Find the Function Rule table[[x,f(x)],[2,25],[3,125],[4,625]]

$\begin{array}{c} x & f (x) \\ 2 & 25 \\ 3 & 125 \\ 4 & 625 \end{array}$

Step 1

Check if the function rule is linear.

To find if the table follows a function rule, check to see if the values follow the linear form $y = a x + b$ .

$y = a x + b$

Build a set of equations from the table such that $f (x) = a x + b$ .

$25 = a (2) + b 125 = a (3) + b 625 = a (4) + b$

Calculate the values of $a$ and $b$ .

Solve for $b$ in $25 = a (2) + b$ .

Rewrite the equation as $a (2) + b = 25$ .

$a (2) + b = 25$

$125 = a (3) + b$

$625 = a (4) + b$

Move $2$ to the left of $a$ .

$2 a + b = 25$

$125 = a (3) + b$

$625 = a (4) + b$

Subtract $2 a$ from both sides of the equation.

$b = 25 - 2 a$

$125 = a (3) + b$

$625 = a (4) + b$

$b = 25 - 2 a$

$125 = a (3) + b$

$625 = a (4) + b$

Replace all occurrences of $b$ with $25 - 2 a$ in each equation.

Replace all occurrences of $b$ in $125 = a (3) + b$ with $25 - 2 a$ .

$125 = a (3) + 25 - 2 a$

$b = 25 - 2 a$

$625 = a (4) + b$

Simplify $125 = a (3) + 25 - 2 a$ .

Simplify the left side.

Remove parentheses.

$125 = a (3) + 25 - 2 a$

$b = 25 - 2 a$

$625 = a (4) + b$

$125 = a (3) + 25 - 2 a$

$b = 25 - 2 a$

$625 = a (4) + b$

Simplify the right side.

Simplify $a (3) + 25 - 2 a$ .

Move $3$ to the left of $a$ .

$125 = 3 a + 25 - 2 a$

$b = 25 - 2 a$

$625 = a (4) + b$

Subtract $2 a$ from $3 a$ .

$125 = a + 25$

$b = 25 - 2 a$

$625 = a (4) + b$

$125 = a + 25$

$b = 25 - 2 a$

$625 = a (4) + b$

$125 = a + 25$

$b = 25 - 2 a$

$625 = a (4) + b$

$125 = a + 25$

$b = 25 - 2 a$

$625 = a (4) + b$

Replace all occurrences of $b$ in $625 = a (4) + b$ with $25 - 2 a$ .

$625 = a (4) + 25 - 2 a$

$125 = a + 25$

$b = 25 - 2 a$

Simplify $625 = a (4) + 25 - 2 a$ .

Simplify the left side.

Remove parentheses.

$625 = a (4) + 25 - 2 a$

$125 = a + 25$

$b = 25 - 2 a$

$625 = a (4) + 25 - 2 a$

$125 = a + 25$

$b = 25 - 2 a$

Simplify the right side.

Simplify $a (4) + 25 - 2 a$ .

Move $4$ to the left of $a$ .

$625 = 4 a + 25 - 2 a$

$125 = a + 25$

$b = 25 - 2 a$

Subtract $2 a$ from $4 a$ .

$625 = 2 a + 25$

$125 = a + 25$

$b = 25 - 2 a$

$625 = 2 a + 25$

$125 = a + 25$

$b = 25 - 2 a$

$625 = 2 a + 25$

$125 = a + 25$

$b = 25 - 2 a$

$625 = 2 a + 25$

$125 = a + 25$

$b = 25 - 2 a$

$625 = 2 a + 25$

$125 = a + 25$

$b = 25 - 2 a$

Solve for $a$ in $625 = 2 a + 25$ .

Rewrite the equation as $2 a + 25 = 625$ .

$2 a + 25 = 625$

$125 = a + 25$

$b = 25 - 2 a$

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

Subtract $25$ from both sides of the equation.

$2 a = 625 - 25$

$125 = a + 25$

$b = 25 - 2 a$

Subtract $25$ from $625$ .

