# 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 , .
Rewrite the equation as .
Simplify each term.
Multiply by 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.
Divide each term in by .
Reduce the expression by cancelling the common factors.
Cancel the common factor.
Divide by to get .
Using , the general equation of the parabola with the vertex and is .
Solve for to get .
Remove the extra parentheses.
Remove the parentheses around the expression .
Simplify each term.
Multiply by to get .
Rewrite as .
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Remove parentheses.
Simplify and combine like terms.
Simplify each term.
Use the power rule to combine exponents.
Move to the left of the expression .
Multiply by to get .
Multiply by to get .
Apply the distributive property.
Simplify.
Simplify .
Write as a fraction with denominator .
Multiply and to get .
Cancel the common factor of .
Rewrite.
Write as a fraction with denominator .
Factor out the greatest common factor .
Factor out of .
Factor out of .
Move .
Multiply by to get .
Cancel the common factor.
Rewrite the expression.
Simplify.
Multiply by to get .
Multiply and to get .
Multiply by to get .
Move the negative in front of the fraction.
Simplify .
Multiply by to get .
Write as a fraction with denominator .
Multiply and to get .
Multiply by to get .
Simplify each term.
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 .
Combine.
Multiply by to get .
Combine the numerators over the common denominator.
Multiply by to get .
Multiply by 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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