Algebra Examples

Find the Function Rule
Step 1
Check if the function rule is linear.
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To find if the table follows a function rule, check to see if the values follow the linear form .
Build a set of equations from the table such that .
Calculate the values of and .
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Solve for in .
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Rewrite the equation as .
Subtract from both sides of the equation.
Replace all occurrences of with in each equation.
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Replace all occurrences of in with .
Simplify the right side.
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Simplify .
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Simplify each term.
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Apply the distributive property.
Multiply by .
Multiply by .
Add and .
Replace all occurrences of in with .
Simplify the right side.
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Simplify .
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Simplify each term.
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Apply the distributive property.
Multiply by .
Multiply by .
Add and .
Replace all occurrences of in with .
Simplify the right side.
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Simplify .
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Simplify each term.
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Apply the distributive property.
Multiply by .
Multiply by .
Add and .
Solve for in .
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Rewrite the equation as .
Move all terms not containing to the right side of the equation.
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Subtract from both sides of the equation.
Subtract from .
Divide each term in by and simplify.
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Divide each term in by .
Simplify the left side.
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Cancel the common factor of .
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Cancel the common factor.
Divide by .
Simplify the right side.
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Divide by .
Replace all occurrences of with in each equation.
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Replace all occurrences of in with .
Simplify the right side.
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Simplify .
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Multiply by .
Add and .
Replace all occurrences of in with .
Simplify the right side.
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Simplify .
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Multiply by .
Add and .
Replace all occurrences of in with .
Simplify the right side.
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Simplify .
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Multiply by .
Add and .
Since is not true, there is no solution.
No solution
No solution
Since for the corresponding values, the function is not linear.
The function is not linear
The function is not linear
Step 2
Check if the function rule is quadratic.
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To find if the table follows a function rule, check whether the function rule could follow the form .
Build a set of equations from the table such that .
Calculate the values of , , and .
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Solve for in .
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Rewrite the equation as .
Move all terms not containing to the right side of the equation.
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Subtract from both sides of the equation.
Subtract from both sides of the equation.
Replace all occurrences of with in each equation.
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Replace all occurrences of in with .
Simplify the right side.
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Simplify .
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Simplify each term.
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Raise to the power of .
Apply the distributive property.
Simplify.
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Multiply by .
Multiply by .
Multiply by .
Move to the left of .
Simplify by adding terms.
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Add and .
Add and .
Replace all occurrences of in with .
Simplify the right side.
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Simplify .
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Simplify each term.
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Raise to the power of .
Apply the distributive property.
Simplify.
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Multiply by .
Multiply by .
Multiply by .
Move to the left of .
Simplify by adding terms.
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Add and .
Add and .
Replace all occurrences of in with .
Simplify the right side.
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Simplify .
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Simplify each term.
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Raise to the power of .
Apply the distributive property.
Simplify.
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Multiply by .
Multiply by .
Multiply by .
Move to the left of .
Simplify by adding terms.
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Add and .
Add and .
Solve for in .
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Rewrite the equation as .
Move all terms not containing to the right side of the equation.
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Subtract from both sides of the equation.
Add to both sides of the equation.
Subtract from .
Add and .
Divide each term in by and simplify.
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Divide each term in by .
Simplify the left side.
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Cancel the common factor of .
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Cancel the common factor.
Divide by .
Simplify the right side.
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Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Move the negative in front of the fraction.
Replace all occurrences of with in each equation.
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Replace all occurrences of in with .
Simplify the right side.
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Simplify .
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Simplify each term.
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Cancel the common factor of .
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Move the leading negative in into the numerator.
Factor out of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Combine and .
Multiply by .
To write as a fraction with a common denominator, multiply by .
Simplify terms.
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Combine and .
Combine the numerators over the common denominator.
Simplify each term.
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Simplify the numerator.
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Factor out of .
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Factor out of .
Factor out of .
Factor out of .
Multiply by .
Subtract from .
Move to the left of .
Move the negative in front of the fraction.
Replace all occurrences of in with .
Simplify the right side.
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Simplify .
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Simplify each term.
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Cancel the common factor of .
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Move the leading negative in into the numerator.
Factor out of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Move the negative in front of the fraction.
Multiply .
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Multiply by .
Multiply by .
To write as a fraction with a common denominator, multiply by .
Simplify terms.
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Combine and .
Combine the numerators over the common denominator.
Simplify each term.
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Simplify the numerator.
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Factor out of .
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Factor out of .
Factor out of .
Factor out of .
Multiply by .
Subtract from .
Move to the left of .
Move the negative in front of the fraction.
Replace all occurrences of in with .
Simplify the right side.
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Simplify .
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Multiply .
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Multiply by .
Multiply by .
To write as a fraction with a common denominator, multiply by .
Simplify terms.
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Combine and .
Combine the numerators over the common denominator.
Simplify each term.
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Simplify the numerator.
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Factor out of .
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Factor out of .
Factor out of .
Factor out of .
Multiply by .
Subtract from .
Multiply by .
Solve for in .
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Rewrite the equation as .
Move all terms not containing to the right side of the equation.
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Subtract from both sides of the equation.
Subtract from .
Set the numerator equal to zero.
Replace all occurrences of with in each equation.
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Replace all occurrences of in with .
Simplify the right side.
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Simplify .
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Divide by .
Add and .
Replace all occurrences of in with .
Simplify the right side.
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Simplify .
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Simplify each term.
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Divide by .
Multiply by .
Add and .
Replace all occurrences of in with .
Simplify the right side.
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Simplify .
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Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply by zero.
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Multiply by .
Multiply by .
Remove any equations from the system that are always true.
List all of the solutions.
Calculate the value of using each value in the table and compare this value to the given value in the table.
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Calculate the value of such that when , , , and .
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Simplify each term.
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Multiply by .
One to any power is one.
Multiply by .
Simplify by adding numbers.
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Add and .
Add and .
If the table has a quadratic function rule, for the corresponding value, . This check passes since and .
Calculate the value of such that when , , , and .
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Simplify each term.
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Multiply by .
Raise to the power of .
Multiply by .
Simplify by adding numbers.
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Add and .
Add and .
If the table has a quadratic function rule, for the corresponding value, . This check passes since and .
Calculate the value of such that when , , , and .
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Simplify each term.
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Multiply by .
Raise to the power of .
Multiply by .
Simplify by adding numbers.
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Add and .
Add and .
If the table has a quadratic function rule, for the corresponding value, . This check passes since and .
Calculate the value of such that when , , , and .
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Simplify each term.
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Multiply by .
Raise to the power of .
Multiply by .
Simplify by adding numbers.
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Add and .
Add and .
If the table has a quadratic function rule, for the corresponding value, . This check passes since and .
Since for the corresponding values, the function is quadratic.
The function is quadratic
The function is quadratic
The function is quadratic
Step 3
Since all , the function is quadratic and follows the form .
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