$2 a = 600$

$125 = a + 25$

$b = 25 - 2 a$

$2 a = 600$

$125 = a + 25$

$b = 25 - 2 a$

Divide each term in $2 a = 600$ by $2$ and simplify.

Divide each term in $2 a = 600$ by $2$ .

$\frac{2 a}{2} = \frac{600}{2}$

$125 = a + 25$

$b = 25 - 2 a$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 a}{2} = \frac{600}{2}$

$125 = a + 25$

$b = 25 - 2 a$

Divide $a$ by $1$ .

$a = \frac{600}{2}$

$125 = a + 25$

$b = 25 - 2 a$

$a = \frac{600}{2}$

$125 = a + 25$

$b = 25 - 2 a$

$a = \frac{600}{2}$

$125 = a + 25$

$b = 25 - 2 a$

Simplify the right side.

Divide $600$ by $2$ .

$a = 300$

$125 = a + 25$

$b = 25 - 2 a$

$a = 300$

$125 = a + 25$

$b = 25 - 2 a$

$a = 300$

$125 = a + 25$

$b = 25 - 2 a$

$a = 300$

$125 = a + 25$

$b = 25 - 2 a$

Replace all occurrences of $a$ with $300$ in each equation.

Replace all occurrences of $a$ in $125 = a + 25$ with $300$ .

$125 = (300) + 25$

$a = 300$

$b = 25 - 2 a$

Simplify the right side.

Add $300$ and $25$ .

$125 = 325$

$a = 300$

$b = 25 - 2 a$

$125 = 325$

$a = 300$

$b = 25 - 2 a$

Replace all occurrences of $a$ in $b = 25 - 2 a$ with $300$ .

$b = 25 - 2 \cdot 300$

$125 = 325$

$a = 300$

Simplify the right side.

Simplify $25 - 2 \cdot 300$ .

Multiply $- 2$ by $300$ .

$b = 25 - 600$

$125 = 325$

$a = 300$

Subtract $600$ from $25$ .

$b = - 575$

$125 = 325$

$a = 300$

$b = - 575$

$125 = 325$

$a = 300$

$b = - 575$

$125 = 325$

$a = 300$

$b = - 575$

$125 = 325$

$a = 300$

Since $125 = 325$ is not true, there is no solution.

No solution

Since $y \neq f (x)$ for the corresponding $x$ values, the function is not linear.

The function is not linear

Step 2

Check if the function rule is quadratic.

To find if the table follows a function rule, check whether the function rule could follow the form $y = a x^{2} + b x + c$ .

$y = a x^{2} + b x + c$

Build a set of $3$ equations from the table such that $f (x) = a x^{2} + b x + c$ .

Calculate the values of $a$ , $b$ , and $c$ .

Solve for $c$ in $25 = a \cdot 2^{2} + b (2) + c$ .

Rewrite the equation as $a \cdot 2^{2} + b (2) + c = 25$ .

$a \cdot 2^{2} + b (2) + c = 25$

$125 = a \cdot 3^{2} + b (3) + c$

$625 = a \cdot 4^{2} + b (4) + c$

Simplify each term.

Raise $2$ to the power of $2$ .

$a \cdot 4 + b (2) + c = 25$

$125 = a \cdot 3^{2} + b (3) + c$

$625 = a \cdot 4^{2} + b (4) + c$

Move $4$ to the left of $a$ .

$4 \cdot a + b (2) + c = 25$

$125 = a \cdot 3^{2} + b (3) + c$

$625 = a \cdot 4^{2} + b (4) + c$

Move $2$ to the left of $b$ .

$4 a + 2 b + c = 25$

$125 = a \cdot 3^{2} + b (3) + c$

$625 = a \cdot 4^{2} + b (4) + c$

$4 a + 2 b + c = 25$

$125 = a \cdot 3^{2} + b (3) + c$

$625 = a \cdot 4^{2} + b (4) + c$

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

Subtract $4 a$ from both sides of the equation.

$2 b + c = 25 - 4 a$

$125 = a \cdot 3^{2} + b (3) + c$

$625 = a \cdot 4^{2} + b (4) + c$

Subtract $2 b$ from both sides of the equation.

