Pre-Algebra Examples

Graph x^2-4x-6y+13=0
Step 1
Solve for .
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Step 1.1
Move all terms not containing to the right side of the equation.
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Step 1.1.1
Subtract from both sides of the equation.
Step 1.1.2
Add to both sides of the equation.
Step 1.1.3
Subtract from both sides of the equation.
Step 1.2
Divide each term in by and simplify.
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Step 1.2.1
Divide each term in by .
Step 1.2.2
Simplify the left side.
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Step 1.2.2.1
Cancel the common factor of .
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Step 1.2.2.1.1
Cancel the common factor.
Step 1.2.2.1.2
Divide by .
Step 1.2.3
Simplify the right side.
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Step 1.2.3.1
Simplify each term.
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Step 1.2.3.1.1
Dividing two negative values results in a positive value.
Step 1.2.3.1.2
Cancel the common factor of and .
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Step 1.2.3.1.2.1
Factor out of .
Step 1.2.3.1.2.2
Cancel the common factors.
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Step 1.2.3.1.2.2.1
Factor out of .
Step 1.2.3.1.2.2.2
Cancel the common factor.
Step 1.2.3.1.2.2.3
Rewrite the expression.
Step 1.2.3.1.3
Move the negative in front of the fraction.
Step 1.2.3.1.4
Dividing two negative values results in a positive value.
Step 2
Find the properties of the given parabola.
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Step 2.1
Rewrite the equation in vertex form.
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Step 2.1.1
Complete the square for .
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Step 2.1.1.1
Use the form , to find the values of , , and .
Step 2.1.1.2
Consider the vertex form of a parabola.
Step 2.1.1.3
Find the value of using the formula .
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Step 2.1.1.3.1
Substitute the values of and into the formula .
Step 2.1.1.3.2
Simplify the right side.
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Step 2.1.1.3.2.1
Multiply the numerator by the reciprocal of the denominator.
Step 2.1.1.3.2.2
Cancel the common factor of .
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Step 2.1.1.3.2.2.1
Move the leading negative in into the numerator.
Step 2.1.1.3.2.2.2
Factor out of .
Step 2.1.1.3.2.2.3
Cancel the common factor.
Step 2.1.1.3.2.2.4
Rewrite the expression.
Step 2.1.1.3.2.3
Multiply by .
Step 2.1.1.3.2.4
Combine and .
Step 2.1.1.3.2.5
Cancel the common factor of and .
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Step 2.1.1.3.2.5.1
Factor out of .
Step 2.1.1.3.2.5.2
Cancel the common factors.
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Step 2.1.1.3.2.5.2.1
Factor out of .
Step 2.1.1.3.2.5.2.2
Cancel the common factor.
Step 2.1.1.3.2.5.2.3
Rewrite the expression.
Step 2.1.1.3.2.6
Multiply the numerator by the reciprocal of the denominator.
Step 2.1.1.3.2.7
Multiply by .
Step 2.1.1.4
Find the value of using the formula .
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Step 2.1.1.4.1
Substitute the values of , and into the formula .
Step 2.1.1.4.2
Simplify the right side.
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Step 2.1.1.4.2.1
Simplify each term.
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Step 2.1.1.4.2.1.1
Simplify the numerator.
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Step 2.1.1.4.2.1.1.1
Apply the product rule to .
Step 2.1.1.4.2.1.1.2
Raise to the power of .
Step 2.1.1.4.2.1.1.3
Apply the product rule to .
Step 2.1.1.4.2.1.1.4
Raise to the power of .
Step 2.1.1.4.2.1.1.5
Raise to the power of .
Step 2.1.1.4.2.1.1.6
Multiply by .
Step 2.1.1.4.2.1.2
Combine and .
Step 2.1.1.4.2.1.3
Cancel the common factor of and .
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Step 2.1.1.4.2.1.3.1
Factor out of .
Step 2.1.1.4.2.1.3.2
Cancel the common factors.
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Step 2.1.1.4.2.1.3.2.1
Factor out of .
Step 2.1.1.4.2.1.3.2.2
Cancel the common factor.
Step 2.1.1.4.2.1.3.2.3
Rewrite the expression.
Step 2.1.1.4.2.1.4
Multiply the numerator by the reciprocal of the denominator.
Step 2.1.1.4.2.1.5
Cancel the common factor of .
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Step 2.1.1.4.2.1.5.1
Factor out of .
Step 2.1.1.4.2.1.5.2
Cancel the common factor.
Step 2.1.1.4.2.1.5.3
Rewrite the expression.
Step 2.1.1.4.2.1.6
Cancel the common factor of .
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Step 2.1.1.4.2.1.6.1
Factor out of .
Step 2.1.1.4.2.1.6.2
Cancel the common factor.
Step 2.1.1.4.2.1.6.3
Rewrite the expression.
Step 2.1.1.4.2.2
To write as a fraction with a common denominator, multiply by .
Step 2.1.1.4.2.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 2.1.1.4.2.3.1
Multiply by .
Step 2.1.1.4.2.3.2
Multiply by .
Step 2.1.1.4.2.4
Combine the numerators over the common denominator.
Step 2.1.1.4.2.5
Simplify the numerator.
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Step 2.1.1.4.2.5.1
Multiply by .
Step 2.1.1.4.2.5.2
Subtract from .
Step 2.1.1.4.2.6
Cancel the common factor of and .
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Step 2.1.1.4.2.6.1
Factor out of .
Step 2.1.1.4.2.6.2
Cancel the common factors.
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Step 2.1.1.4.2.6.2.1
Factor out of .
Step 2.1.1.4.2.6.2.2
Cancel the common factor.
Step 2.1.1.4.2.6.2.3
Rewrite the expression.
