Examples

Find the Function Rule
Step 1
Check if the function rule is linear.
Tap for more steps...
Step 1.1
To find if the table follows a function rule, check to see if the values follow the linear form .
Step 1.2
Build a set of equations from the table such that .
Step 1.3
Calculate the values of and .
Tap for more steps...
Step 1.3.1
Solve for in .
Tap for more steps...
Step 1.3.1.1
Rewrite the equation as .
Step 1.3.1.2
Subtract from both sides of the equation.
Step 1.3.2
Replace all occurrences of with in each equation.
Tap for more steps...
Step 1.3.2.1
Replace all occurrences of in with .
Step 1.3.2.2
Simplify the right side.
Tap for more steps...
Step 1.3.2.2.1
Simplify .
Tap for more steps...
Step 1.3.2.2.1.1
Simplify each term.
Tap for more steps...
Step 1.3.2.2.1.1.1
Apply the distributive property.
Step 1.3.2.2.1.1.2
Multiply by .
Step 1.3.2.2.1.1.3
Multiply by .
Step 1.3.2.2.1.2
Add and .
Step 1.3.2.3
Replace all occurrences of in with .
Step 1.3.2.4
Simplify the right side.
Tap for more steps...
Step 1.3.2.4.1
Simplify .
Tap for more steps...
Step 1.3.2.4.1.1
Simplify each term.
Tap for more steps...
Step 1.3.2.4.1.1.1
Apply the distributive property.
Step 1.3.2.4.1.1.2
Multiply by .
Step 1.3.2.4.1.1.3
Multiply by .
Step 1.3.2.4.1.2
Add and .
Step 1.3.2.5
Replace all occurrences of in with .
Step 1.3.2.6
Simplify the right side.
Tap for more steps...
Step 1.3.2.6.1
Simplify .
Tap for more steps...
Step 1.3.2.6.1.1
Simplify each term.
Tap for more steps...
Step 1.3.2.6.1.1.1
Apply the distributive property.
Step 1.3.2.6.1.1.2
Multiply by .
Step 1.3.2.6.1.1.3
Multiply by .
Step 1.3.2.6.1.2
Add and .
Step 1.3.3
Solve for in .
Tap for more steps...
Step 1.3.3.1
Rewrite the equation as .
Step 1.3.3.2
Move all terms not containing to the right side of the equation.
Tap for more steps...
Step 1.3.3.2.1
Subtract from both sides of the equation.
Step 1.3.3.2.2
Subtract from .
Step 1.3.3.3
Divide each term in by and simplify.
Tap for more steps...
Step 1.3.3.3.1
Divide each term in by .
Step 1.3.3.3.2
Simplify the left side.
Tap for more steps...
Step 1.3.3.3.2.1
Cancel the common factor of .
Tap for more steps...
Step 1.3.3.3.2.1.1
Cancel the common factor.
Step 1.3.3.3.2.1.2
Divide by .
Step 1.3.3.3.3
Simplify the right side.
Tap for more steps...
Step 1.3.3.3.3.1
Divide by .
Step 1.3.4
Replace all occurrences of with in each equation.
Tap for more steps...
Step 1.3.4.1
Replace all occurrences of in with .
Step 1.3.4.2
Simplify the right side.
Tap for more steps...
Step 1.3.4.2.1
Simplify .
Tap for more steps...
Step 1.3.4.2.1.1
Multiply by .
Step 1.3.4.2.1.2
Add and .
Step 1.3.4.3
Replace all occurrences of in with .
Step 1.3.4.4
Simplify the right side.
Tap for more steps...
Step 1.3.4.4.1
Simplify .
Tap for more steps...
Step 1.3.4.4.1.1
Multiply by .
Step 1.3.4.4.1.2
Add and .
Step 1.3.4.5
Replace all occurrences of in with .
Step 1.3.4.6
Simplify the right side.
Tap for more steps...
Step 1.3.4.6.1
Simplify .