$c = 25 - 4 a - 2 b$

$125 = a \cdot 3^{2} + b (3) + c$

$625 = a \cdot 4^{2} + b (4) + c$

$c = 25 - 4 a - 2 b$

$125 = a \cdot 3^{2} + b (3) + c$

$625 = a \cdot 4^{2} + b (4) + c$

$c = 25 - 4 a - 2 b$

$125 = a \cdot 3^{2} + b (3) + c$

$625 = a \cdot 4^{2} + b (4) + c$

Replace all occurrences of $c$ with $25 - 4 a - 2 b$ in each equation.

Replace all occurrences of $c$ in $125 = a \cdot 3^{2} + b (3) + c$ with $25 - 4 a - 2 b$ .

$125 = a \cdot 3^{2} + b (3) + 25 - 4 a - 2 b$

$c = 25 - 4 a - 2 b$

$625 = a \cdot 4^{2} + b (4) + c$

Simplify $125 = a \cdot 3^{2} + b (3) + 25 - 4 a - 2 b$ .

Simplify the left side.

Remove parentheses.

$125 = a \cdot 3^{2} + b (3) + 25 - 4 a - 2 b$

$c = 25 - 4 a - 2 b$

$625 = a \cdot 4^{2} + b (4) + c$

$125 = a \cdot 3^{2} + b (3) + 25 - 4 a - 2 b$

$c = 25 - 4 a - 2 b$

$625 = a \cdot 4^{2} + b (4) + c$

Simplify the right side.

Simplify $a \cdot 3^{2} + b (3) + 25 - 4 a - 2 b$ .

Simplify each term.

Raise $3$ to the power of $2$ .

$125 = a \cdot 9 + b (3) + 25 - 4 a - 2 b$

$c = 25 - 4 a - 2 b$

$625 = a \cdot 4^{2} + b (4) + c$

Move $9$ to the left of $a$ .

$125 = 9 \cdot a + b (3) + 25 - 4 a - 2 b$

$c = 25 - 4 a - 2 b$

$625 = a \cdot 4^{2} + b (4) + c$

Move $3$ to the left of $b$ .

$125 = 9 a + 3 b + 25 - 4 a - 2 b$

$c = 25 - 4 a - 2 b$

$625 = a \cdot 4^{2} + b (4) + c$

$125 = 9 a + 3 b + 25 - 4 a - 2 b$

$c = 25 - 4 a - 2 b$

$625 = a \cdot 4^{2} + b (4) + c$

Simplify by adding terms.

Subtract $4 a$ from $9 a$ .

$125 = 5 a + 3 b + 25 - 2 b$

$c = 25 - 4 a - 2 b$

$625 = a \cdot 4^{2} + b (4) + c$

Subtract $2 b$ from $3 b$ .

$125 = 5 a + b + 25$

$c = 25 - 4 a - 2 b$

$625 = a \cdot 4^{2} + b (4) + c$

$125 = 5 a + b + 25$

$c = 25 - 4 a - 2 b$

$625 = a \cdot 4^{2} + b (4) + c$

$125 = 5 a + b + 25$

$c = 25 - 4 a - 2 b$

$625 = a \cdot 4^{2} + b (4) + c$

$125 = 5 a + b + 25$

$c = 25 - 4 a - 2 b$

$625 = a \cdot 4^{2} + b (4) + c$

$125 = 5 a + b + 25$

$c = 25 - 4 a - 2 b$

$625 = a \cdot 4^{2} + b (4) + c$

Replace all occurrences of $c$ in $625 = a \cdot 4^{2} + b (4) + c$ with $25 - 4 a - 2 b$ .

$625 = a \cdot 4^{2} + b (4) + 25 - 4 a - 2 b$

$125 = 5 a + b + 25$

$c = 25 - 4 a - 2 b$

Simplify $625 = a \cdot 4^{2} + b (4) + 25 - 4 a - 2 b$ .

Simplify the left side.

Remove parentheses.

$625 = a \cdot 4^{2} + b (4) + 25 - 4 a - 2 b$

$125 = 5 a + b + 25$

$c = 25 - 4 a - 2 b$

$625 = a \cdot 4^{2} + b (4) + 25 - 4 a - 2 b$

$125 = 5 a + b + 25$

$c = 25 - 4 a - 2 b$

Simplify the right side.

