Precalculus Examples

Find the properties of the given parabola.
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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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Multiply by to get .
Multiply by to get .
Find the value of using the formula .
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Simplify each term.
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Multiply by to get .
Raise to the power of 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 .
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Combine.
Multiply by to get .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by to get .
Multiply by to get .
Subtract from to get .
Move the negative in front of the fraction.
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 adding to both sides.
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.
Reduce the expression by cancelling the common factors.
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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:
Direction: Opens Up
Vertex:
Focus:
Axis of Symmetry:
Directrix:
Select a few values, and plug them into the equation to find the corresponding values. The values should be selected around the vertex.
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Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Remove parentheses around .
One to any power is one.
Multiply by to get .
Simplify by adding and subtracting.
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Subtract from to get .
Add and to get .
The final answer is .
The value at is .
Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Remove parentheses around .
Raising to any positive power yields .
Multiply by to get .
Simplify by adding zeros.
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Add and to get .
Add and to get .
The final answer is .
The value at is .
Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Remove parentheses around .
Raise to the power of to get .
Multiply by to get .
Simplify by adding and subtracting.
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Subtract from to get .
Add and to get .
The final answer is .
The value at is .
Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Remove parentheses around .
Raise to the power of to get .
Multiply by to get .
Simplify by adding and subtracting.
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Subtract from to get .
Add and to get .
The final answer is .
The value at is .
Graph the parabola using its properties and the selected points.
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex:
Focus:
Axis of Symmetry:
Directrix:
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