Algebra Examples

$\begin{array}{c} x & y \\ 0 & 0 \\ 1 & 1 \\ 2 & 4 \\ 3 & 9 \end{array}$

Step 1

Check if the function rule is linear.

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Step 1.1

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

$y = a x + b$

Step 1.2

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

$0 = a (0) + b 1 = a (1) + b 4 = a (2) + b 9 = a (3) + b$

Step 1.3

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

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Step 1.3.1

Rewrite the equation as $b = 0$ .

$b = 0$

$1 = a + b$

$4 = a (2) + b$

$9 = a (3) + b$

Step 1.3.2

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

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Step 1.3.2.1

Replace all occurrences of $b$ in $1 = a + b$ with $0$ .

$1 = a + 0$

$b = 0$

$4 = a (2) + b$

$9 = a (3) + b$

Step 1.3.2.2

Simplify $1 = a + 0$ .

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Step 1.3.2.2.1

Simplify the left side.

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Step 1.3.2.2.1.1

Remove parentheses.

$1 = a + 0$

$b = 0$

$4 = a (2) + b$

$9 = a (3) + b$

$1 = a + 0$

$b = 0$

$4 = a (2) + b$

$9 = a (3) + b$

Step 1.3.2.2.2

Simplify the right side.

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Step 1.3.2.2.2.1

Add $a$ and $0$ .

$1 = a$

$b = 0$

$4 = a (2) + b$

$9 = a (3) + b$

$1 = a$

$b = 0$

$4 = a (2) + b$

$9 = a (3) + b$

$1 = a$

$b = 0$

$4 = a (2) + b$

$9 = a (3) + b$

Step 1.3.2.3

Replace all occurrences of $b$ in $4 = a (2) + b$ with $0$ .

$4 = a (2) + 0$

$1 = a$

$b = 0$

$9 = a (3) + b$

Step 1.3.2.4

Simplify $4 = a (2) + 0$ .

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Step 1.3.2.4.1

Simplify the left side.

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Step 1.3.2.4.1.1

Remove parentheses.

$4 = a (2) + 0$

$1 = a$

$b = 0$

$9 = a (3) + b$

$4 = a (2) + 0$

$1 = a$

$b = 0$

$9 = a (3) + b$

Step 1.3.2.4.2

Simplify the right side.

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Step 1.3.2.4.2.1

Simplify $a (2) + 0$ .

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Step 1.3.2.4.2.1.1

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

$4 = a (2)$

$1 = a$

$b = 0$

$9 = a (3) + b$

Step 1.3.2.4.2.1.2

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

$4 = 2 a$

$1 = a$

$b = 0$

$9 = a (3) + b$

$4 = 2 a$

$1 = a$

$b = 0$

$9 = a (3) + b$

$4 = 2 a$

$1 = a$

$b = 0$

$9 = a (3) + b$

$4 = 2 a$

$1 = a$

$b = 0$

$9 = a (3) + b$

Step 1.3.2.5

Replace all occurrences of $b$ in $9 = a (3) + b$ with $0$ .

$9 = a (3) + 0$

$4 = 2 a$

$1 = a$

$b = 0$

Step 1.3.2.6

Simplify $9 = a (3) + 0$ .

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Step 1.3.2.6.1

Simplify the left side.

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Step 1.3.2.6.1.1

Remove parentheses.

$9 = a (3) + 0$

$4 = 2 a$

$1 = a$

$b = 0$

$9 = a (3) + 0$

$4 = 2 a$

$1 = a$

$b = 0$

Step 1.3.2.6.2

Simplify the right side.

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Step 1.3.2.6.2.1

Simplify $a (3) + 0$ .

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Step 1.3.2.6.2.1.1

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

$9 = a (3)$

$4 = 2 a$

$1 = a$

$b = 0$

Step 1.3.2.6.2.1.2

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

$9 = 3 a$

$4 = 2 a$

$1 = a$

$b = 0$

$9 = 3 a$

$4 = 2 a$

$1 = a$

$b = 0$

$9 = 3 a$

$4 = 2 a$

$1 = a$

$b = 0$

$9 = 3 a$

$4 = 2 a$

$1 = a$

$b = 0$

$9 = 3 a$

$4 = 2 a$

$1 = a$

$b = 0$

Step 1.3.3

Solve for $a$ in $9 = 3 a$ .

