Find the Function Rule table[[x,y],[0,0],[1,41],[2,164],[3,369],[4,656],[5,1025]]

$\begin{array}{c} x & y \\ 0 & 0 \\ 1 & 41 \\ 2 & 164 \\ 3 & 369 \\ 4 & 656 \\ 5 & 1025 \end{array}$

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 $y = a x + b$ .

$0 = a (0) + b 41 = a (1) + b 164 = a (2) + b 369 = a (3) + b 656 = a (4) + b 1025 = a (5) + b$

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

Rewrite the equation as $b = 0$ .

$b = 0$

$41 = a + b$

$164 = a (2) + b$

$369 = a (3) + b$

$656 = a (4) + b$

$1025 = a (5) + b$

Replace all occurrences of $b$ with $0$ in each equation.

Replace all occurrences of $b$ in $41 = a + b$ with $0$ .

$41 = a + 0$

$b = 0$

$164 = a (2) + b$

$369 = a (3) + b$

$656 = a (4) + b$

$1025 = a (5) + b$

Simplify $41 = a + 0$ .

Simplify the left side.

Remove parentheses.

$41 = a + 0$

$b = 0$

$164 = a (2) + b$

$369 = a (3) + b$

$656 = a (4) + b$

$1025 = a (5) + b$

$41 = a + 0$

$b = 0$

$164 = a (2) + b$

$369 = a (3) + b$

$656 = a (4) + b$

$1025 = a (5) + b$

Simplify the right side.

Add $a$ and $0$ .

$41 = a$

$b = 0$

$164 = a (2) + b$

$369 = a (3) + b$

$656 = a (4) + b$

$1025 = a (5) + b$

$41 = a$

$b = 0$

$164 = a (2) + b$

$369 = a (3) + b$

$656 = a (4) + b$

$1025 = a (5) + b$

$41 = a$

$b = 0$

$164 = a (2) + b$

$369 = a (3) + b$

$656 = a (4) + b$

$1025 = a (5) + b$

Replace all occurrences of $b$ in $164 = a (2) + b$ with $0$ .

$164 = a (2) + 0$

$41 = a$

$b = 0$

$369 = a (3) + b$

$656 = a (4) + b$

$1025 = a (5) + b$

Simplify $164 = a (2) + 0$ .

Simplify the left side.

Remove parentheses.

$164 = a (2) + 0$

$41 = a$

$b = 0$

$369 = a (3) + b$

$656 = a (4) + b$

$1025 = a (5) + b$

$164 = a (2) + 0$

$41 = a$

$b = 0$

$369 = a (3) + b$

$656 = a (4) + b$

$1025 = a (5) + b$

Simplify the right side.

Simplify $a (2) + 0$ .

Add $a (2)$ and $0$ .

$164 = a (2)$

$41 = a$

$b = 0$

$369 = a (3) + b$

$656 = a (4) + b$

$1025 = a (5) + b$

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

$164 = 2 a$

$41 = a$

$b = 0$

$369 = a (3) + b$

$656 = a (4) + b$

$1025 = a (5) + b$

$164 = 2 a$

$41 = a$

$b = 0$

$369 = a (3) + b$

$656 = a (4) + b$

$1025 = a (5) + b$

$164 = 2 a$

$41 = a$

$b = 0$

$369 = a (3) + b$

$656 = a (4) + b$

$1025 = a (5) + b$

$164 = 2 a$

$41 = a$

$b = 0$

$369 = a (3) + b$

$656 = a (4) + b$

$1025 = a (5) + b$

Replace all occurrences of $b$ in $369 = a (3) + b$ with $0$ .

$369 = a (3) + 0$

$164 = 2 a$

$41 = a$

$b = 0$

$656 = a (4) + b$

$1025 = a (5) + b$

Simplify $369 = a (3) + 0$ .

Simplify the left side.

Remove parentheses.

$369 = a (3) + 0$

$164 = 2 a$

$41 = a$

$b = 0$

$656 = a (4) + b$

$1025 = a (5) + b$

$369 = a (3) + 0$

$164 = 2 a$

$41 = a$

$b = 0$

$656 = a (4) + b$

$1025 = a (5) + b$

Simplify the right side.

Simplify $a (3) + 0$ .

Add $a (3)$ and $0$ .

