Find the Function Rule table[[x,y],[0,1.6],[1,3.2],[2,6.4]]

$\begin{array}{c} x & y \\ 0 & 1.6 \\ 1 & 3.2 \\ 2 & 6.4 \end{array}$

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 $y = a x + b$ .

$1.6 = a (0) + b 3.2 = a (1) + b 6.4 = a (2) + b$

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

Rewrite the equation as $b = 1.6$ .

$b = 1.6$

$3.2 = a + b$

$6.4 = a (2) + b$

Replace all occurrences of $b$ with $1.6$ in each equation.

Replace all occurrences of $b$ in $3.2 = a + b$ with $1.6$ .

$3.2 = a + 1.6$

$b = 1.6$

$6.4 = a (2) + b$

Simplify the left side.

Remove parentheses.

$3.2 = a + 1.6$

$b = 1.6$

$6.4 = a (2) + b$

$3.2 = a + 1.6$

$b = 1.6$

$6.4 = a (2) + b$

Replace all occurrences of $b$ in $6.4 = a (2) + b$ with $1.6$ .

$6.4 = a (2) + 1.6$

$3.2 = a + 1.6$

$b = 1.6$

Simplify $6.4 = a (2) + 1.6$ .

Simplify the left side.

Remove parentheses.

$6.4 = a (2) + 1.6$

$3.2 = a + 1.6$

$b = 1.6$

$6.4 = a (2) + 1.6$

$3.2 = a + 1.6$

$b = 1.6$

Simplify the right side.

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

$6.4 = 2 a + 1.6$

$3.2 = a + 1.6$

$b = 1.6$

$6.4 = 2 a + 1.6$

$3.2 = a + 1.6$

$b = 1.6$

$6.4 = 2 a + 1.6$

$3.2 = a + 1.6$

$b = 1.6$

$6.4 = 2 a + 1.6$

$3.2 = a + 1.6$

$b = 1.6$

Solve for $a$ in $6.4 = 2 a + 1.6$ .

Rewrite the equation as $2 a + 1.6 = 6.4$ .

$2 a + 1.6 = 6.4$

$3.2 = a + 1.6$

$b = 1.6$

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

Subtract $1.6$ from both sides of the equation.

$2 a = 6.4 - 1.6$

$3.2 = a + 1.6$

$b = 1.6$

Subtract $1.6$ from $6.4$ .

$2 a = 4.8$

$3.2 = a + 1.6$

$b = 1.6$

$2 a = 4.8$

$3.2 = a + 1.6$

$b = 1.6$

Divide each term in $2 a = 4.8$ by $2$ and simplify.

Divide each term in $2 a = 4.8$ by $2$ .

$\frac{2 a}{2} = \frac{4.8}{2}$

$3.2 = a + 1.6$

$b = 1.6$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 a}{2} = \frac{4.8}{2}$

$3.2 = a + 1.6$

$b = 1.6$

Divide $a$ by $1$ .

$a = \frac{4.8}{2}$

$3.2 = a + 1.6$

$b = 1.6$

$a = \frac{4.8}{2}$

$3.2 = a + 1.6$

$b = 1.6$

$a = \frac{4.8}{2}$

$3.2 = a + 1.6$

$b = 1.6$

Simplify the right side.

Divide $4.8$ by $2$ .

$a = 2.4$

$3.2 = a + 1.6$

$b = 1.6$

$a = 2.4$

$3.2 = a + 1.6$

$b = 1.6$

$a = 2.4$

$3.2 = a + 1.6$

$b = 1.6$

$a = 2.4$

$3.2 = a + 1.6$

$b = 1.6$

Replace all occurrences of $a$ with $2.4$ in each equation.

Replace all occurrences of $a$ in $3.2 = a + 1.6$ with $2.4$ .

$3.2 = (2.4) + 1.6$

$a = 2.4$

$b = 1.6$

Simplify the right side.

Add $2.4$ and $1.6$ .

$3.2 = 4$

$a = 2.4$

$b = 1.6$

$3.2 = 4$

$a = 2.4$

$b = 1.6$

$3.2 = 4$

$a = 2.4$

$b = 1.6$

Since $3.2 = 4$ 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 $1.6 = a \cdot 0^{2} + c$ .

Rewrite the equation as $a \cdot 0^{2} + c = 1.6$ .

$a \cdot 0^{2} + c = 1.6$

$3.2 = a + b + c$