Simplify $a \cdot 4^{2} + b (4) + 25 - 4 a - 2 b$ .

Simplify each term.

Raise $4$ to the power of $2$ .

$625 = a \cdot 16 + b (4) + 25 - 4 a - 2 b$

$125 = 5 a + b + 25$

$c = 25 - 4 a - 2 b$

Move $16$ to the left of $a$ .

$625 = 16 \cdot a + b (4) + 25 - 4 a - 2 b$

$125 = 5 a + b + 25$

$c = 25 - 4 a - 2 b$

Move $4$ to the left of $b$ .

$625 = 16 a + 4 b + 25 - 4 a - 2 b$

$125 = 5 a + b + 25$

$c = 25 - 4 a - 2 b$

$625 = 16 a + 4 b + 25 - 4 a - 2 b$

$125 = 5 a + b + 25$

$c = 25 - 4 a - 2 b$

Simplify by adding terms.

Subtract $4 a$ from $16 a$ .

$625 = 12 a + 4 b + 25 - 2 b$

$125 = 5 a + b + 25$

$c = 25 - 4 a - 2 b$

Subtract $2 b$ from $4 b$ .

$625 = 12 a + 2 b + 25$

$125 = 5 a + b + 25$

$c = 25 - 4 a - 2 b$

$625 = 12 a + 2 b + 25$

$125 = 5 a + b + 25$

$c = 25 - 4 a - 2 b$

$625 = 12 a + 2 b + 25$

$125 = 5 a + b + 25$

$c = 25 - 4 a - 2 b$

$625 = 12 a + 2 b + 25$

$125 = 5 a + b + 25$

$c = 25 - 4 a - 2 b$

$625 = 12 a + 2 b + 25$

$125 = 5 a + b + 25$

$c = 25 - 4 a - 2 b$

$625 = 12 a + 2 b + 25$

$125 = 5 a + b + 25$

$c = 25 - 4 a - 2 b$

Solve for $b$ in $125 = 5 a + b + 25$ .

Rewrite the equation as $5 a + b + 25 = 125$ .

$5 a + b + 25 = 125$

$625 = 12 a + 2 b + 25$

$c = 25 - 4 a - 2 b$

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

Subtract $5 a$ from both sides of the equation.

$b + 25 = 125 - 5 a$

$625 = 12 a + 2 b + 25$

$c = 25 - 4 a - 2 b$

Subtract $25$ from both sides of the equation.

$b = 125 - 5 a - 25$

$625 = 12 a + 2 b + 25$

$c = 25 - 4 a - 2 b$

Subtract $25$ from $125$ .

$b = - 5 a + 100$

$625 = 12 a + 2 b + 25$

$c = 25 - 4 a - 2 b$

$b = - 5 a + 100$

$625 = 12 a + 2 b + 25$

$c = 25 - 4 a - 2 b$

$b = - 5 a + 100$

$625 = 12 a + 2 b + 25$

$c = 25 - 4 a - 2 b$

Replace all occurrences of $b$ with $- 5 a + 100$ in each equation.

Replace all occurrences of $b$ in $625 = 12 a + 2 b + 25$ with $- 5 a + 100$ .

$625 = 12 a + 2 (- 5 a + 100) + 25$

$b = - 5 a + 100$

$c = 25 - 4 a - 2 b$

Simplify the right side.

Simplify $12 a + 2 (- 5 a + 100) + 25$ .

Simplify each term.

Apply the distributive property.

$625 = 12 a + 2 (- 5 a) + 2 \cdot 100 + 25$

$b = - 5 a + 100$

$c = 25 - 4 a - 2 b$

Multiply $- 5$ by $2$ .

$625 = 12 a - 10 a + 2 \cdot 100 + 25$

$b = - 5 a + 100$

$c = 25 - 4 a - 2 b$

Multiply $2$ by $100$ .

$625 = 12 a - 10 a + 200 + 25$

$b = - 5 a + 100$

$c = 25 - 4 a - 2 b$

$625 = 12 a - 10 a + 200 + 25$

$b = - 5 a + 100$

$c = 25 - 4 a - 2 b$

Simplify by adding terms.