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Step 1.3.3.1

Rewrite the equation as $3 a = 9$ .

$3 a = 9$

$4 = 2 a$

$1 = a$

$b = 0$

Step 1.3.3.2

Divide each term in $3 a = 9$ by $3$ and simplify.

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Step 1.3.3.2.1

Divide each term in $3 a = 9$ by $3$ .

$\frac{3 a}{3} = \frac{9}{3}$

$4 = 2 a$

$1 = a$

$b = 0$

Step 1.3.3.2.2

Simplify the left side.

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Step 1.3.3.2.2.1

Cancel the common factor of $3$ .

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Step 1.3.3.2.2.1.1

Cancel the common factor.

$\frac{3 a}{3} = \frac{9}{3}$

$4 = 2 a$

$1 = a$

$b = 0$

Step 1.3.3.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{9}{3}$

$4 = 2 a$

$1 = a$

$b = 0$

$a = \frac{9}{3}$

$4 = 2 a$

$1 = a$

$b = 0$

$a = \frac{9}{3}$

$4 = 2 a$

$1 = a$

$b = 0$

Step 1.3.3.2.3

Simplify the right side.

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Step 1.3.3.2.3.1

Divide $9$ by $3$ .

$a = 3$

$4 = 2 a$

$1 = a$

$b = 0$

$a = 3$

$4 = 2 a$

$1 = a$

$b = 0$

$a = 3$

$4 = 2 a$

$1 = a$

$b = 0$

$a = 3$

$4 = 2 a$

$1 = a$

$b = 0$

Step 1.3.4

Replace all occurrences of $a$ with $3$ in each equation.

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Step 1.3.4.1

Replace all occurrences of $a$ in $4 = 2 a$ with $3$ .

$4 = 2 (3)$

$a = 3$

$1 = a$

$b = 0$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Multiply $2$ by $3$ .

$4 = 6$

$a = 3$

$1 = a$

$b = 0$

$4 = 6$

$a = 3$

$1 = a$

$b = 0$

Step 1.3.4.3

Replace all occurrences of $a$ in $1 = a$ with $3$ .

$1 = 3$

$4 = 6$

$a = 3$

$b = 0$

Step 1.3.4.4

Simplify the left side.

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Step 1.3.4.4.1

Remove parentheses.

$1 = 3$

$4 = 6$

$a = 3$

$b = 0$

$1 = 3$

$4 = 6$

$a = 3$

$b = 0$

$1 = 3$

$4 = 6$

$a = 3$

$b = 0$

Step 1.3.5

Since $1 = 3$ is not true, there is no solution.

No solution

Step 1.4

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.

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Step 2.1

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

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

Step 2.2

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

Step 2.3

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

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Step 2.3.1

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

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Step 2.3.1.1

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

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

$1 = a + b + c$

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

$9 = a \cdot 3^{2} + b (3) + c$

Step 2.3.1.2

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

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Step 2.3.1.2.1

Simplify each term.

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Step 2.3.1.2.1.1

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

$a \cdot 0 + c = 0$

$1 = a + b + c$

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

$9 = a \cdot 3^{2} + b (3) + c$

Step 2.3.1.2.1.2

Multiply $a$ by $0$ .

$0 + c = 0$

$1 = a + b + c$

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

$9 = a \cdot 3^{2} + b (3) + c$

$0 + c = 0$

$1 = a + b + c$

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

$9 = a \cdot 3^{2} + b (3) + c$

Step 2.3.1.2.2

Add $0$ and $c$ .

$c = 0$

$1 = a + b + c$

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

$9 = a \cdot 3^{2} + b (3) + c$

$c = 0$

$1 = a + b + c$

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

$9 = a \cdot 3^{2} + b (3) + c$

$c = 0$

$1 = a + b + c$

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

$9 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2

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

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Step 2.3.2.1

Replace all occurrences of $c$ in $1 = a + b + c$ with $0$ .

$1 = a + b + 0$

$c = 0$

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

$9 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.2

Simplify $1 = a + b + 0$ .

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Step 2.3.2.2.1

Simplify the left side.

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Step 2.3.2.2.1.1

Remove parentheses.

