine and .
Step 1.2.4.2
Cancel the common factor of .
Step 1.2.4.2.1
Factor out of .
Step 1.2.4.2.2
Factor out of .
Step 1.2.4.2.3
Cancel the common factor.
Step 1.2.4.2.4
Rewrite the expression.
Step 1.2.4.3
Combine and .
Step 1.2.4.4
Cancel the common factor of .
Step 1.2.4.4.1
Factor out of .
Step 1.2.4.4.2
Factor out of .
Step 1.2.4.4.3
Cancel the common factor.
Step 1.2.4.4.4
Rewrite the expression.
Step 1.2.4.5
Combine and .
Step 1.2.4.6
Multiply by .
Step 1.2.5
Move the negative in front of the fraction.
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
Multiply the numerator by the reciprocal of the denominator.
Step 2.1.1.3.2.2
Combine and .
Step 2.1.1.3.2.3
Multiply by .
Step 2.1.1.3.2.4
Cancel the common factor of and .
Step 2.1.1.3.2.4.1
Factor out of .
Step 2.1.1.3.2.4.2
Cancel the common factors.
Step 2.1.1.3.2.4.2.1
Factor out of .
Step 2.1.1.3.2.4.2.2
Cancel the common factor.
Step 2.1.1.3.2.4.2.3
Rewrite the expression.
Step 2.1.1.3.2.5
Multiply the numerator by the reciprocal of the denominator.
Step 2.1.1.3.2.6
Multiply by .
Step 2.1.1.3.2.7
Cancel the common factor of .
Step 2.1.1.3.2.7.1
Move the leading negative in into the numerator.
Step 2.1.1.3.2.7.2
Factor out of .
Step 2.1.1.3.2.7.3
Cancel the common factor.
Step 2.1.1.3.2.7.4
Rewrite the expression.
Step 2.1.1.3.2.8
Cancel the common factor of .
Step 2.1.1.3.2.8.1
Factor out of .
Step 2.1.1.3.2.8.2
Factor out of .
Step 2.1.1.3.2.8.3
Cancel the common factor.
Step 2.1.1.3.2.8.4
Rewrite the expression.
Step 2.1.1.3.2.9
Combine and .
Step 2.1.1.3.2.10
Multiply by .
Step 2.1.1.3.2.11
Move the negative in front of the fraction.
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
Simplify the numerator.
Step 2.1.1.4.2.1.1.1
Apply the product rule to .
Step 2.1.1.4.2.1.1.2
Raise to the power of .
Step 2.1.1.4.2.1.1.3
Apply the product rule to .
Step 2.1.1.4.2.1.1.4
Raise to the power of .
Step 2.1.1.4.2.1.1.5
Raise to the power of .
Step 2.1.1.4.2.1.1.6
Multiply by .
Step 2.1.1.4.2.1.2
Combine and .
Step 2.1.1.4.2.1.3
Multiply by .
Step 2.1.1.4.2.1.4
Cancel the common factor of and .
Step 2.1.1.4.2.1.4.1
Factor out of .
Step 2.1.1.4.2.1.4.2
Cancel the common factors.
Step 2.1.1.4.2.1.4.2.1
Factor out of .
Step 2.1.1.4.2.1.4.2.2
Cancel the common factor.
Step 2.1.1.4.2.1.4.2.3
Rewrite the expression.
Step 2.1.1.4.2.1.5
Multiply the numerator by the reciprocal of the denominator.
Step 2.1.1.4.2.1.6
Cancel the common factor of .
Step 2.1.1.4.2.1.6.1
Factor out of .
Step 2.1.1.4.2.1.6.2
Cancel the common factor.
Step 2.1.1.4.2.1.6.3
Rewrite the expression.
Step 2.1.1.4.2.1.7
Cancel the common factor of .
Step 2.1.1.4.2.1.7.1
Factor out of .
Step 2.1.1.4.2.1.7.2
Factor out of .
Step 2.1.1.4.2.1.7.3
Cancel the common factor.
Step 2.1.1.4.2.1.7.4
Rewrite the expression.
Step 2.1.1.4.2.1.8
Combine and .
Step 2.1.1.4.2.1.9
Multiply by .
Step 2.1.1.4.2.2
To write as a fraction with a common denominator, multiply by .
Step 2.1.1.4.2.3
To write as a fraction with a common denominator, multiply by .
Step 2.1.1.4.2.4
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 2.1.1.4.2.4.1
Multiply by .
Step 2.1.1.4.2.4.2
Multiply by .
Step 2.1.1.4.2.4.3
Multiply by .
Step 2.1.1.4.2.4.4
Multiply by .
Step 2.1.1.4.2.5
Combine the numerators over the common denominator.
Step 2.1.1.4.2.6
Simplify the numerator.
Step 2.1.1.4.2.6.1
Multiply by .
Step 2.1.1.4.2.6.2
Multiply by .
Step 2.1.1.4.2.6.3
Subtract from .
Step 2.1.1.4.2.7
Move the negative in front of the fraction.
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 positive, the parabola opens up.
Opens Up
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
Simplify.
Step 2.5.3.1
Combine and .
Step 2.5.3.2
Reduce the expression by cancelling the common factors.
Step 2.5.3.2.1
Multiply by .
Step 2.5.3.2.2
Cancel the common factor of and .
Step 2.5.3.2.2.1
Factor out of .
Step 2.5.3.2.2.2
Cancel the common factors.