Tap for more steps...
Step 1.3.4.6.1.1
Multiply by .
Step 1.3.4.6.1.2
Add and .
Step 1.3.5
Since is not true, there is no solution.
No solution
No solution
Step 1.4
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.
Tap for more steps...
Step 2.1
To find if the table follows a function rule, check whether the function rule could follow the form .
Step 2.2
Build a set of equations from the table such that .
Step 2.3
Calculate the values of , , and .
Tap for more steps...
Step 2.3.1
Solve for in .
Tap for more steps...
Step 2.3.1.1
Rewrite the equation as .
Step 2.3.1.2
Move all terms not containing to the right side of the equation.
Tap for more steps...
Step 2.3.1.2.1
Subtract from both sides of the equation.
Step 2.3.1.2.2
Subtract from both sides of the equation.
Step 2.3.2
Replace all occurrences of with in each equation.
Tap for more steps...
Step 2.3.2.1
Replace all occurrences of in with .
Step 2.3.2.2
Simplify the right side.
Tap for more steps...
Step 2.3.2.2.1
Simplify .
Tap for more steps...
Step 2.3.2.2.1.1
Simplify each term.
Tap for more steps...
Step 2.3.2.2.1.1.1
Raise to the power of .
Step 2.3.2.2.1.1.2
Apply the distributive property.
Step 2.3.2.2.1.1.3
Simplify.
Tap for more steps...
Step 2.3.2.2.1.1.3.1
Multiply by .
Step 2.3.2.2.1.1.3.2
Multiply by .
Step 2.3.2.2.1.1.3.3
Multiply by .
Step 2.3.2.2.1.1.4
Move to the left of .
Step 2.3.2.2.1.2
Simplify by adding terms.
Tap for more steps...
Step 2.3.2.2.1.2.1
Add and .
Step 2.3.2.2.1.2.2
Add and .
Step 2.3.2.3
Replace all occurrences of in with .
Step 2.3.2.4
Simplify the right side.
Tap for more steps...
Step 2.3.2.4.1
Simplify .
Tap for more steps...
Step 2.3.2.4.1.1
Simplify each term.
Tap for more steps...
Step 2.3.2.4.1.1.1
Raise to the power of .
Step 2.3.2.4.1.1.2
Apply the distributive property.
Step 2.3.2.4.1.1.3
Simplify.
Tap for more steps...
Step 2.3.2.4.1.1.3.1
Multiply by .
Step 2.3.2.4.1.1.3.2
Multiply by .
Step 2.3.2.4.1.1.3.3
Multiply by .
Step 2.3.2.4.1.1.4
Move to the left of .
Step 2.3.2.4.1.2
Simplify by adding terms.
Tap for more steps...
Step 2.3.2.4.1.2.1
Add and .
Step 2.3.2.4.1.2.2
Add and .
Step 2.3.2.5
Replace all occurrences of in with .
Step 2.3.2.6
Simplify the right side.
Tap for more steps...
Step 2.3.2.6.1
Simplify .
Tap for more steps...
Step 2.3.2.6.1.1
Simplify each term.
Tap for more steps...
Step 2.3.2.6.1.1.1
Raise to the power of .
Step 2.3.2.6.1.1.2
Apply the distributive property.
Step 2.3.2.6.1.1.3
Simplify.
Tap for more steps...
Step 2.3.2.6.1.1.3.1
Multiply by .
Step 2.3.2.6.1.1.3.2
Multiply by .
Step 2.3.2.6.1.1.3.3
Multiply by .
Step 2.3.2.6.1.1.4
Move to the left of .
Step 2.3.2.6.1.2
Simplify by adding terms.
Tap for more steps...
Step 2.3.2.6.1.2.1
Add and .
Step 2.3.2.6.1.2.2
Add and .
Step 2.3.3
Solve for in .
Tap for more steps...
Step 2.3.3.1
Rewrite the equation as .
