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Trigonometry Examples
Step 1
Rewrite the equation as .
Step 2
Multiply both sides of the equation by .
Step 3
Step 3.1
Simplify .
Step 3.1.1
Apply the distributive property.
Step 3.1.2
Combine and .
Step 3.1.3
Multiply .
Step 3.1.3.1
Multiply by .
Step 3.1.3.2
Combine and .
Step 3.1.4
Apply the distributive property.
Step 3.1.5
Cancel the common factor of .
Step 3.1.5.1
Move the leading negative in into the numerator.
Step 3.1.5.2
Factor out of .
Step 3.1.5.3
Cancel the common factor.
Step 3.1.5.4
Rewrite the expression.
Step 3.1.6
Multiply.
Step 3.1.6.1
Multiply by .
Step 3.1.6.2
Multiply by .
Step 3.1.7
Cancel the common factor of .
Step 3.1.7.1
Factor out of .
Step 3.1.7.2
Cancel the common factor.
Step 3.1.7.3
Rewrite the expression.
Step 3.1.8
Multiply by .
Step 4
Step 4.1
Rewrite.
Step 4.2
Rewrite as .
Step 4.3
Expand using the FOIL Method.
Step 4.3.1
Apply the distributive property.
Step 4.3.2
Apply the distributive property.
Step 4.3.3
Apply the distributive property.
Step 4.4
Simplify and combine like terms.
Step 4.4.1
Simplify each term.
Step 4.4.1.1
Multiply by .
Step 4.4.1.2
Move to the left of .
Step 4.4.1.3
Multiply by .
Step 4.4.2
Subtract from .
Step 4.5
Apply the distributive property.
Step 4.6
Simplify.
Step 4.6.1
Multiply by .
Step 4.6.2
Multiply by .
Step 5
Step 5.1
Add to both sides of the equation.
Step 5.2
Add and .
Step 6
Step 6.1
Rewrite the equation in vertex form.
Step 6.1.1
Complete the square for .
Step 6.1.1.1
Use the form , to find the values of , , and .
Step 6.1.1.2
Consider the vertex form of a parabola.
Step 6.1.1.3
Find the value of using the formula .
Step 6.1.1.3.1
Substitute the values of and into the formula .
Step 6.1.1.3.2
Simplify the right side.
Step 6.1.1.3.2.1
Cancel the common factor of and .
Step 6.1.1.3.2.1.1
Factor out of .
Step 6.1.1.3.2.1.2
Cancel the common factors.
Step 6.1.1.3.2.1.2.1
Factor out of .
Step 6.1.1.3.2.1.2.2
Cancel the common factor.
Step 6.1.1.3.2.1.2.3
Rewrite the expression.
Step 6.1.1.3.2.2
Cancel the common factor of and .
Step 6.1.1.3.2.2.1
Factor out of .
Step 6.1.1.3.2.2.2
Move the negative one from the denominator of .
Step 6.1.1.3.2.3
Multiply by .
Step 6.1.1.4
Find the value of using the formula .
Step 6.1.1.4.1
Substitute the values of , and into the formula .
Step 6.1.1.4.2
Simplify the right side.
Step 6.1.1.4.2.1
Simplify each term.
Step 6.1.1.4.2.1.1
Raise to the power of .
Step 6.1.1.4.2.1.2
Multiply by .
Step 6.1.1.4.2.1.3
Divide by .
Step 6.1.1.4.2.1.4
Multiply by .
Step 6.1.1.4.2.2
Add and .
Step 6.1.1.5
Substitute the values of , , and into the vertex form .
Step 6.1.2
Set equal to the new right side.
Step 6.2
Use the vertex form, , to determine the values of , , and .
Step 6.3
Since the value of is negative, the parabola opens left.
Opens Left
Step 6.4
Find the vertex .
Step 6.5
Find , the distance from the vertex to the focus.
Step 6.5.1
Find the distance from the vertex to a focus of the parabola by using the following formula.
