Precalculus Examples

Find the Parabola Through (3,0) with Vertex (-5,9)
,
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 to get .
Using to solve for , .
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Rewrite the equation as .
Simplify each term.
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Multiply by to get .
Add and to get .
Remove parentheses around .
Raise to the power of to get .
Move to the left of the expression .
Multiply by to get .
Since does not contain the variable to solve for, move it to the right side of the equation by subtracting from both sides.
Divide each term by and simplify.
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Divide each term in by .
Reduce the expression by cancelling the common factors.
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Cancel the common factor.
Divide by to get .
Using , the general equation of the parabola with the vertex and is .
Solve for to get .
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Remove the extra parentheses.
Remove the parentheses around the expression .
Simplify each term.
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Multiply by to get .
Rewrite as .
Expand using the FOIL Method.
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Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Remove parentheses.
Simplify and combine like terms.
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Simplify each term.
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Use the power rule to combine exponents.
Add and to get .
Move to the left of the expression .
Multiply by to get .
Multiply by to get .
Add and to get .
Apply the distributive property.
Simplify.
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Simplify .
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Write as a fraction with denominator .
Multiply and to get .
Cancel the common factor of .
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Rewrite.
Write as a fraction with denominator .
Factor out the greatest common factor .
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Factor out of .
Factor out of .
Move .
Multiply by to get .
Cancel the common factor.
Rewrite the expression.
Simplify.
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Multiply by to get .
Multiply and to get .
Multiply by to get .
Move the negative in front of the fraction.
Simplify .
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Multiply by to get .
Write as a fraction with denominator .
Multiply and to get .
Multiply by to get .
Simplify each term.
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Move to the left of the expression .
Multiply by to get .
Multiply by to get .
Move the negative in front of the fraction.
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Combine.
Multiply by to get .
Combine the numerators over the common denominator.
Multiply by to get .
Multiply by to get .
Add and to get .
The standard form and vertex form are as follows.
Standard Form:
Vertex Form:
Simplify the standard form.
Standard Form:
Vertex Form:
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