Step 2.5.3.2.2.2.1
Factor out of .
Step 2.5.3.2.2.2.2
Cancel the common factor.
Step 2.5.3.2.2.2.3
Rewrite the expression.
Step 2.5.3.3
Multiply the numerator by the reciprocal of the denominator.
Step 2.5.3.4
Multiply by .
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 Up
Vertex:
Focus:
Axis of Symmetry:
Directrix:
Direction: Opens Up
Vertex:
Focus:
Axis of Symmetry:
Directrix:
Step 3
Step 3.1
Replace the variable with in the expression.
Step 3.2
Simplify the result.
Step 3.2.1
Simplify each term.
Step 3.2.1.1
One to any power is one.
Step 3.2.1.2
Multiply by .
Step 3.2.1.3
Multiply by .
Step 3.2.2
Find the common denominator.
Step 3.2.2.1
Multiply by .
Step 3.2.2.2
Multiply by .
Step 3.2.2.3
Multiply by .
Step 3.2.2.4
Multiply by .
Step 3.2.2.5
Reorder the factors of .
Step 3.2.2.6
Multiply by .
Step 3.2.2.7
Reorder the factors of .
Step 3.2.2.8
Multiply by .
Step 3.2.3
Combine the numerators over the common denominator.
Step 3.2.4
Simplify each term.
Step 3.2.4.1
Multiply by .
Step 3.2.4.2
Multiply by .
Step 3.2.5
Reduce the expression by cancelling the common factors.
Step 3.2.5.1
Subtract from .
Step 3.2.5.2
Subtract from .
Step 3.2.5.3
Cancel the common factor of and .
Step 3.2.5.3.1
Factor out of .
Step 3.2.5.3.2
Cancel the common factors.
Step 3.2.5.3.2.1
Factor out of .
Step 3.2.5.3.2.2
Cancel the common factor.
Step 3.2.5.3.2.3
Rewrite the expression.
Step 3.2.5.4
Move the negative in front of the fraction.
Step 3.2.6
The final answer is .
Step 3.3
The value at is .
Step 3.4
Replace the variable with in the expression.
Step 3.5
Simplify the result.
Step 3.5.1
Simplify each term.
Step 3.5.1.1
Raising to any positive power yields .
Step 3.5.1.2
Multiply by .
Step 3.5.1.3
Divide by .
Step 3.5.1.4
Cancel the common factor of and .
Step 3.5.1.4.1
Factor out of .
Step 3.5.1.4.2
Cancel the common factors.
Step 3.5.1.4.2.1
Factor out of .
Step 3.5.1.4.2.2
Cancel the common factor.
Step 3.5.1.4.2.3
Rewrite the expression.
Step 3.5.1.4.2.4
Divide by .
Step 3.5.1.5
Multiply .
Step 3.5.1.5.1
Multiply by .
Step 3.5.1.5.2
Multiply by .
Step 3.5.2
Simplify by adding numbers.
Step 3.5.2.1
Add and .
Step 3.5.2.2
Subtract from .
Step 3.5.3
The final answer is .
Step 3.6
The value at is .
Step 3.7
Replace the variable with in the expression.
Step 3.8
Simplify the result.
Step 3.8.1
Simplify each term.
Step 3.8.1.1
Raise to the power of .
Step 3.8.1.2
Multiply by .
Step 3.8.1.3
Multiply by .
Step 3.8.2
Find the common denominator.
Step 3.8.2.1
Multiply by .
Step 3.8.2.2
Multiply by .
Step 3.8.2.3
Multiply by .
Step 3.8.2.4
Multiply by .
Step 3.8.2.5
Reorder the factors of .
Step 3.8.2.6
Multiply by .
Step 3.8.2.7
Reorder the factors of .
Step 3.8.2.8
Multiply by .
Step 3.8.3
Combine the numerators over the common denominator.
Step 3.8.4
Simplify each term.
Step 3.8.4.1
Multiply by .
Step 3.8.4.2
Multiply by .
Step 3.8.5
Simplify the expression.
Step 3.8.5.1
Subtract from .
Step 3.8.5.2
Subtract from .
Step 3.8.5.3
Divide by .
Step 3.8.6
The final answer is .
Step 3.9
The value at is .
Step 3.10
Replace the variable with in the expression.
Step 3.11
Simplify the result.
Step 3.11.1
Simplify each term.
Step 3.11.1.1
Raise to the power of .
Step 3.11.1.2
Multiply by .
Step 3.11.1.3
Cancel the common factor of and .
Step 3.11.1.3.1
Factor out of .
Step 3.11.1.3.2
Cancel the common factors.
Step 3.11.1.3.2.1
Factor out of .
Step 3.11.1.3.2.2
Cancel the common factor.
Step 3.11.1.3.2.3
Rewrite the expression.
Step 3.11.1.4
Multiply by .
Step 3.11.2
Combine fractions.
Step 3.11.2.1
Combine the numerators over the common denominator.
Step 3.11.2.2
Subtract from .
Step 3.11.3
Simplify each term.
Step 3.11.3.1
Divide by .
Step 3.11.3.2
Move the negative in front of the fraction.
Step 3.11.4
Subtract from .
Step 3.11.5
The final answer is .
Step 3.12
The value at is .
Step 3.13
Graph the parabola using its properties and the selected points.
Step 4
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex:
Focus:
Axis of Symmetry:
Directrix:
Step 5