Pre-Algebra Examples

Graph p(x)=(k-1)x^2-(5-2k)x+4k+5
Step 1
Move all terms containing variables to the left side of the equation.
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Step 1.1
Subtract from both sides of the equation.
Step 1.2
Add to both sides of the equation.
Step 1.3
Subtract from both sides of the equation.
Step 1.4
Simplify each term.
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Step 1.4.1
Apply the distributive property.
Step 1.4.2
Multiply by .
Step 1.4.3
Apply the distributive property.
Step 1.4.4
Multiply by .
Step 1.4.5
Apply the distributive property.
Step 1.5
Move .
Step 1.6
Move .
Step 1.7
Move .
Step 2
This is the form of a hyperbola. Use this form to determine the values used to find vertices and asymptotes of the hyperbola.
Step 3
Match the values in this hyperbola to those of the standard form. The variable represents the x-offset from the origin, represents the y-offset from origin, .
Step 4
The center of a hyperbola follows the form of . Substitute in the values of and .
Step 5
Find , the distance from the center to a focus.
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Step 5.1
Find the distance from the center to a focus of the hyperbola by using the following formula.
Step 5.2
Substitute the values of and in the formula.
Step 5.3
Simplify.
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Step 5.3.1
One to any power is one.
Step 5.3.2
One to any power is one.
Step 5.3.3
Add and .
Step 6
Find the vertices.
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Step 6.1
The first vertex of a hyperbola can be found by adding to .
Step 6.2
Substitute the known values of , , and into the formula and simplify.
Step 6.3
The second vertex of a hyperbola can be found by subtracting from .
Step 6.4
Substitute the known values of , , and into the formula and simplify.
Step 6.5
The vertices of a hyperbola follow the form of . Hyperbolas have two vertices.
Step 7
Find the foci.
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Step 7.1
The first focus of a hyperbola can be found by adding to .
Step 7.2
Substitute the known values of , , and into the formula and simplify.
Step 7.3
The second focus of a hyperbola can be found by subtracting from .
Step 7.4
Substitute the known values of , , and into the formula and simplify.
Step 7.5
The foci of a hyperbola follow the form of . Hyperbolas have two foci.
Step 8
Find the focal parameter.
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Step 8.1
Find the value of the focal parameter of the hyperbola by using the following formula.
Step 8.2
Substitute the values of and in the formula.
Step 8.3
Simplify.
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Step 8.3.1
One to any power is one.
Step 8.3.2
Multiply by .
Step 8.3.3
Combine and simplify the denominator.
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Step 8.3.3.1
Multiply by .
Step 8.3.3.2
Raise to the power of .
Step 8.3.3.3
Raise to the power of .
Step 8.3.3.4
Use the power rule to combine exponents.
Step 8.3.3.5
Add and .
Step 8.3.3.6
Rewrite as .
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Step 8.3.3.6.1
Use to rewrite as .
Step 8.3.3.6.2
Apply the power rule and multiply exponents, .
Step 8.3.3.6.3
Combine and .
Step 8.3.3.6.4
Cancel the common factor of .
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Step 8.3.3.6.4.1
Cancel the common factor.
Step 8.3.3.6.4.2
Rewrite the expression.
Step 8.3.3.6.5
Evaluate the exponent.
Step 9
The asymptotes follow the form because this hyperbola opens left and right.
Step 10
Simplify .
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Step 10.1
Add and .
Step 10.2
Multiply by .
Step 11
Simplify .
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Step 11.1
Add and .
Step 11.2
Rewrite as .
Step 12
This hyperbola has two asymptotes.
Step 13
These values represent the important values for graphing and analyzing a hyperbola.
Center:
Vertices:
Foci:
Eccentricity:
Focal Parameter:
Asymptotes: ,
Step 14