Step 2.3.3.2
Move all terms not containing to the right side of the equation.
Tap for more steps...
Step 2.3.3.2.1
Subtract from both sides of the equation.
Step 2.3.3.2.2
Add to both sides of the equation.
Step 2.3.3.2.3
Subtract from .
Step 2.3.3.3
Divide each term in by and simplify.
Tap for more steps...
Step 2.3.3.3.1
Divide each term in by .
Step 2.3.3.3.2
Simplify the left side.
Tap for more steps...
Step 2.3.3.3.2.1
Cancel the common factor of .
Tap for more steps...
Step 2.3.3.3.2.1.1
Cancel the common factor.
Step 2.3.3.3.2.1.2
Divide by .
Step 2.3.3.3.3
Simplify the right side.
Tap for more steps...
Step 2.3.3.3.3.1
Simplify each term.
Tap for more steps...
Step 2.3.3.3.3.1.1
Cancel the common factor of and .
Tap for more steps...
Step 2.3.3.3.3.1.1.1
Factor out of .
Step 2.3.3.3.3.1.1.2
Cancel the common factors.
Tap for more steps...
Step 2.3.3.3.3.1.1.2.1
Factor out of .
Step 2.3.3.3.3.1.1.2.2
Cancel the common factor.
Step 2.3.3.3.3.1.1.2.3
Rewrite the expression.
Step 2.3.3.3.3.1.2
Cancel the common factor of and .
Tap for more steps...
Step 2.3.3.3.3.1.2.1
Factor out of .
Step 2.3.3.3.3.1.2.2
Cancel the common factors.
Tap for more steps...
Step 2.3.3.3.3.1.2.2.1
Factor out of .
Step 2.3.3.3.3.1.2.2.2
Cancel the common factor.
Step 2.3.3.3.3.1.2.2.3
Rewrite the expression.
Step 2.3.3.3.3.1.3
Move the negative in front of the fraction.
Step 2.3.4
Replace all occurrences of with in each equation.
Tap for more steps...
Step 2.3.4.1
Replace all occurrences of in with .
Step 2.3.4.2
Simplify the right side.
Tap for more steps...
Step 2.3.4.2.1
Simplify .
Tap for more steps...
Step 2.3.4.2.1.1
Simplify each term.
Tap for more steps...
Step 2.3.4.2.1.1.1
Apply the distributive property.
Step 2.3.4.2.1.1.2
Cancel the common factor of .
Tap for more steps...
Step 2.3.4.2.1.1.2.1
Factor out of .
Step 2.3.4.2.1.1.2.2
Cancel the common factor.
Step 2.3.4.2.1.1.2.3
Rewrite the expression.
Step 2.3.4.2.1.1.3
Multiply by .
Step 2.3.4.2.1.1.4
Cancel the common factor of .
Tap for more steps...
Step 2.3.4.2.1.1.4.1
Move the leading negative in into the numerator.
Step 2.3.4.2.1.1.4.2
Factor out of .
Step 2.3.4.2.1.1.4.3
Factor out of .
Step 2.3.4.2.1.1.4.4
Cancel the common factor.
Step 2.3.4.2.1.1.4.5
Rewrite the expression.
Step 2.3.4.2.1.1.5
Combine and .
Step 2.3.4.2.1.1.6
Multiply by .
Step 2.3.4.2.1.2
Subtract from .
Step 2.3.4.2.1.3
To write as a fraction with a common denominator, multiply by .
Step 2.3.4.2.1.4
Simplify terms.
Tap for more steps...
Step 2.3.4.2.1.4.1
Combine and .
Step 2.3.4.2.1.4.2
Combine the numerators over the common denominator.
Step 2.3.4.2.1.5
Simplify each term.
Tap for more steps...
Step 2.3.4.2.1.5.1
Simplify the numerator.
Tap for more steps...
Step 2.3.4.2.1.5.1.1
Factor out of .
Tap for more steps...
