Trigonometry Examples

Graph y=1/2*((x-5)(x+2))
Step 1
Solve for .
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Step 1.1
Remove parentheses.
Step 1.2
Simplify .
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Step 1.2.1
Expand using the FOIL Method.
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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.
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Step 1.2.2.1
Simplify each term.
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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.
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Step 1.2.4.1
Combine and .
Step 1.2.4.2
Multiply .
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Step 1.2.4.2.1
Combine and .
Step 1.2.4.2.2
Combine and .
Step 1.2.4.3
Cancel the common factor of .
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Step 1.2.4.3.1
Factor out of .
Step 1.2.4.3.2
Cancel the common factor.
Step 1.2.4.3.3
Rewrite the expression.
Step 1.2.5
Move the negative in front of the fraction.
Step 2
Find the properties of the given parabola.
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Step 2.1
Rewrite the equation in vertex form.
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Step 2.1.1
Complete the square for .
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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 .
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Step 2.1.1.3.1
Substitute the values of and into the formula .
Step 2.1.1.3.2
Simplify the right side.
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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
Divide by .
Step 2.1.1.3.2.4
Cancel the common factor of .
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Step 2.1.1.3.2.4.1
Cancel the common factor.
Step 2.1.1.3.2.4.2
Rewrite the expression.
Step 2.1.1.3.2.5
Multiply by .
Step 2.1.1.4
Find the value of using the formula .
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Step 2.1.1.4.1
Substitute the values of , and into the formula .
Step 2.1.1.4.2
Simplify the right side.
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Step 2.1.1.4.2.1
Simplify each term.
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Step 2.1.1.4.2.1.1
Simplify the numerator.
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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
Divide by .
Step 2.1.1.4.2.1.4
Multiply the numerator by the reciprocal of the denominator.
Step 2.1.1.4.2.1.5
Multiply .
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Step 2.1.1.4.2.1.5.1
Multiply by .
Step 2.1.1.4.2.1.5.2
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
Combine and .
Step 2.1.1.4.2.4
Combine the numerators over the common denominator.
Step 2.1.1.4.2.5
Simplify the numerator.
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Step 2.1.1.4.2.5.1
Multiply by .
Step 2.1.1.4.2.5.2
Subtract from .
Step 2.1.1.4.2.6
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.
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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.
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Step 2.5.3.1
Combine and .
Step 2.5.3.2
Divide by .
Step 2.6
Find the focus.
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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.
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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
Select a few values, and plug them into the equation to find the corresponding values. The values should be selected around the vertex.
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Step 3.1
Replace the variable with in the expression.
Step 3.2
Simplify the result.
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Step 3.2.1
Combine the numerators over the common denominator.
Step 3.2.2
Simplify each term.
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Step 3.2.2.1
One to any power is one.
Step 3.2.2.2
Multiply by .
Step 3.2.3
Simplify the expression.
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Step 3.2.3.1
Subtract from .
Step 3.2.3.2
Divide by .
Step 3.2.3.3
Subtract from .
Step 3.2.4
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.
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Step 3.5.1
Combine the numerators over the common denominator.
Step 3.5.2
Simplify each term.
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Step 3.5.2.1
Raising to any positive power yields .
Step 3.5.2.2
Multiply by .
Step 3.5.3
Simplify the expression.
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Step 3.5.3.1
Add and .
Step 3.5.3.2
Divide by .
Step 3.5.3.3
Add and .
Step 3.5.4
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.
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Step 3.8.1
Combine the numerators over the common denominator.
Step 3.8.2
Simplify each term.
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Step 3.8.2.1
Raise to the power of .
Step 3.8.2.2
Multiply by .
Step 3.8.3
Simplify the expression.
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Step 3.8.3.1
Subtract from .
Step 3.8.3.2
Divide by .
Step 3.8.3.3
Add and .
Step 3.8.4
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.
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Step 3.11.1
Combine the numerators over the common denominator.
Step 3.11.2
Simplify each term.
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Step 3.11.2.1
Raise to the power of .
Step 3.11.2.2
Multiply by .
Step 3.11.3
Simplify the expression.
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Step 3.11.3.1
Subtract from .
Step 3.11.3.2
Divide by .
Step 3.11.3.3
Add and .
Step 3.11.4
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