# Precalculus Examples

Consider the vertex form of the parabola.
Rewrite the function in terms of and .
Complete the square on the right side of the equation.
Use the form , to find the values of , , and .
Consider the vertex form of a parabola.
Find the value of using the formula .
Multiply by to get .
Multiply by to get .
Move the negative in front of the fraction.
Simplify .
Multiply by to get .
Multiply by to get .
Find the value of using the formula .
Simplify each term.
Multiply by to get .
Raise to the power of to get .
Multiply by to get .
Move the negative in front of the fraction.
Simplify .
Multiply by to get .
Multiply by to get .
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.
Simplify the numerator.
Multiply by to get .
Substitute the values of , , and into the vertex form .
Reorder the right side of the equation to match the vertex form of a parabola.
Since does not contain the variable to solve for, move it to the right side of the equation by subtracting from both sides.
Use the vertex form, , to determine the values of , , and .
Since the value of is negative, the parabola opens down.
Opens Down
Find the vertex .
Find , the distance from the vertex to the focus.
Find the distance from the vertex to a focus of the parabola by using the following formula.
Substitute the value of into the formula.
Simplify.
Reduce the expression by cancelling the common factors.
Rewrite as .
Multiply by to get .
Cancel the common factor.
Rewrite the expression.
Move the negative in front of the fraction.
Find the focus.
The focus of a parabola can be found by adding to the y-coordinate if the parabola opens up or down.
Substitute the known values of , , and into the formula and simplify.
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
Find the directrix.
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.
Substitute the known values of and into the formula and simplify.
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Down
Vertex:
Focus:
Axis of Symmetry:
Directrix:

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