Precalculus Examples

Arrange all the terms containing on the left side and all other terms on the right hand side.
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Subtract from both sides of the equation.
Subtract from both sides of the equation.
Complete the square on the right side of the equation.
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Use the form , to find the values of , , and .
Consider the vertex form of a parabola.
Find the value of using the formula .
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Reduce the expression by cancelling the common factors.
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Cancel the common factor.
Rewrite the expression.
Multiply the numerator by the reciprocal of the denominator.
Multiply by .
Find the value of using the formula .
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Simplify each term.
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Simplify the numerator.
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Remove parentheses around .
Raising to any positive power yields .
Simplify the denominator.
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Remove parentheses.
Write as a fraction with denominator .
Multiply and .
Multiply the numerator by the reciprocal of the denominator.
Reduce the expression by cancelling the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply by .
Multiply by .
Add and .
Substitute the values of , , and into the vertex form .
Use the vertex form, , to determine the values of , , and .
Since the value of is positive, the parabola opens up.
Opens Up
Find the vertex .
Find , the distance from the vertex to the focus.
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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.
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Simplify the denominator.
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Multiply by .
Write as a fraction with denominator .
Multiply and .
Multiply the numerator by the reciprocal of the denominator.
Multiply by .
Reduce the expression by cancelling the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Find the focus.
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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.
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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 Up
Vertex:
Focus:
Axis of Symmetry:
Directrix:
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