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 .

Simplify each equation.

Move to the left of .

Move to the left of .

Move to the left of .

Solve for in the first equation.

Rewrite the equation as .

Subtract from both sides of the equation.

Replace all occurrences of with in each equation.

Replace all occurrences of in with .

Replace all occurrences of in with .

Simplify each equation.

Remove parentheses.

Subtract from .

Remove parentheses.

Subtract from .

Solve for in the second equation.

Rewrite the equation as .

Move all terms not containing to the right side of the equation.

Subtract from both sides of the equation.

Subtract from .

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Divide by .

Replace all occurrences of with in each equation.

Replace all occurrences of in with .

Replace all occurrences of in with .

Simplify.

Simplify .

Multiply by .

Subtract from .

Simplify .

Multiply by .

Add and .

Since , there are no solutions.

No solution

No solution

Calculate the value of using each value in the relation and compare this value to the given value in the relation.

Calculate the value of when , , and .

Multiply by .

Subtract from .

If the table has a linear function rule, for the corresponding value, . This check passes since and .

Calculate the value of when , , and .

Multiply by .

Subtract from .

If the table has a linear function rule, for the corresponding value, . This check passes since and .

Calculate the value of when , , and .

Multiply by .

Subtract from .

If the table has a linear function rule, for the corresponding value, . This check does not pass, since and . The function rule can’t be linear.

Since for the corresponding values, the function is not linear.

The function is not linear

The function is not linear

The function is not linear

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 .

Simplify each equation.

Simplify each term.

Raise to the power of .

Move to the left of .

Move to the left of .

Simplify each term.

Raise to the power of .

Move to the left of .

Move to the left of .

Simplify each term.

Raise to the power of .

Move to the left of .

Move to the left of .

Solve for in the first equation.

Rewrite the equation as .

Move all terms not containing to the right side of the equation.

Subtract from both sides of the equation.

Subtract from both sides of the equation.

Replace all occurrences of with in each equation.

Replace all occurrences of in with .

Replace all occurrences of in with .

Simplify each equation.

Remove parentheses.

Simplify .

Subtract from .

Subtract from .

Remove parentheses.

Simplify .

Subtract from .

Subtract from .

Solve for in the second equation.

Rewrite the equation as .

Move all terms not containing to the right side of the equation.

Subtract from both sides of the equation.

Subtract from both sides of the equation.

Subtract from .

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Simplify each term.

Cancel the common factor of and .

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Move the negative in front of the fraction.

Divide by .

Replace all occurrences of with in each equation.

Replace all occurrences of in with .

Replace all occurrences of in with .

Simplify each equation.

Simplify .

Simplify each term.

Apply the distributive property.

Cancel the common factor of .

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 .

Multiply by .

Subtract from .

To write as a fraction with a common denominator, multiply by .

Simplify terms.

Combine and .

Combine the numerators over the common denominator.

Simplify each term.

Simplify the numerator.

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Multiply by .

Subtract from .

Multiply by .

Move the negative in front of the fraction.

Simplify .

Simplify each term.

Apply the distributive property.

Cancel the common factor of .

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 .

Multiply by .

Move the negative in front of the fraction.

To write as a fraction with a common denominator, multiply by .

Combine and .

Combine the numerators over the common denominator.

Find the common denominator.

Write as a fraction with denominator .

Multiply by .

Multiply and .

Write as a fraction with denominator .

Multiply by .

Multiply and .

Combine fractions with similar denominators.

Simplify the expression.

Multiply by .

Multiply by .

Add and .

Multiply by .

Add and .

Factor out of .

Rewrite as .

Factor out of .

Simplify the expression.

Rewrite as .

Move the negative in front of the fraction.

Solve for in the third equation.

Rewrite the equation as .

Multiply both sides of the equation by .

Simplify both sides of the equation.

Simplify .

Cancel the common factor of .

Move the leading negative in into the numerator.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Multiply.

Multiply by .

Multiply by .

Multiply by .

Move all terms not containing to the right side of the equation.

Add to both sides of the equation.

Add and .

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Divide by .

Replace all occurrences of with in each equation.

Replace all occurrences of in with .

Replace all occurrences of in with .

Simplify.

Simplify .

Simplify each term.

Cancel the common factor of and .

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Divide by .

Multiply by .

Multiply by .

Subtract from .

Simplify .

Simplify each term.

Divide by .

Multiply by .

Add and .

Calculate the value of using each value in the table and compare this value to the given value in the table.

Calculate the value of such that when , , , and .

Simplify each term.

Raise to the power of .

Multiply by .

Multiply by .

Simplify by adding and subtracting.

Subtract from .

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 .

Simplify each term.

Raise to the power of .

Multiply by .

Multiply by .

Simplify by adding and subtracting.

Subtract from .

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 .

Simplify each term.

Raise to the power of .

Multiply by .

Multiply by .

Simplify by adding and subtracting.

Subtract from .

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

Since all , the function is quadratic and follows the form .

Find the Function Rule table[[x,y],[2,9],[4,81],[6,729]]