$369 = a (3)$

$164 = 2 a$

$41 = a$

$b = 0$

$656 = a (4) + b$

$1025 = a (5) + b$

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

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

$656 = a (4) + b$

$1025 = a (5) + b$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

$656 = a (4) + b$

$1025 = a (5) + b$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

$656 = a (4) + b$

$1025 = a (5) + b$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

$656 = a (4) + b$

$1025 = a (5) + b$

Replace all occurrences of $b$ in $656 = a (4) + b$ with $0$ .

$656 = a (4) + 0$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

$1025 = a (5) + b$

Simplify $656 = a (4) + 0$ .

Simplify the left side.

Remove parentheses.

$656 = a (4) + 0$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

$1025 = a (5) + b$

$656 = a (4) + 0$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

$1025 = a (5) + b$

Simplify the right side.

Simplify $a (4) + 0$ .

Add $a (4)$ and $0$ .

$656 = a (4)$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

$1025 = a (5) + b$

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

$656 = 4 a$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

$1025 = a (5) + b$

$656 = 4 a$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

$1025 = a (5) + b$

$656 = 4 a$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

$1025 = a (5) + b$

$656 = 4 a$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

$1025 = a (5) + b$

Replace all occurrences of $b$ in $1025 = a (5) + b$ with $0$ .

$1025 = a (5) + 0$

$656 = 4 a$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

Simplify $1025 = a (5) + 0$ .

Simplify the left side.

Remove parentheses.

$1025 = a (5) + 0$

$656 = 4 a$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

$1025 = a (5) + 0$

$656 = 4 a$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

Simplify the right side.

Simplify $a (5) + 0$ .

Add $a (5)$ and $0$ .

$1025 = a (5)$

$656 = 4 a$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

Move $5$ to the left of $a$ .

$1025 = 5 a$

$656 = 4 a$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

$1025 = 5 a$

$656 = 4 a$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

$1025 = 5 a$

$656 = 4 a$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

$1025 = 5 a$

$656 = 4 a$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

$1025 = 5 a$

$656 = 4 a$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

Solve for $a$ in $1025 = 5 a$ .

Rewrite the equation as $5 a = 1025$ .

$5 a = 1025$

$656 = 4 a$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

Divide each term in $5 a = 1025$ by $5$ and simplify.

Divide each term in $5 a = 1025$ by $5$ .

$\frac{5 a}{5} = \frac{1025}{5}$

$656 = 4 a$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 a}{5} = \frac{1025}{5}$

$656 = 4 a$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

Divide $a$ by $1$ .

$a = \frac{1025}{5}$

$656 = 4 a$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

$a = \frac{1025}{5}$

$656 = 4 a$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

$a = \frac{1025}{5}$

$656 = 4 a$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

Simplify the right side.

Divide $1025$ by $5$ .

$a = 205$

$656 = 4 a$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

$a = 205$

$656 = 4 a$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

$a = 205$

$656 = 4 a$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

$a = 205$

$656 = 4 a$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

Replace all occurrences of $a$ with $205$ in each equation.

Replace all occurrences of $a$ in $656 = 4 a$ with $205$ .

$656 = 4 (205)$

$a = 205$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

Simplify the right side.

Multiply $4$ by $205$ .

$656 = 820$

$a = 205$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

$656 = 820$

$a = 205$

$369 = 3 a$

$164 = 2 a$

$41 = a$

$b = 0$

Replace all occurrences of $a$ in $369 = 3 a$ with $205$ .

$369 = 3 (205)$

$656 = 820$

$a = 205$

$164 = 2 a$

$41 = a$

$b = 0$

Simplify the right side.

Multiply $3$ by $205$ .

$369 = 615$

$656 = 820$

$a = 205$

$164 = 2 a$

$41 = a$

$b = 0$

$369 = 615$

$656 = 820$

$a = 205$

$164 = 2 a$

$41 = a$

$b = 0$

Replace all occurrences of $a$ in $164 = 2 a$ with $205$ .

$164 = 2 (205)$

$369 = 615$

$656 = 820$

$a = 205$

$41 = a$

$b = 0$

Simplify the right side.

Multiply $2$ by $205$ .

$164 = 410$

$369 = 615$

$656 = 820$

$a = 205$

$41 = a$

$b = 0$

$164 = 410$

$369 = 615$

$656 = 820$

$a = 205$

$41 = a$

$b = 0$

Replace all occurrences of $a$ in $41 = a$ with $205$ .