Step 6.5.2
Substitute the value of into the formula.
Step 6.5.3
Simplify.
Step 6.5.3.1
Multiply by .
Step 6.5.3.2
Move the negative in front of the fraction.
Step 6.6
Find the focus.
Step 6.6.1
The focus of a parabola can be found by adding to the x-coordinate if the parabola opens left or right.
Step 6.6.2
Substitute the known values of , , and into the formula and simplify.
Step 6.7
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
Step 6.8
Find the directrix.
Step 6.8.1
The directrix of a parabola is the vertical line found by subtracting from the x-coordinate of the vertex if the parabola opens left or right.
Step 6.8.2
Substitute the known values of and into the formula and simplify.
Step 6.9
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Left
Vertex:
Focus:
Axis of Symmetry:
Directrix:
Direction: Opens Left
Vertex:
Focus:
Axis of Symmetry:
Directrix:
Step 7
Step 7.1
Substitute the value into . In this case, the point is .
Step 7.1.1
Replace the variable with in the expression.
Step 7.1.2
Simplify the result.
Step 7.1.2.1
Simplify each term.
Step 7.1.2.1.1
Subtract from .
Step 7.1.2.1.2
Combine exponents.
Step 7.1.2.1.2.1
Multiply by .
Step 7.1.2.1.2.2
Combine and .
Step 7.1.2.1.3
Rewrite as .
Step 7.1.2.1.4
Multiply by .
Step 7.1.2.1.5
Combine and simplify the denominator.
Step 7.1.2.1.5.1
Multiply by .
Step 7.1.2.1.5.2
Raise to the power of .
Step 7.1.2.1.5.3
Raise to the power of .
Step 7.1.2.1.5.4
Use the power rule to combine exponents.
Step 7.1.2.1.5.5
Add and .
Step 7.1.2.1.5.6
Rewrite as .
Step 7.1.2.1.5.6.1
Use to rewrite as .
Step 7.1.2.1.5.6.2
Apply the power rule and multiply exponents, .
Step 7.1.2.1.5.6.3
Combine and .
Step 7.1.2.1.5.6.4
Cancel the common factor of .
Step 7.1.2.1.5.6.4.1
Cancel the common factor.
Step 7.1.2.1.5.6.4.2
Rewrite the expression.
Step 7.1.2.1.5.6.5
Evaluate the exponent.
Step 7.1.2.1.6
Simplify the numerator.
Step 7.1.2.1.6.1
Combine using the product rule for radicals.
Step 7.1.2.1.6.2
Multiply by .
Step 7.1.2.2
The final answer is .
Step 7.1.3
Convert to decimal.
Step 7.2
Substitute the value into . In this case, the point is .
Step 7.2.1
Replace the variable with in the expression.
Step 7.2.2
Simplify the result.
Step 7.2.2.1
Simplify each term.
Step 7.2.2.1.1
Subtract from .
Step 7.2.2.1.2
Combine exponents.
Step 7.2.2.1.2.1
Multiply by .
Step 7.2.2.1.2.2
Combine and .
Step 7.2.2.1.3
Rewrite as .
Step 7.2.2.1.4
Multiply by .
Step 7.2.2.1.5
Combine and simplify the denominator.
Step 7.2.2.1.5.1
Multiply by .
Step 7.2.2.1.5.2
Raise to the power of .
Step 7.2.2.1.5.3
Raise to the power of .
Step 7.2.2.1.5.4
Use the power rule to combine exponents.
Step 7.2.2.1.5.5
Add and .
Step 7.2.2.1.5.6
Rewrite as .
Step 7.2.2.1.5.6.1
Use to rewrite as .
Step 7.2.2.1.5.6.2
Apply the power rule and multiply exponents, .
Step 7.2.2.1.5.6.3
Combine and .
Step 7.2.2.1.5.6.4
Cancel the common factor of .
Step 7.2.2.1.5.6.4.1
Cancel the common factor.