Step 2.3.4.2.1.5.1.1.1
Factor out of .
Step 2.3.4.2.1.5.1.1.2
Factor out of .
Step 2.3.4.2.1.5.1.1.3
Factor out of .
Step 2.3.4.2.1.5.1.2
Multiply by .
Step 2.3.4.2.1.5.1.3
Subtract from .
Step 2.3.4.2.1.5.2
Move to the left of .
Step 2.3.4.2.1.5.3
Move the negative in front of the fraction.
Step 2.3.4.3
Replace all occurrences of in with .
Step 2.3.4.4
Simplify the right side.
Tap for more steps...
Step 2.3.4.4.1
Simplify .
Tap for more steps...
Step 2.3.4.4.1.1
Simplify each term.
Tap for more steps...
Step 2.3.4.4.1.1.1
Apply the distributive property.
Step 2.3.4.4.1.1.2
Cancel the common factor of .
Tap for more steps...
Step 2.3.4.4.1.1.2.1
Factor out of .
Step 2.3.4.4.1.1.2.2
Cancel the common factor.
Step 2.3.4.4.1.1.2.3
Rewrite the expression.
Step 2.3.4.4.1.1.3
Multiply by .
Step 2.3.4.4.1.1.4
Cancel the common factor of .
Tap for more steps...
Step 2.3.4.4.1.1.4.1
Move the leading negative in into the numerator.
Step 2.3.4.4.1.1.4.2
Factor out of .
Step 2.3.4.4.1.1.4.3
Factor out of .
Step 2.3.4.4.1.1.4.4
Cancel the common factor.
Step 2.3.4.4.1.1.4.5
Rewrite the expression.
Step 2.3.4.4.1.1.5
Simplify each term.
Tap for more steps...
Step 2.3.4.4.1.1.5.1
Move the negative in front of the fraction.
Step 2.3.4.4.1.1.5.2
Multiply .
Tap for more steps...
Step 2.3.4.4.1.1.5.2.1
Multiply by .
Step 2.3.4.4.1.1.5.2.2
Multiply by .
Step 2.3.4.4.1.2
Subtract from .
Step 2.3.4.4.1.3
To write as a fraction with a common denominator, multiply by .
Step 2.3.4.4.1.4
Simplify terms.
Tap for more steps...
Step 2.3.4.4.1.4.1
Combine and .
Step 2.3.4.4.1.4.2
Combine the numerators over the common denominator.
Step 2.3.4.4.1.5
Simplify each term.
Tap for more steps...
Step 2.3.4.4.1.5.1
Simplify the numerator.
Tap for more steps...
Step 2.3.4.4.1.5.1.1
Factor out of .
Tap for more steps...
Step 2.3.4.4.1.5.1.1.1
Factor out of .
Step 2.3.4.4.1.5.1.1.2
Factor out of .
Step 2.3.4.4.1.5.1.1.3
Factor out of .
Step 2.3.4.4.1.5.1.2
Multiply by .
Step 2.3.4.4.1.5.1.3
Subtract from .
Step 2.3.4.4.1.5.2
Move to the left of .
Step 2.3.4.4.1.5.3
Move the negative in front of the fraction.
Step 2.3.4.5
Replace all occurrences of in with .
Step 2.3.4.6
Simplify the right side.
Tap for more steps...
Step 2.3.4.6.1
Simplify .
Tap for more steps...
Step 2.3.4.6.1.1
Simplify each term.
Tap for more steps...
Step 2.3.4.6.1.1.1
Apply the distributive property.
Step 2.3.4.6.1.1.2
Multiply .
Tap for more steps...
Step 2.3.4.6.1.1.2.1
Multiply by .
Step 2.3.4.6.1.1.2.2
Multiply by .
Step 2.3.4.6.1.2
To write as a fraction with a common denominator, multiply by .
Step 2.3.4.6.1.3
Combine and .
Step 2.3.4.6.1.4
Combine the numerators over the common denominator.