$41 = 205$

$164 = 410$

$369 = 615$

$656 = 820$

$a = 205$

$b = 0$

Simplify the left side.

Remove parentheses.

$41 = 205$

$164 = 410$

$369 = 615$

$656 = 820$

$a = 205$

$b = 0$

$41 = 205$

$164 = 410$

$369 = 615$

$656 = 820$

$a = 205$

$b = 0$

$41 = 205$

$164 = 410$

$369 = 615$

$656 = 820$

$a = 205$

$b = 0$

Since $41 = 205$ is not true, there is no solution.

No solution

Since $y \neq y$ 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 $y = a x^{2} + b x + c$ .

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

Solve for $c$ in $0 = a \cdot 0^{2} + c$ .

Rewrite the equation as $a \cdot 0^{2} + c = 0$ .

$a \cdot 0^{2} + c = 0$

$41 = a + b + c$

$164 = a \cdot 2^{2} + b (2) + c$

$369 = a \cdot 3^{2} + b (3) + c$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

Simplify $a \cdot 0^{2} + c$ .

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$a \cdot 0 + c = 0$

$41 = a + b + c$

$164 = a \cdot 2^{2} + b (2) + c$

$369 = a \cdot 3^{2} + b (3) + c$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

Multiply $a$ by $0$ .

$0 + c = 0$

$41 = a + b + c$

$164 = a \cdot 2^{2} + b (2) + c$

$369 = a \cdot 3^{2} + b (3) + c$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

$0 + c = 0$

$41 = a + b + c$

$164 = a \cdot 2^{2} + b (2) + c$

$369 = a \cdot 3^{2} + b (3) + c$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

Add $0$ and $c$ .

$c = 0$

$41 = a + b + c$

$164 = a \cdot 2^{2} + b (2) + c$

$369 = a \cdot 3^{2} + b (3) + c$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

$c = 0$

$41 = a + b + c$

$164 = a \cdot 2^{2} + b (2) + c$

$369 = a \cdot 3^{2} + b (3) + c$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

$c = 0$

$41 = a + b + c$

$164 = a \cdot 2^{2} + b (2) + c$

$369 = a \cdot 3^{2} + b (3) + c$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

Replace all occurrences of $c$ with $0$ in each equation.

Replace all occurrences of $c$ in $41 = a + b + c$ with $0$ .

$41 = a + b + 0$

$c = 0$

$164 = a \cdot 2^{2} + b (2) + c$

$369 = a \cdot 3^{2} + b (3) + c$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

Simplify $41 = a + b + 0$ .

Simplify the left side.

Remove parentheses.

$41 = a + b + 0$

$c = 0$

$164 = a \cdot 2^{2} + b (2) + c$

$369 = a \cdot 3^{2} + b (3) + c$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

$41 = a + b + 0$

$c = 0$

$164 = a \cdot 2^{2} + b (2) + c$

$369 = a \cdot 3^{2} + b (3) + c$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

Simplify the right side.

Add $a + b$ and $0$ .

$41 = a + b$

$c = 0$

$164 = a \cdot 2^{2} + b (2) + c$

$369 = a \cdot 3^{2} + b (3) + c$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

$41 = a + b$

$c = 0$

$164 = a \cdot 2^{2} + b (2) + c$

$369 = a \cdot 3^{2} + b (3) + c$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

$41 = a + b$

$c = 0$

$164 = a \cdot 2^{2} + b (2) + c$

$369 = a \cdot 3^{2} + b (3) + c$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

Replace all occurrences of $c$ in $164 = a \cdot 2^{2} + b (2) + c$ with $0$ .

$164 = a \cdot 2^{2} + b (2) + 0$

$41 = a + b$

$c = 0$

$369 = a \cdot 3^{2} + b (3) + c$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

Simplify $164 = a \cdot 2^{2} + b (2) + 0$ .

Simplify the left side.

Remove parentheses.

$164 = a \cdot 2^{2} + b (2) + 0$

$41 = a + b$

$c = 0$

$369 = a \cdot 3^{2} + b (3) + c$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

$164 = a \cdot 2^{2} + b (2) + 0$

$41 = a + b$

$c = 0$

$369 = a \cdot 3^{2} + b (3) + c$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

Simplify the right side.