Subtract $10 a$ from $12 a$ .

$625 = 2 a + 200 + 25$

$b = - 5 a + 100$

$c = 25 - 4 a - 2 b$

Add $200$ and $25$ .

$625 = 2 a + 225$

$b = - 5 a + 100$

$c = 25 - 4 a - 2 b$

$625 = 2 a + 225$

$b = - 5 a + 100$

$c = 25 - 4 a - 2 b$

$625 = 2 a + 225$

$b = - 5 a + 100$

$c = 25 - 4 a - 2 b$

$625 = 2 a + 225$

$b = - 5 a + 100$

$c = 25 - 4 a - 2 b$

Replace all occurrences of $b$ in $c = 25 - 4 a - 2 b$ with $- 5 a + 100$ .

$c = 25 - 4 a - 2 (- 5 a + 100)$

$625 = 2 a + 225$

$b = - 5 a + 100$

Simplify the right side.

Simplify $25 - 4 a - 2 (- 5 a + 100)$ .

Simplify each term.

Apply the distributive property.

$c = 25 - 4 a - 2 (- 5 a) - 2 \cdot 100$

$625 = 2 a + 225$

$b = - 5 a + 100$

Multiply $- 5$ by $- 2$ .

$c = 25 - 4 a + 10 a - 2 \cdot 100$

$625 = 2 a + 225$

$b = - 5 a + 100$

Multiply $- 2$ by $100$ .

$c = 25 - 4 a + 10 a - 200$

$625 = 2 a + 225$

$b = - 5 a + 100$

$c = 25 - 4 a + 10 a - 200$

$625 = 2 a + 225$

$b = - 5 a + 100$

Simplify by adding terms.

Subtract $200$ from $25$ .

$c = - 4 a + 10 a - 175$

$625 = 2 a + 225$

$b = - 5 a + 100$

Add $- 4 a$ and $10 a$ .

$c = 6 a - 175$

$625 = 2 a + 225$

$b = - 5 a + 100$

$c = 6 a - 175$

$625 = 2 a + 225$

$b = - 5 a + 100$

$c = 6 a - 175$

$625 = 2 a + 225$

$b = - 5 a + 100$

$c = 6 a - 175$

$625 = 2 a + 225$

$b = - 5 a + 100$

$c = 6 a - 175$

$625 = 2 a + 225$

$b = - 5 a + 100$

Solve for $a$ in $625 = 2 a + 225$ .

Rewrite the equation as $2 a + 225 = 625$ .

$2 a + 225 = 625$

$c = 6 a - 175$

$b = - 5 a + 100$

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

Subtract $225$ from both sides of the equation.

$2 a = 625 - 225$

$c = 6 a - 175$

$b = - 5 a + 100$

Subtract $225$ from $625$ .

$2 a = 400$

$c = 6 a - 175$

$b = - 5 a + 100$

$2 a = 400$

$c = 6 a - 175$

$b = - 5 a + 100$

Divide each term in $2 a = 400$ by $2$ and simplify.

Divide each term in $2 a = 400$ by $2$ .

$\frac{2 a}{2} = \frac{400}{2}$

$c = 6 a - 175$

$b = - 5 a + 100$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 a}{2} = \frac{400}{2}$

$c = 6 a - 175$

$b = - 5 a + 100$

Divide $a$ by $1$ .

$a = \frac{400}{2}$

$c = 6 a - 175$

$b = - 5 a + 100$

$a = \frac{400}{2}$

$c = 6 a - 175$

$b = - 5 a + 100$

$a = \frac{400}{2}$

$c = 6 a - 175$

$b = - 5 a + 100$

Simplify the right side.

Divide $400$ by $2$ .

$a = 200$

$c = 6 a - 175$

$b = - 5 a + 100$

$a = 200$

$c = 6 a - 175$

$b = - 5 a + 100$

$a = 200$

$c = 6 a - 175$

$b = - 5 a + 100$

$a = 200$

$c = 6 a - 175$

$b = - 5 a + 100$

Replace all occurrences of $a$ with $200$ in each equation.