Step 2.3.4.6.1.5
Simplify the numerator.
Tap for more steps...
Step 2.3.4.6.1.5.1
Multiply by .
Step 2.3.4.6.1.5.2
Subtract from .
Step 2.3.4.6.1.6
To write as a fraction with a common denominator, multiply by .
Step 2.3.4.6.1.7
Simplify terms.
Tap for more steps...
Step 2.3.4.6.1.7.1
Combine and .
Step 2.3.4.6.1.7.2
Combine the numerators over the common denominator.
Step 2.3.4.6.1.8
Simplify each term.
Tap for more steps...
Step 2.3.4.6.1.8.1
Simplify the numerator.
Tap for more steps...
Step 2.3.4.6.1.8.1.1
Factor out of .
Tap for more steps...
Step 2.3.4.6.1.8.1.1.1
Factor out of .
Step 2.3.4.6.1.8.1.1.2
Factor out of .
Step 2.3.4.6.1.8.1.1.3
Factor out of .
Step 2.3.4.6.1.8.1.2
Multiply by .
Step 2.3.4.6.1.8.1.3
Subtract from .
Step 2.3.4.6.1.8.2
Multiply by .
Step 2.3.5
Solve for in .
Tap for more steps...
Step 2.3.5.1
Rewrite the equation as .
Step 2.3.5.2
Move all terms not containing to the right side of the equation.
Tap for more steps...
Step 2.3.5.2.1
Subtract from both sides of the equation.
Step 2.3.5.2.2
Subtract from .
Step 2.3.5.3
Multiply both sides of the equation by .
Step 2.3.5.4
Simplify both sides of the equation.
Tap for more steps...
Step 2.3.5.4.1
Simplify the left side.
Tap for more steps...
Step 2.3.5.4.1.1
Simplify .
Tap for more steps...
Step 2.3.5.4.1.1.1
Cancel the common factor of .
Tap for more steps...
Step 2.3.5.4.1.1.1.1
Move the leading negative in into the numerator.
Step 2.3.5.4.1.1.1.2
Factor out of .
Step 2.3.5.4.1.1.1.3
Cancel the common factor.
Step 2.3.5.4.1.1.1.4
Rewrite the expression.
Step 2.3.5.4.1.1.2
Multiply.
Tap for more steps...
Step 2.3.5.4.1.1.2.1
Multiply by .
Step 2.3.5.4.1.1.2.2
Multiply by .
Step 2.3.5.4.2
Simplify the right side.
Tap for more steps...
Step 2.3.5.4.2.1
Multiply by .
Step 2.3.6
Replace all occurrences of with in each equation.
Tap for more steps...
Step 2.3.6.1
Replace all occurrences of in with .
Step 2.3.6.2
Simplify the right side.
Tap for more steps...
Step 2.3.6.2.1
Simplify .
Tap for more steps...
Step 2.3.6.2.1.1
Cancel the common factor of and .
Tap for more steps...
Step 2.3.6.2.1.1.1
Factor out of .
Step 2.3.6.2.1.1.2
Cancel the common factors.
Tap for more steps...
Step 2.3.6.2.1.1.2.1
Factor out of .
Step 2.3.6.2.1.1.2.2
Cancel the common factor.
Step 2.3.6.2.1.1.2.3
Rewrite the expression.
Step 2.3.6.2.1.2
Combine the numerators over the common denominator.
Step 2.3.6.2.1.3
Simplify the expression.
Tap for more steps...
Step 2.3.6.2.1.3.1
Add and .
Step 2.3.6.2.1.3.2
Divide by .
Step 2.3.6.3
Replace all occurrences of in with .
Step 2.3.6.4
Simplify the right side.
Tap for more steps...
Step 2.3.6.4.1
Simplify .
Tap for more steps...
Step 2.3.6.4.1.1
Simplify each term.
Tap for more steps...