Simplify $a \cdot 2^{2} + b (2) + 0$ .

Add $a \cdot 2^{2} + b (2)$ and $0$ .

$164 = a \cdot 2^{2} + b (2)$

$41 = a + b$

$c = 0$

$369 = a \cdot 3^{2} + b (3) + c$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

Simplify each term.

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

$164 = a \cdot 4 + b (2)$

$41 = a + b$

$c = 0$

$369 = a \cdot 3^{2} + b (3) + c$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

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

$164 = 4 \cdot a + b (2)$

$41 = a + b$

$c = 0$

$369 = a \cdot 3^{2} + b (3) + c$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

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

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$369 = a \cdot 3^{2} + b (3) + c$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$369 = a \cdot 3^{2} + b (3) + c$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$369 = a \cdot 3^{2} + b (3) + c$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$369 = a \cdot 3^{2} + b (3) + c$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$369 = a \cdot 3^{2} + b (3) + c$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

Replace all occurrences of $c$ in $369 = a \cdot 3^{2} + b (3) + c$ with $0$ .

$369 = a \cdot 3^{2} + b (3) + 0$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

Simplify $369 = a \cdot 3^{2} + b (3) + 0$ .

Simplify the left side.

Remove parentheses.

$369 = a \cdot 3^{2} + b (3) + 0$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

$369 = a \cdot 3^{2} + b (3) + 0$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

Simplify the right side.

Simplify $a \cdot 3^{2} + b (3) + 0$ .

Add $a \cdot 3^{2} + b (3)$ and $0$ .

$369 = a \cdot 3^{2} + b (3)$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

Simplify each term.

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

$369 = a \cdot 9 + b (3)$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

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

$369 = 9 \cdot a + b (3)$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

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

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$656 = a \cdot 4^{2} + b (4) + c$

$1025 = a \cdot 5^{2} + b (5) + c$

Replace all occurrences of $c$ in $656 = a \cdot 4^{2} + b (4) + c$ with $0$ .

$656 = a \cdot 4^{2} + b (4) + 0$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$1025 = a \cdot 5^{2} + b (5) + c$

Simplify $656 = a \cdot 4^{2} + b (4) + 0$ .

Simplify the left side.

Remove parentheses.

$656 = a \cdot 4^{2} + b (4) + 0$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$1025 = a \cdot 5^{2} + b (5) + c$

$656 = a \cdot 4^{2} + b (4) + 0$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$1025 = a \cdot 5^{2} + b (5) + c$

Simplify the right side.

Simplify $a \cdot 4^{2} + b (4) + 0$ .

Add $a \cdot 4^{2} + b (4)$ and $0$ .

$656 = a \cdot 4^{2} + b (4)$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$1025 = a \cdot 5^{2} + b (5) + c$

Simplify each term.

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

$656 = a \cdot 16 + b (4)$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$1025 = a \cdot 5^{2} + b (5) + c$

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

$656 = 16 \cdot a + b (4)$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$1025 = a \cdot 5^{2} + b (5) + c$

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

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$1025 = a \cdot 5^{2} + b (5) + c$

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$1025 = a \cdot 5^{2} + b (5) + c$

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$1025 = a \cdot 5^{2} + b (5) + c$

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$1025 = a \cdot 5^{2} + b (5) + c$

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$1025 = a \cdot 5^{2} + b (5) + c$

Replace all occurrences of $c$ in $1025 = a \cdot 5^{2} + b (5) + c$ with $0$ .

$1025 = a \cdot 5^{2} + b (5) + 0$

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

Simplify $1025 = a \cdot 5^{2} + b (5) + 0$ .

Simplify the left side.

Remove parentheses.

$1025 = a \cdot 5^{2} + b (5) + 0$

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$1025 = a \cdot 5^{2} + b (5) + 0$

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

Simplify the right side.

Simplify $a \cdot 5^{2} + b (5) + 0$ .

Add $a \cdot 5^{2} + b (5)$ and $0$ .

$1025 = a \cdot 5^{2} + b (5)$

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

Simplify each term.

Raise $5$ to the power of $2$ .

$1025 = a \cdot 25 + b (5)$

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

Move $25$ to the left of $a$ .

$1025 = 25 \cdot a + b (5)$

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

Move $5$ to the left of $b$ .

