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Trigonometry Examples
Step 1
Step 1.1
Use the vertex form, , to determine the values of , , and .
Step 1.2
Since the value of is negative, the parabola opens down.
Opens Down
Step 1.3
Find the vertex .
Step 1.4
Find , the distance from the vertex to the focus.
Step 1.4.1
Find the distance from the vertex to a focus of the parabola by using the following formula.
Step 1.4.2
Substitute the value of into the formula.
Step 1.4.3
Simplify.
Step 1.4.3.1
Cancel the common factor of and .
Step 1.4.3.1.1
Rewrite as .
Step 1.4.3.1.2
Move the negative in front of the fraction.
Step 1.4.3.2
Combine and .
Step 1.4.3.3
Cancel the common factor of and .
Step 1.4.3.3.1
Factor out of .
Step 1.4.3.3.2
Cancel the common factors.
Step 1.4.3.3.2.1
Factor out of .
Step 1.4.3.3.2.2
Cancel the common factor.
Step 1.4.3.3.2.3
Rewrite the expression.
Step 1.4.3.4
Multiply the numerator by the reciprocal of the denominator.
Step 1.4.3.5
Multiply .
Step 1.4.3.5.1
Multiply by .
Step 1.4.3.5.2
Multiply by .
Step 1.5
Find the focus.
Step 1.5.1
The focus of a parabola can be found by adding to the y-coordinate if the parabola opens up or down.
Step 1.5.2
Substitute the known values of , , and into the formula and simplify.
Step 1.6
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
Step 1.7
Find the directrix.
Step 1.7.1
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 1.7.2
Substitute the known values of and into the formula and simplify.
Step 1.8
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 2
Step 2.1
Replace the variable with in the expression.
Step 2.2
Simplify the result.
Step 2.2.1
Combine the numerators over the common denominator.
Step 2.2.2
Simplify each term.
Step 2.2.2.1
Raise to the power of .
Step 2.2.2.2
Multiply by .
Step 2.2.3
Simplify the expression.
Step 2.2.3.1
Subtract from .
Step 2.2.3.2
Multiply by .
Step 2.2.4
Simplify each term.
Step 2.2.4.1
Move the negative in front of the fraction.
Step 2.2.4.2
Cancel the common factor of and .
Step 2.2.4.2.1
Factor out of .
Step 2.2.4.2.2
Cancel the common factors.
Step 2.2.4.2.2.1
Factor out of .
Step 2.2.4.2.2.2
Cancel the common factor.
Step 2.2.4.2.2.3
Rewrite the expression.
Step 2.2.5
To write as a fraction with a common denominator, multiply by .
Step 2.2.6
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 2.2.6.1
Multiply by .
Step 2.2.6.2
Multiply by .
Step 2.2.7
Combine the numerators over the common denominator.
Step 2.2.8
Simplify the numerator.
Step 2.2.8.1
Multiply by .
Step 2.2.8.2
Add and .
Step 2.2.9
Move the negative in front of the fraction.
Step 2.2.10
The final answer is .
Step 2.3
The value at is .
Step 2.4
Replace the variable with in the expression.
Step 2.5
Simplify the result.
Step 2.5.1
Combine the numerators over the common denominator.
Step 2.5.2
Simplify each term.
Step 2.5.2.1
Raise to the power of .
Step 2.5.2.2
Multiply by .
Step 2.5.3
Simplify the expression.
Step 2.5.3.1
Subtract from .
Step 2.5.3.2
Multiply by .
Step 2.5.4
Simplify each term.
Step 2.5.4.1
Cancel the common factor of and .
Step 2.5.4.1.1
Factor out of .
Step 2.5.4.1.2
Cancel the common factors.
Step 2.5.4.1.2.1
Factor out of .
Step 2.5.4.1.2.2
Cancel the common factor.
Step 2.5.4.1.2.3
Rewrite the expression.
Step 2.5.4.2
Move the negative in front of the fraction.
Step 2.5.5
Simplify terms.