Replace all occurrences of $a$ in $c = 6 a - 175$ with $200$ .

$c = 6 (200) - 175$

$a = 200$

$b = - 5 a + 100$

Simplify the right side.

Simplify $6 (200) - 175$ .

Multiply $6$ by $200$ .

$c = 1200 - 175$

$a = 200$

$b = - 5 a + 100$

Subtract $175$ from $1200$ .

$c = 1025$

$a = 200$

$b = - 5 a + 100$

$c = 1025$

$a = 200$

$b = - 5 a + 100$

$c = 1025$

$a = 200$

$b = - 5 a + 100$

Replace all occurrences of $a$ in $b = - 5 a + 100$ with $200$ .

$b = - 5 \cdot 200 + 100$

$c = 1025$

$a = 200$

Simplify the right side.

Simplify $- 5 \cdot 200 + 100$ .

Multiply $- 5$ by $200$ .

$b = - 1000 + 100$

$c = 1025$

$a = 200$

Add $- 1000$ and $100$ .

$b = - 900$

$c = 1025$

$a = 200$

$b = - 900$

$c = 1025$

$a = 200$

$b = - 900$

$c = 1025$

$a = 200$

$b = - 900$

$c = 1025$

$a = 200$

List all of the solutions.

$b = - 900, c = 1025, a = 200$

Calculate the value of $y$ using each $x$ value in the table and compare this value to the given $f (x)$ value in the table.

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 200$ , $b = - 900$ , $c = 1025$ , and $x = 2$ .

Simplify each term.

Raise $2$ to the power of $2$ .

$y = 200 \cdot 4 + (- 900) \cdot (2) + 1025$

Multiply $200$ by $4$ .

$y = 800 + (- 900) \cdot (2) + 1025$

Multiply $- 900$ by $2$ .

$y = 800 - 1800 + 1025$

Simplify by adding and subtracting.

Subtract $1800$ from $800$ .

$y = - 1000 + 1025$

Add $- 1000$ and $1025$ .

$y = 25$

If the table has a quadratic function rule, $y = f (x)$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = 25$ and $f (x) = 25$ .

$25 = 25$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 200$ , $b = - 900$ , $c = 1025$ , and $x = 3$ .

Simplify each term.

Raise $3$ to the power of $2$ .

$y = 200 \cdot 9 + (- 900) \cdot (3) + 1025$

Multiply $200$ by $9$ .

$y = 1800 + (- 900) \cdot (3) + 1025$

Multiply $- 900$ by $3$ .

$y = 1800 - 2700 + 1025$

Simplify by adding and subtracting.

Subtract $2700$ from $1800$ .

$y = - 900 + 1025$

Add $- 900$ and $1025$ .

$y = 125$

If the table has a quadratic function rule, $y = f (x)$ for the corresponding $x$ value, $x = 3$ . This check passes since $y = 125$ and $f (x) = 125$ .

$125 = 125$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 200$ , $b = - 900$ , $c = 1025$ , and $x = 4$ .

Simplify each term.

Raise $4$ to the power of $2$ .

$y = 200 \cdot 16 + (- 900) \cdot (4) + 1025$

Multiply $200$ by $16$ .

$y = 3200 + (- 900) \cdot (4) + 1025$

Multiply $- 900$ by $4$ .

$y = 3200 - 3600 + 1025$

Simplify by adding and subtracting.

Subtract $3600$ from $3200$ .

$y = - 400 + 1025$

Add $- 400$ and $1025$ .

$y = 625$

If the table has a quadratic function rule, $y = f (x)$ for the corresponding $x$ value, $x = 4$ . This check passes since $y = 625$ and $f (x) = 625$ .

$625 = 625$

Since $y = f (x)$ for the corresponding $x$ values, the function is quadratic.

The function is quadratic

Step 3

Since all $y = f (x)$ , the function is quadratic and follows the form $y = 200 x^{2} - 900 x + 1025$ .

$y = 200 x^{2} - 900 x + 1025$

Find the Function Rule table[[x,f(x)],[2,25],[3,125],[4,625]]

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