$1025 = 25 a + 5 b$

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$1025 = 25 a + 5 b$

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$1025 = 25 a + 5 b$

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$1025 = 25 a + 5 b$

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$1025 = 25 a + 5 b$

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

$1025 = 25 a + 5 b$

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$41 = a + b$

$c = 0$

Solve for $a$ in $41 = a + b$ .

Rewrite the equation as $a + b = 41$ .

$a + b = 41$

$1025 = 25 a + 5 b$

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$c = 0$

Subtract $b$ from both sides of the equation.

$a = 41 - b$

$1025 = 25 a + 5 b$

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$c = 0$

$a = 41 - b$

$1025 = 25 a + 5 b$

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$c = 0$

Replace all occurrences of $a$ with $41 - b$ in each equation.

Replace all occurrences of $a$ in $1025 = 25 a + 5 b$ with $41 - b$ .

$1025 = 25 (41 - b) + 5 b$

$a = 41 - b$

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$c = 0$

Simplify the right side.

Simplify $25 (41 - b) + 5 b$ .

Simplify each term.

Apply the distributive property.

$1025 = 25 \cdot 41 + 25 (- b) + 5 b$

$a = 41 - b$

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$c = 0$

Multiply $25$ by $41$ .

$1025 = 1025 + 25 (- b) + 5 b$

$a = 41 - b$

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$c = 0$

Multiply $- 1$ by $25$ .

$1025 = 1025 - 25 b + 5 b$

$a = 41 - b$

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$c = 0$

$1025 = 1025 - 25 b + 5 b$

$a = 41 - b$

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$c = 0$

Add $- 25 b$ and $5 b$ .

$1025 = 1025 - 20 b$

$a = 41 - b$

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$c = 0$

$1025 = 1025 - 20 b$

$a = 41 - b$

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$c = 0$

$1025 = 1025 - 20 b$

$a = 41 - b$

$656 = 16 a + 4 b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$c = 0$

Replace all occurrences of $a$ in $656 = 16 a + 4 b$ with $41 - b$ .

$656 = 16 (41 - b) + 4 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$c = 0$

Simplify the right side.

Simplify $16 (41 - b) + 4 b$ .

Simplify each term.

Apply the distributive property.

$656 = 16 \cdot 41 + 16 (- b) + 4 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$c = 0$

Multiply $16$ by $41$ .

$656 = 656 + 16 (- b) + 4 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$c = 0$

Multiply $- 1$ by $16$ .

$656 = 656 - 16 b + 4 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$c = 0$

$656 = 656 - 16 b + 4 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$c = 0$

Add $- 16 b$ and $4 b$ .

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$c = 0$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$c = 0$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$369 = 9 a + 3 b$

$164 = 4 a + 2 b$

$c = 0$

Replace all occurrences of $a$ in $369 = 9 a + 3 b$ with $41 - b$ .

$369 = 9 (41 - b) + 3 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$164 = 4 a + 2 b$

$c = 0$

Simplify the right side.

Simplify $9 (41 - b) + 3 b$ .

Simplify each term.

Apply the distributive property.

$369 = 9 \cdot 41 + 9 (- b) + 3 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$164 = 4 a + 2 b$

$c = 0$

Multiply $9$ by $41$ .

$369 = 369 + 9 (- b) + 3 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$164 = 4 a + 2 b$

$c = 0$

Multiply $- 1$ by $9$ .

$369 = 369 - 9 b + 3 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$164 = 4 a + 2 b$

$c = 0$

$369 = 369 - 9 b + 3 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$164 = 4 a + 2 b$

$c = 0$

Add $- 9 b$ and $3 b$ .

$369 = 369 - 6 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$164 = 4 a + 2 b$

$c = 0$

$369 = 369 - 6 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$164 = 4 a + 2 b$

$c = 0$

$369 = 369 - 6 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$164 = 4 a + 2 b$

$c = 0$

Replace all occurrences of $a$ in $164 = 4 a + 2 b$ with $41 - b$ .

$164 = 4 (41 - b) + 2 b$

$369 = 369 - 6 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

Simplify the right side.

Simplify $4 (41 - b) + 2 b$ .

Simplify each term.

Apply the distributive property.

$164 = 4 \cdot 41 + 4 (- b) + 2 b$

$369 = 369 - 6 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

Multiply $4$ by $41$ .