Step 7.2.2.1.5.6.4.2
Rewrite the expression.
Step 7.2.2.1.5.6.5
Evaluate the exponent.
Step 7.2.2.1.6
Simplify the numerator.
Step 7.2.2.1.6.1
Combine using the product rule for radicals.
Step 7.2.2.1.6.2
Multiply by .
Step 7.2.2.2
The final answer is .
Step 7.2.3
Convert to decimal.
Step 7.3
Substitute the value into . In this case, the point is .
Step 7.3.1
Replace the variable with in the expression.
Step 7.3.2
Simplify the result.
Step 7.3.2.1
Simplify each term.
Step 7.3.2.1.1
Subtract from .
Step 7.3.2.1.2
Combine exponents.
Step 7.3.2.1.2.1
Factor out negative.
Step 7.3.2.1.2.2
Multiply by .
Step 7.3.2.1.2.3
Multiply by .
Step 7.3.2.1.3
Rewrite as .
Step 7.3.2.1.4
Any root of is .
Step 7.3.2.1.5
Multiply by .
Step 7.3.2.1.6
Combine and simplify the denominator.
Step 7.3.2.1.6.1
Multiply by .
Step 7.3.2.1.6.2
Raise to the power of .
Step 7.3.2.1.6.3
Raise to the power of .
Step 7.3.2.1.6.4
Use the power rule to combine exponents.
Step 7.3.2.1.6.5
Add and .
Step 7.3.2.1.6.6
Rewrite as .
Step 7.3.2.1.6.6.1
Use to rewrite as .
Step 7.3.2.1.6.6.2
Apply the power rule and multiply exponents, .
Step 7.3.2.1.6.6.3
Combine and .
Step 7.3.2.1.6.6.4
Cancel the common factor of .
Step 7.3.2.1.6.6.4.1
Cancel the common factor.
Step 7.3.2.1.6.6.4.2
Rewrite the expression.
Step 7.3.2.1.6.6.5
Evaluate the exponent.
Step 7.3.2.2
The final answer is .
Step 7.3.3
Convert to decimal.
Step 7.4
Substitute the value into . In this case, the point is .
Step 7.4.1
Replace the variable with in the expression.
Step 7.4.2
Simplify the result.
Step 7.4.2.1
Simplify each term.
Step 7.4.2.1.1
Subtract from .
Step 7.4.2.1.2
Combine exponents.
Step 7.4.2.1.2.1
Factor out negative.
Step 7.4.2.1.2.2
Multiply by .
Step 7.4.2.1.2.3
Multiply by .
Step 7.4.2.1.3
Rewrite as .
Step 7.4.2.1.4
Any root of is .
Step 7.4.2.1.5
Multiply by .
Step 7.4.2.1.6
Combine and simplify the denominator.
Step 7.4.2.1.6.1
Multiply by .
Step 7.4.2.1.6.2
Raise to the power of .
Step 7.4.2.1.6.3
Raise to the power of .
Step 7.4.2.1.6.4
Use the power rule to combine exponents.
Step 7.4.2.1.6.5
Add and .
Step 7.4.2.1.6.6
Rewrite as .
Step 7.4.2.1.6.6.1
Use to rewrite as .
Step 7.4.2.1.6.6.2
Apply the power rule and multiply exponents, .
Step 7.4.2.1.6.6.3
Combine and .
Step 7.4.2.1.6.6.4
Cancel the common factor of .
Step 7.4.2.1.6.6.4.1
Cancel the common factor.
Step 7.4.2.1.6.6.4.2
Rewrite the expression.
Step 7.4.2.1.6.6.5
Evaluate the exponent.
Step 7.4.2.2
The final answer is .
Step 7.4.3
Convert to decimal.
Step 7.5
Graph the parabola using its properties and the selected points.
Step 8
Graph the parabola using its properties and the selected points.
Direction: Opens Left
Vertex:
Focus:
Axis of Symmetry:
Directrix:
Step 9