Precalculus Examples

Find the Parabola Through (-1,-5) with Vertex (2,3)
,
Step 1
The general equation of a parabola with vertex is . In this case we have as the vertex and is a point on the parabola. To find , substitute the two points in .
Step 2
Using to solve for , .
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Step 2.1
Rewrite the equation as .
Step 2.2
Simplify each term.
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Step 2.2.1
Multiply by .
Step 2.2.2
Subtract from .
Step 2.2.3
Raise to the power of .
Step 2.2.4
Move to the left of .
Step 2.3
Move all terms not containing to the right side of the equation.
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Step 2.3.1
Subtract from both sides of the equation.
Step 2.3.2
Subtract from .
Step 2.4
Divide each term in by and simplify.
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Step 2.4.1
Divide each term in by .
Step 2.4.2
Simplify the left side.
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Step 2.4.2.1
Cancel the common factor of .
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Step 2.4.2.1.1
Cancel the common factor.
Step 2.4.2.1.2
Divide by .
Step 2.4.3
Simplify the right side.
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Step 2.4.3.1
Move the negative in front of the fraction.
Step 3
Using , the general equation of the parabola with the vertex and is .
Step 4
Solve for .
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Step 4.1
Remove parentheses.
Step 4.2
Multiply by .
Step 4.3
Remove parentheses.
Step 4.4
Simplify .
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Step 4.4.1
Simplify each term.
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Step 4.4.1.1
Multiply by .
Step 4.4.1.2
Rewrite as .
Step 4.4.1.3
Expand using the FOIL Method.
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Step 4.4.1.3.1
Apply the distributive property.
Step 4.4.1.3.2
Apply the distributive property.
Step 4.4.1.3.3
Apply the distributive property.
Step 4.4.1.4
Simplify and combine like terms.
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Step 4.4.1.4.1
Simplify each term.
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Step 4.4.1.4.1.1
Multiply by .
Step 4.4.1.4.1.2
Move to the left of .
Step 4.4.1.4.1.3
Multiply by .
Step 4.4.1.4.2
Subtract from .
Step 4.4.1.5
Apply the distributive property.
Step 4.4.1.6
Simplify.
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Step 4.4.1.6.1
Combine and .
Step 4.4.1.6.2
Multiply .
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Step 4.4.1.6.2.1
Multiply by .
Step 4.4.1.6.2.2
Combine and .
Step 4.4.1.6.2.3
Multiply by .
Step 4.4.1.6.2.4
Combine and .
Step 4.4.1.6.3
Multiply .
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Step 4.4.1.6.3.1
Multiply by .
Step 4.4.1.6.3.2
Combine and .
Step 4.4.1.6.3.3
Multiply by .
Step 4.4.1.7
Simplify each term.
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Step 4.4.1.7.1
Move to the left of .
Step 4.4.1.7.2
Move the negative in front of the fraction.
Step 4.4.2
To write as a fraction with a common denominator, multiply by .
Step 4.4.3
Combine and .
Step 4.4.4
Combine the numerators over the common denominator.
Step 4.4.5
Simplify the numerator.
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Step 4.4.5.1
Multiply by .
Step 4.4.5.2
Add and .
Step 4.4.6
Move the negative in front of the fraction.
Step 5
The standard form and vertex form are as follows.
Standard Form:
Vertex Form:
Step 6
Simplify the standard form.
Standard Form:
Vertex Form:
Step 7
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