$164 = 164 + 4 (- b) + 2 b$

$369 = 369 - 6 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

Multiply $- 1$ by $4$ .

$164 = 164 - 4 b + 2 b$

$369 = 369 - 6 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

$164 = 164 - 4 b + 2 b$

$369 = 369 - 6 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

Add $- 4 b$ and $2 b$ .

$164 = 164 - 2 b$

$369 = 369 - 6 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

$164 = 164 - 2 b$

$369 = 369 - 6 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

$164 = 164 - 2 b$

$369 = 369 - 6 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

$164 = 164 - 2 b$

$369 = 369 - 6 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

Solve for $b$ in $164 = 164 - 2 b$ .

Rewrite the equation as $164 - 2 b = 164$ .

$164 - 2 b = 164$

$369 = 369 - 6 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

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

Subtract $164$ from both sides of the equation.

$- 2 b = 164 - 164$

$369 = 369 - 6 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

Subtract $164$ from $164$ .

$- 2 b = 0$

$369 = 369 - 6 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

$- 2 b = 0$

$369 = 369 - 6 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

Divide each term in $- 2 b = 0$ by $- 2$ and simplify.

Divide each term in $- 2 b = 0$ by $- 2$ .

$\frac{- 2 b}{- 2} = \frac{0}{- 2}$

$369 = 369 - 6 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 b}{- 2} = \frac{0}{- 2}$

$369 = 369 - 6 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

Divide $b$ by $1$ .

$b = \frac{0}{- 2}$

$369 = 369 - 6 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

$b = \frac{0}{- 2}$

$369 = 369 - 6 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

$b = \frac{0}{- 2}$

$369 = 369 - 6 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

Simplify the right side.

Divide $0$ by $- 2$ .

$b = 0$

$369 = 369 - 6 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

$b = 0$

$369 = 369 - 6 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

$b = 0$

$369 = 369 - 6 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

$b = 0$

$369 = 369 - 6 b$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

Replace all occurrences of $b$ with $0$ in each equation.

Replace all occurrences of $b$ in $369 = 369 - 6 b$ with $0$ .

$369 = 369 - 6 \cdot 0$

$b = 0$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

Simplify the right side.

Simplify $369 - 6 \cdot 0$ .

Multiply $- 6$ by $0$ .

$369 = 369 + 0$

$b = 0$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

Add $369$ and $0$ .

$369 = 369$

$b = 0$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

$369 = 369$

$b = 0$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

$369 = 369$

$b = 0$

$656 = 656 - 12 b$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

Replace all occurrences of $b$ in $656 = 656 - 12 b$ with $0$ .

$656 = 656 - 12 \cdot 0$

$369 = 369$

$b = 0$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

Simplify the right side.

Simplify $656 - 12 \cdot 0$ .

Multiply $- 12$ by $0$ .

$656 = 656 + 0$

$369 = 369$

$b = 0$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

Add $656$ and $0$ .

$656 = 656$

$369 = 369$

$b = 0$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

$656 = 656$

$369 = 369$

$b = 0$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

$656 = 656$

$369 = 369$

$b = 0$

$1025 = 1025 - 20 b$

$a = 41 - b$

$c = 0$

Replace all occurrences of $b$ in $1025 = 1025 - 20 b$ with $0$ .

$1025 = 1025 - 20 \cdot 0$

$656 = 656$

$369 = 369$

$b = 0$

$a = 41 - b$

$c = 0$

Simplify the right side.

Simplify $1025 - 20 \cdot 0$ .

Multiply $- 20$ by $0$ .

$1025 = 1025 + 0$

$656 = 656$

$369 = 369$

$b = 0$

$a = 41 - b$

$c = 0$

Add $1025$ and $0$ .

$1025 = 1025$

$656 = 656$

$369 = 369$

$b = 0$

$a = 41 - b$

$c = 0$

$1025 = 1025$

$656 = 656$

$369 = 369$

$b = 0$

$a = 41 - b$

$c = 0$

$1025 = 1025$

$656 = 656$

$369 = 369$

$b = 0$

$a = 41 - b$

$c = 0$

Replace all occurrences of $b$ in $a = 41 - b$ with $0$ .

$a = 41 - (0)$

$1025 = 1025$

$656 = 656$

$369 = 369$

$b = 0$

$c = 0$

Simplify the right side.