Step 2.3.6.4.1.1.1
Cancel the common factor of .
Tap for more steps...
Step 2.3.6.4.1.1.1.1
Cancel the common factor.
Step 2.3.6.4.1.1.1.2
Rewrite the expression.
Step 2.3.6.4.1.1.2
Multiply by .
Step 2.3.6.4.1.2
Subtract from .
Step 2.3.6.5
Replace all occurrences of in with .
Step 2.3.6.6
Simplify the right side.
Tap for more steps...
Step 2.3.6.6.1
Simplify .
Tap for more steps...
Step 2.3.6.6.1.1
Cancel the common factor of and .
Tap for more steps...
Step 2.3.6.6.1.1.1
Factor out of .
Step 2.3.6.6.1.1.2
Cancel the common factors.
Tap for more steps...
Step 2.3.6.6.1.1.2.1
Factor out of .
Step 2.3.6.6.1.1.2.2
Cancel the common factor.
Step 2.3.6.6.1.1.2.3
Rewrite the expression.
Step 2.3.6.6.1.2
Combine the numerators over the common denominator.
Step 2.3.6.6.1.3
Simplify the expression.
Tap for more steps...
Step 2.3.6.6.1.3.1
Subtract from .
Step 2.3.6.6.1.3.2
Divide by .
Step 2.3.7
Remove any equations from the system that are always true.
Step 2.3.8
List all of the solutions.
Step 2.4
Calculate the value of using each value in the table and compare this value to the given value in the table.
Tap for more steps...
Step 2.4.1
Calculate the value of such that when , , , and .
Tap for more steps...
Step 2.4.1.1
Simplify each term.
Tap for more steps...
Step 2.4.1.1.1
Multiply by .
Step 2.4.1.1.2
One to any power is one.
Step 2.4.1.1.3
Multiply by .
Step 2.4.1.2
Simplify by adding numbers.
Tap for more steps...
Step 2.4.1.2.1
Add and .
Step 2.4.1.2.2
Add and .
Step 2.4.2
If the table has a quadratic function rule, for the corresponding value, . This check passes since and .
Step 2.4.3
Calculate the value of such that when , , , and .
Tap for more steps...
Step 2.4.3.1
Simplify each term.
Tap for more steps...
Step 2.4.3.1.1
Multiply by .
Step 2.4.3.1.2
Raise to the power of .
Step 2.4.3.1.3
Multiply by .
Step 2.4.3.2
Simplify by adding numbers.
Tap for more steps...
Step 2.4.3.2.1
Add and .
Step 2.4.3.2.2
Add and .
Step 2.4.4
If the table has a quadratic function rule, for the corresponding value, . This check passes since and .
Step 2.4.5
Calculate the value of such that when , , , and .
Tap for more steps...
Step 2.4.5.1
Simplify each term.
Tap for more steps...
Step 2.4.5.1.1
Multiply by .
Step 2.4.5.1.2
Raise to the power of .
Step 2.4.5.1.3
Multiply by .
Step 2.4.5.2
Simplify by adding numbers.
Tap for more steps...
Step 2.4.5.2.1
Add and .
Step 2.4.5.2.2
Add and .
Step 2.4.6
If the table has a quadratic function rule, for the corresponding value, . This check passes since and .
Step 2.4.7
Calculate the value of such that when , , , and .
Tap for more steps...
Step 2.4.7.1
Simplify each term.
Tap for more steps...
Step 2.4.7.1.1
Multiply by .
Step 2.4.7.1.2
Raise to the power of .
Step 2.4.7.1.3
Multiply by .
Step 2.4.7.2
Simplify by adding numbers.
Tap for more steps...
Step 2.4.7.2.1
Add and .
Step 2.4.7.2.2
Add and .
Step 2.4.8
If the table has a quadratic function rule, for the corresponding value, . This check passes since and .
Step 2.4.9
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 .
Enter YOUR Problem
Mathway requires javascript and a modern browser.