Step 2.5.5.1
Combine the numerators over the common denominator.
Step 2.5.5.2
Add and .
Step 2.5.5.3
Cancel the common factor of and .
Step 2.5.5.3.1
Factor out of .
Step 2.5.5.3.2
Cancel the common factors.
Step 2.5.5.3.2.1
Factor out of .
Step 2.5.5.3.2.2
Cancel the common factor.
Step 2.5.5.3.2.3
Rewrite the expression.
Step 2.5.5.4
Move the negative in front of the fraction.
Step 2.5.6
The final answer is .
Step 2.6
The value at is .
Step 2.7
Replace the variable with in the expression.
Step 2.8
Simplify the result.
Step 2.8.1
Combine the numerators over the common denominator.
Step 2.8.2
Simplify each term.
Step 2.8.2.1
Raise to the power of .
Step 2.8.2.2
Multiply by .
Step 2.8.3
Simplify the expression.
Step 2.8.3.1
Subtract from .
Step 2.8.3.2
Multiply by .
Step 2.8.4
Simplify each term.
Step 2.8.4.1
Move the negative in front of the fraction.
Step 2.8.4.2
Cancel the common factor of and .
Step 2.8.4.2.1
Factor out of .
Step 2.8.4.2.2
Cancel the common factors.
Step 2.8.4.2.2.1
Factor out of .
Step 2.8.4.2.2.2
Cancel the common factor.
Step 2.8.4.2.2.3
Rewrite the expression.
Step 2.8.5
To write as a fraction with a common denominator, multiply by .
Step 2.8.6
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 2.8.6.1
Multiply by .
Step 2.8.6.2
Multiply by .
Step 2.8.7
Combine the numerators over the common denominator.
Step 2.8.8
Simplify the numerator.
Step 2.8.8.1
Multiply by .
Step 2.8.8.2
Add and .
Step 2.8.9
Move the negative in front of the fraction.
Step 2.8.10
The final answer is .
Step 2.9
The value at is .
Step 2.10
Replace the variable with in the expression.
Step 2.11
Simplify the result.
Step 2.11.1
Combine the numerators over the common denominator.
Step 2.11.2
Simplify each term.
Step 2.11.2.1
Multiply by by adding the exponents.
Step 2.11.2.1.1
Multiply by .
Step 2.11.2.1.1.1
Raise to the power of .
Step 2.11.2.1.1.2
Use the power rule to combine exponents.
Step 2.11.2.1.2
Add and .
Step 2.11.2.2
Raise to the power of .
Step 2.11.3
Simplify the expression.
Step 2.11.3.1
Subtract from .
Step 2.11.3.2
Multiply by .
Step 2.11.4
Simplify each term.
Step 2.11.4.1
Cancel the common factor of and .
Step 2.11.4.1.1
Factor out of .
Step 2.11.4.1.2
Cancel the common factors.
Step 2.11.4.1.2.1
Factor out of .
Step 2.11.4.1.2.2
Cancel the common factor.
Step 2.11.4.1.2.3
Rewrite the expression.
Step 2.11.4.2
Move the negative in front of the fraction.
Step 2.11.5
Simplify terms.
Step 2.11.5.1
Combine the numerators over the common denominator.
Step 2.11.5.2
Add and .
Step 2.11.5.3
Cancel the common factor of and .
Step 2.11.5.3.1
Factor out of .
Step 2.11.5.3.2
Cancel the common factors.
Step 2.11.5.3.2.1
Factor out of .
Step 2.11.5.3.2.2
Cancel the common factor.
Step 2.11.5.3.2.3
Rewrite the expression.
Step 2.11.5.4
Move the negative in front of the fraction.
Step 2.11.6
The final answer is .
Step 2.12
The value at is .
Step 2.13
Graph the parabola using its properties and the selected points.
Step 3
Graph the parabola using its properties and the selected points.
Direction: Opens Down
Vertex:
Focus:
Axis of Symmetry:
Directrix:
Step 4