Subtract $0$ from $41$ .

$a = 41$

$1025 = 1025$

$656 = 656$

$369 = 369$

$b = 0$

$c = 0$

$a = 41$

$1025 = 1025$

$656 = 656$

$369 = 369$

$b = 0$

$c = 0$

$a = 41$

$1025 = 1025$

$656 = 656$

$369 = 369$

$b = 0$

$c = 0$

Remove any equations from the system that are always true.

$a = 41$

$b = 0$

$c = 0$

List all of the solutions.

$a = 41, b = 0, c = 0$

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

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 41$ , $b = 0$ , $c = 0$ , and $x = 0$ .

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$y = 41 \cdot 0 + (0) \cdot (0) + 0$

Multiply $41$ by $0$ .

$y = 0 + (0) \cdot (0) + 0$

Multiply $0$ by $0$ .

$y = 0 + 0 + 0$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 0$

Add $0$ and $0$ .

$y = 0$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 0$ . This check passes since $y = 0$ and $y = 0$ .

$0 = 0$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 41$ , $b = 0$ , $c = 0$ , and $x = 1$ .

Simplify each term.

One to any power is one.

$y = 41 \cdot 1 + (0) \cdot (1) + 0$

Multiply $41$ by $1$ .

$y = 41 + (0) \cdot (1) + 0$

Multiply $0$ by $1$ .

$y = 41 + 0 + 0$

Simplify by adding numbers.

Add $41$ and $0$ .

$y = 41 + 0$

Add $41$ and $0$ .

$y = 41$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 1$ . This check passes since $y = 41$ and $y = 41$ .

$41 = 41$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 41$ , $b = 0$ , $c = 0$ , and $x = 2$ .

Simplify each term.

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

$y = 41 \cdot 4 + (0) \cdot (2) + 0$

Multiply $41$ by $4$ .

$y = 164 + (0) \cdot (2) + 0$

Multiply $0$ by $2$ .

$y = 164 + 0 + 0$

Simplify by adding numbers.

Add $164$ and $0$ .

$y = 164 + 0$

Add $164$ and $0$ .

$y = 164$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = 164$ and $y = 164$ .

$164 = 164$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 41$ , $b = 0$ , $c = 0$ , and $x = 3$ .

Simplify each term.

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

$y = 41 \cdot 9 + (0) \cdot (3) + 0$

Multiply $41$ by $9$ .

$y = 369 + (0) \cdot (3) + 0$

Multiply $0$ by $3$ .

$y = 369 + 0 + 0$

Simplify by adding numbers.

Add $369$ and $0$ .

$y = 369 + 0$

Add $369$ and $0$ .

$y = 369$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 3$ . This check passes since $y = 369$ and $y = 369$ .

$369 = 369$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 41$ , $b = 0$ , $c = 0$ , and $x = 4$ .

Simplify each term.

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

$y = 41 \cdot 16 + (0) \cdot (4) + 0$

Multiply $41$ by $16$ .

$y = 656 + (0) \cdot (4) + 0$

Multiply $0$ by $4$ .

$y = 656 + 0 + 0$

Simplify by adding numbers.

Add $656$ and $0$ .

$y = 656 + 0$

Add $656$ and $0$ .

$y = 656$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 4$ . This check passes since $y = 656$ and $y = 656$ .

$656 = 656$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 41$ , $b = 0$ , $c = 0$ , and $x = 5$ .

Simplify each term.

Raise $5$ to the power of $2$ .

$y = 41 \cdot 25 + (0) \cdot (5) + 0$

Multiply $41$ by $25$ .

$y = 1025 + (0) \cdot (5) + 0$

Multiply $0$ by $5$ .

$y = 1025 + 0 + 0$

Simplify by adding numbers.

Add $1025$ and $0$ .

$y = 1025 + 0$

Add $1025$ and $0$ .

$y = 1025$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 5$ . This check passes since $y = 1025$ and $y = 1025$ .

$1025 = 1025$

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

The function is quadratic

Step 3

Since all $y = y$ , the function is quadratic and follows the form $y = 41 x^{2}$ .

$y = 41 x^{2}$

Find the Function Rule table[[x,y],[0,0],[1,41],[2,164],[3,369],[4,656],[5,1025]]

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