Precalculus Examples

Find the Properties (y-1)^2=16(x+1)
Step 1
Rewrite the equation in vertex form.
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Step 1.1
Isolate to the left side of the equation.
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Step 1.1.1
Rewrite the equation as .
Step 1.1.2
Divide each term in by and simplify.
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Step 1.1.2.1
Divide each term in by .
Step 1.1.2.2
Simplify the left side.
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Step 1.1.2.2.1
Cancel the common factor of .
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Step 1.1.2.2.1.1
Cancel the common factor.
Step 1.1.2.2.1.2
Divide by .
Step 1.1.3
Subtract from both sides of the equation.
Step 1.1.4
Reorder terms.
Step 1.2
Complete the square for .
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Step 1.2.1
Simplify the expression.
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Step 1.2.1.1
Simplify each term.
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Step 1.2.1.1.1
Rewrite as .
Step 1.2.1.1.2
Expand using the FOIL Method.
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Step 1.2.1.1.2.1
Apply the distributive property.
Step 1.2.1.1.2.2
Apply the distributive property.
Step 1.2.1.1.2.3
Apply the distributive property.
Step 1.2.1.1.3
Simplify and combine like terms.
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Step 1.2.1.1.3.1
Simplify each term.
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Step 1.2.1.1.3.1.1
Multiply by .
Step 1.2.1.1.3.1.2
Move to the left of .
Step 1.2.1.1.3.1.3
Rewrite as .
Step 1.2.1.1.3.1.4
Rewrite as .
Step 1.2.1.1.3.1.5
Multiply by .
Step 1.2.1.1.3.2
Subtract from .
Step 1.2.1.1.4
Apply the distributive property.
Step 1.2.1.1.5
Simplify.
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Step 1.2.1.1.5.1
Combine and .
Step 1.2.1.1.5.2
Cancel the common factor of .
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Step 1.2.1.1.5.2.1
Factor out of .
Step 1.2.1.1.5.2.2
Factor out of .
Step 1.2.1.1.5.2.3
Cancel the common factor.
Step 1.2.1.1.5.2.4
Rewrite the expression.
Step 1.2.1.1.5.3
Combine and .
Step 1.2.1.1.5.4
Multiply by .
Step 1.2.1.2
To write as a fraction with a common denominator, multiply by .
Step 1.2.1.3
Combine and .
Step 1.2.1.4
Combine the numerators over the common denominator.
Step 1.2.1.5
Simplify the numerator.
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Step 1.2.1.5.1
Multiply by .
Step 1.2.1.5.2
Subtract from .
Step 1.2.1.6
Move the negative in front of the fraction.
Step 1.2.2
Use the form , to find the values of , , and .
Step 1.2.3
Consider the vertex form of a parabola.
Step 1.2.4
Find the value of using the formula .
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Step 1.2.4.1
Substitute the values of and into the formula .
Step 1.2.4.2
Simplify the right side.
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Step 1.2.4.2.1
Multiply the numerator by the reciprocal of the denominator.
Step 1.2.4.2.2
Combine and .
Step 1.2.4.2.3
Cancel the common factor of and .
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Step 1.2.4.2.3.1
Factor out of .
Step 1.2.4.2.3.2
Cancel the common factors.
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Step 1.2.4.2.3.2.1
Factor out of .
Step 1.2.4.2.3.2.2
Cancel the common factor.
Step 1.2.4.2.3.2.3
Rewrite the expression.
Step 1.2.4.2.4
Multiply the numerator by the reciprocal of the denominator.
Step 1.2.4.2.5
Cancel the common factor of .
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Step 1.2.4.2.5.1
Move the leading negative in into the numerator.
Step 1.2.4.2.5.2
Factor out of .
Step 1.2.4.2.5.3
Cancel the common factor.
Step 1.2.4.2.5.4
Rewrite the expression.
Step 1.2.5
Find the value of using the formula .
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Step 1.2.5.1
Substitute the values of , and into the formula .
Step 1.2.5.2
Simplify the right side.
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Step 1.2.5.2.1
Simplify each term.
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Step 1.2.5.2.1.1
Simplify the numerator.
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Step 1.2.5.2.1.1.1
Apply the product rule to .
Step 1.2.5.2.1.1.2
Raise to the power of .
Step 1.2.5.2.1.1.3
Apply the product rule to .
Step 1.2.5.2.1.1.4
One to any power is one.
Step 1.2.5.2.1.1.5
Raise to the power of .
Step 1.2.5.2.1.1.6
Multiply by .
Step 1.2.5.2.1.2
Combine and .
Step 1.2.5.2.1.3
Cancel the common factor of and .
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Step 1.2.5.2.1.3.1
Factor out of .
Step 1.2.5.2.1.3.2
Cancel the common factors.
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Step 1.2.5.2.1.3.2.1
Factor out of .
Step 1.2.5.2.1.3.2.2
Cancel the common factor.
Step 1.2.5.2.1.3.2.3
Rewrite the expression.
Step 1.2.5.2.1.4
Multiply the numerator by the reciprocal of the denominator.
Step 1.2.5.2.1.5
Cancel the common factor of .
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Step 1.2.5.2.1.5.1
Factor out of .
Step 1.2.5.2.1.5.2
Cancel the common factor.
Step 1.2.5.2.1.5.3
Rewrite the expression.
Step 1.2.5.2.2
Combine the numerators over the common denominator.
Step 1.2.5.2.3
Subtract from .
Step 1.2.5.2.4
Divide by .
Step 1.2.6
Substitute the values of , , and into the vertex form .
Step 1.3
Set equal to the new right side.
Step 2
Use the vertex form, , to determine the values of , , and .
Step 3
Since the value of is positive, the parabola opens right.
Opens Right
Step 4
Find the vertex .
Step 5
Find , the distance from the vertex to the focus.
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Step 5.1
Find the distance from the vertex to a focus of the parabola by using the following formula.
Step 5.2
Substitute the value of into the formula.
Step 5.3
Simplify.
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Step 5.3.1
Combine and .
Step 5.3.2
Cancel the common factor of and .
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Step 5.3.2.1
Factor out of .
Step 5.3.2.2
Cancel the common factors.
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Step 5.3.2.2.1
Factor out of .
Step 5.3.2.2.2
Cancel the common factor.
Step 5.3.2.2.3
Rewrite the expression.
Step 5.3.3
Multiply the numerator by the reciprocal of the denominator.
Step 5.3.4
Multiply by .
Step 6
Find the focus.
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Step 6.1
The focus of a parabola can be found by adding to the x-coordinate if the parabola opens left or right.
Step 6.2
Substitute the known values of , , and into the formula and simplify.
Step 7
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
Step 8
Find the directrix.
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Step 8.1
The directrix of a parabola is the vertical line found by subtracting from the x-coordinate of the vertex if the parabola opens left or right.
Step 8.2
Substitute the known values of and into the formula and simplify.
Step 9
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Right
Vertex:
Focus:
Axis of Symmetry:
Directrix:
Step 10