Step 2.1.1.5
Substitute the values of , , and into the vertex form .
Step 2.1.2
Set equal to the new right side.
Step 2.2
Use the vertex form, , to determine the values of , , and .
Step 2.3
Since the value of is positive, the parabola opens up.
Opens Up
Step 2.4
Find the vertex .
Step 2.5
Find , the distance from the vertex to the focus.
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Step 2.5.1
Find the distance from the vertex to a focus of the parabola by using the following formula.
Step 2.5.2
Substitute the value of into the formula.
Step 2.5.3
Simplify.
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Step 2.5.3.1
Combine and .
Step 2.5.3.2
Cancel the common factor of and .
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Step 2.5.3.2.1
Factor out of .
Step 2.5.3.2.2
Cancel the common factors.
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Step 2.5.3.2.2.1
Factor out of .
Step 2.5.3.2.2.2
Cancel the common factor.
Step 2.5.3.2.2.3
Rewrite the expression.
Step 2.5.3.3
Multiply the numerator by the reciprocal of the denominator.
Step 2.5.3.4
Multiply by .
Step 2.6
Find the focus.
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Step 2.6.1
The focus of a parabola can be found by adding to the y-coordinate if the parabola opens up or down.
Step 2.6.2
Substitute the known values of , , and into the formula and simplify.
Step 2.7
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
Step 2.8
Find the directrix.
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Step 2.8.1
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.
Step 2.8.2
Substitute the known values of and into the formula and simplify.
Step 2.9
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:
Step 3
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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Step 3.1
Replace the variable with in the expression.
Step 3.2
Simplify the result.
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Step 3.2.1
Combine fractions.
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Step 3.2.1.1
Combine the numerators over the common denominator.
Step 3.2.1.2
Simplify the expression.
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Step 3.2.1.2.1
One to any power is one.
Step 3.2.1.2.2
Add and .
Step 3.2.1.2.3
Multiply by .
Step 3.2.2
Simplify each term.
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Step 3.2.2.1
Cancel the common factor of and .
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Step 3.2.2.1.1
Factor out of .
Step 3.2.2.1.2
Cancel the common factors.
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Step 3.2.2.1.2.1
Factor out of .
Step 3.2.2.1.2.2
Cancel the common factor.
Step 3.2.2.1.2.3
Rewrite the expression.
Step 3.2.2.2
Move the negative in front of the fraction.
Step 3.2.3
Combine fractions.
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Step 3.2.3.1
Combine the numerators over the common denominator.
Step 3.2.3.2
Subtract from .
Step 3.2.4
The final answer is .
Step 3.3
The value at is .
Step 3.4
Replace the variable with in the expression.
Step 3.5
Simplify the result.
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Step 3.5.1
Combine the numerators over the common denominator.
Step 3.5.2
Simplify the expression.
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Step 3.5.2.1
Raising to any positive power yields .
Step 3.5.2.2
Add and .
Step 3.5.2.3
Multiply by .
Step 3.5.2.4
Divide by .
Step 3.5.2.5
Add and .
Step 3.5.3
The final answer is .
Step 3.6
The value at is .
Step 3.7
Replace the variable with in the expression.
Step 3.8
Simplify the result.
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Step 3.8.1
Combine fractions.
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Step 3.8.1.1
Combine the numerators over the common denominator.
Step 3.8.1.2
Simplify the expression.
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Step 3.8.1.2.1
Raise to the power of .
Step 3.8.1.2.2
Add and .
Step 3.8.1.2.3
Multiply by .
Step 3.8.2
Simplify each term.
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Step 3.8.2.1
Cancel the common factor of and .
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Step 3.8.2.1.1
Factor out of .
Step 3.8.2.1.2
Cancel the common factors.
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Step 3.8.2.1.2.1
Factor out of .
Step 3.8.2.1.2.2
Cancel the common factor.
Step 3.8.2.1.2.3
Rewrite the expression.
Step 3.8.2.2
Move the negative in front of the fraction.
Step 3.8.3
Combine fractions.
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Step 3.8.3.1
Combine the numerators over the common denominator.
Step 3.8.3.2
Simplify the expression.
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Step 3.8.3.2.1
Subtract from .
Step 3.8.3.2.2
Divide by .
Step 3.8.4
The final answer is .
Step 3.9
The value at is .
Step 3.10
Replace the variable with in the expression.
Step 3.11
Simplify the result.
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Step 3.11.1
Combine fractions.
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Step 3.11.1.1
Combine the numerators over the common denominator.
Step 3.11.1.2
Simplify the expression.
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Step 3.11.1.2.1
Raise to the power of .
Step 3.11.1.2.2
Add and .
Step 3.11.1.2.3
Multiply by .
Step 3.11.2
Simplify each term.
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Step 3.11.2.1
Cancel the common factor of and .
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Step 3.11.2.1.1
Factor out of .
Step 3.11.2.1.2
Cancel the common factors.
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Step 3.11.2.1.2.1
Factor out of .
Step 3.11.2.1.2.2
Cancel the common factor.
Step 3.11.2.1.2.3
Rewrite the expression.
Step 3.11.2.2
Divide by .
Step 3.11.3
To write as a fraction with a common denominator, multiply by .
Step 3.11.4
Combine and .
Step 3.11.5
Combine the numerators over the common denominator.
Step 3.11.6
Simplify the numerator.
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Step 3.11.6.1
Multiply by .
Step 3.11.6.2
Subtract from .
Step 3.11.7
The final answer is .
Step 3.12
The value at is .
Step 3.13
Graph the parabola using its properties and the selected points.
Step 4
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex:
Focus:
Axis of Symmetry:
Directrix:
Step 5