Pre-Algebra Examples

Find the Function Rule table[[x,y],[1,5],[2,20],[3,45]]
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 1.3.3
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 1.3.5
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 2.3.3
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 2.3.3.2.4
Add and .
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
Cancel the common factor of and .
Tap for more steps...
Step 2.3.3.3.3.1.1
Factor out of .
Step 2.3.3.3.3.1.2
Cancel the common factors.
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 factor.
Step 2.3.3.3.3.1.2.3
Rewrite the expression.
Step 2.3.3.3.3.2
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
Multiply .
Tap for more steps...
Step 2.3.4.2.1.1.1
Multiply by .
Step 2.3.4.2.1.1.2
Combine and .
Step 2.3.4.2.1.1.3
Multiply by .
Step 2.3.4.2.1.2
To write as a fraction with a common denominator, multiply by .
Step 2.3.4.2.1.3
Simplify terms.
Tap for more steps...
Step 2.3.4.2.1.3.1
Combine and .
Step 2.3.4.2.1.3.2
Combine the numerators over the common denominator.
Step 2.3.4.2.1.4
Simplify each term.
Tap for more steps...
Step 2.3.4.2.1.4.1
Simplify the numerator.
Tap for more steps...
Step 2.3.4.2.1.4.1.1
Factor out of .
Tap for more steps...
Step 2.3.4.2.1.4.1.1.1
Factor out of .
Step 2.3.4.2.1.4.1.1.2
Factor out of .
Step 2.3.4.2.1.4.1.1.3
Factor out of .
Step 2.3.4.2.1.4.1.2
Multiply by .
Step 2.3.4.2.1.4.1.3
Subtract from .
Step 2.3.4.2.1.4.2
Move to the left of .
Step 2.3.4.2.1.4.3
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
Multiply .
Tap for more steps...
Step 2.3.4.4.1.1.1
Multiply by .
Step 2.3.4.4.1.1.2
Multiply by .
Step 2.3.4.4.1.2
To write as a fraction with a common denominator, multiply by .
Step 2.3.4.4.1.3
Simplify terms.
Tap for more steps...
Step 2.3.4.4.1.3.1
Combine and .
Step 2.3.4.4.1.3.2
Combine the numerators over the common denominator.
Step 2.3.4.4.1.4
Simplify each term.
Tap for more steps...
Step 2.3.4.4.1.4.1
Simplify the numerator.
Tap for more steps...
Step 2.3.4.4.1.4.1.1
Factor out of .
Tap for more steps...
Step 2.3.4.4.1.4.1.1.1
Factor out of .
Step 2.3.4.4.1.4.1.1.2
Factor out of .
Step 2.3.4.4.1.4.1.1.3
Factor out of .
Step 2.3.4.4.1.4.1.2
Multiply by .
Step 2.3.4.4.1.4.1.3
Subtract from .
Step 2.3.4.4.1.4.2
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
Set the numerator equal to zero.
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
Divide by .
Step 2.3.6.2.1.2
Add and .
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
Cancel the common factor of and .
Tap for more steps...
Step 2.3.6.4.1.1.1
Factor out of .
Step 2.3.6.4.1.1.2
Cancel the common factors.
Tap for more steps...
Step 2.3.6.4.1.1.2.1
Factor out of .
Step 2.3.6.4.1.1.2.2
Cancel the common factor.
Step 2.3.6.4.1.1.2.3
Rewrite the expression.
Step 2.3.6.4.1.1.2.4
Divide by .
Step 2.3.6.4.1.2
Multiply .
Tap for more steps...
Step 2.3.6.4.1.2.1
Multiply by .
Step 2.3.6.4.1.2.2
Multiply by .
Step 2.3.7
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
One to any power is one.
Step 2.4.1.1.2
Multiply by .
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
Raise to the power of .
Step 2.4.3.1.2
Multiply by .
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
Raise to the power of .
Step 2.4.5.1.2
Multiply by .
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
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 .