$1 = a + b + 0$

$c = 0$

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

$9 = a \cdot 3^{2} + b (3) + c$

$1 = a + b + 0$

$c = 0$

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

$9 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.2.2

Simplify the right side.

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Step 2.3.2.2.2.1

Add $a + b$ and $0$ .

$1 = a + b$

$c = 0$

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

$9 = a \cdot 3^{2} + b (3) + c$

$1 = a + b$

$c = 0$

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

$9 = a \cdot 3^{2} + b (3) + c$

$1 = a + b$

$c = 0$

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

$9 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.3

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

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

$1 = a + b$

$c = 0$

$9 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.4

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

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Step 2.3.2.4.1

Simplify the left side.

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Step 2.3.2.4.1.1

Remove parentheses.

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

$1 = a + b$

$c = 0$

$9 = a \cdot 3^{2} + b (3) + c$

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

$1 = a + b$

$c = 0$

$9 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.4.2

Simplify the right side.

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Step 2.3.2.4.2.1

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

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Step 2.3.2.4.2.1.1

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

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

$1 = a + b$

$c = 0$

$9 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.4.2.1.2

Simplify each term.

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Step 2.3.2.4.2.1.2.1

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

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

$1 = a + b$

$c = 0$

$9 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.4.2.1.2.2

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

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

$1 = a + b$

$c = 0$

$9 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.4.2.1.2.3

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

$4 = 4 a + 2 b$

$1 = a + b$

$c = 0$

$9 = a \cdot 3^{2} + b (3) + c$

$4 = 4 a + 2 b$

$1 = a + b$

$c = 0$

$9 = a \cdot 3^{2} + b (3) + c$

$4 = 4 a + 2 b$

$1 = a + b$

$c = 0$

$9 = a \cdot 3^{2} + b (3) + c$

$4 = 4 a + 2 b$

$1 = a + b$

$c = 0$

$9 = a \cdot 3^{2} + b (3) + c$

$4 = 4 a + 2 b$

$1 = a + b$

$c = 0$

$9 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.5

Replace all occurrences of $c$ in $9 = a \cdot 3^{2} + b (3) + c$ with $0$ .

$9 = a \cdot 3^{2} + b (3) + 0$

$4 = 4 a + 2 b$

$1 = a + b$

$c = 0$

Step 2.3.2.6

Simplify $9 = a \cdot 3^{2} + b (3) + 0$ .

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Step 2.3.2.6.1

Simplify the left side.

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Step 2.3.2.6.1.1

Remove parentheses.

$9 = a \cdot 3^{2} + b (3) + 0$

$4 = 4 a + 2 b$

$1 = a + b$

$c = 0$

$9 = a \cdot 3^{2} + b (3) + 0$

$4 = 4 a + 2 b$

$1 = a + b$

$c = 0$

Step 2.3.2.6.2

Simplify the right side.

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Step 2.3.2.6.2.1

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

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Step 2.3.2.6.2.1.1

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

$9 = a \cdot 3^{2} + b (3)$

$4 = 4 a + 2 b$

$1 = a + b$

$c = 0$

Step 2.3.2.6.2.1.2

Simplify each term.

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Step 2.3.2.6.2.1.2.1

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

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

$4 = 4 a + 2 b$

$1 = a + b$

$c = 0$

Step 2.3.2.6.2.1.2.2

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

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

$4 = 4 a + 2 b$

$1 = a + b$

$c = 0$

Step 2.3.2.6.2.1.2.3

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

$9 = 9 a + 3 b$

$4 = 4 a + 2 b$

$1 = a + b$

$c = 0$

$9 = 9 a + 3 b$

$4 = 4 a + 2 b$

$1 = a + b$

$c = 0$

$9 = 9 a + 3 b$

$4 = 4 a + 2 b$

$1 = a + b$

$c = 0$

$9 = 9 a + 3 b$

$4 = 4 a + 2 b$

$1 = a + b$

$c = 0$

$9 = 9 a + 3 b$

$4 = 4 a + 2 b$

$1 = a + b$

$c = 0$

$9 = 9 a + 3 b$

$4 = 4 a + 2 b$

$1 = a + b$

$c = 0$

Step 2.3.3

Solve for $a$ in $1 = a + b$ .

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Step 2.3.3.1

Rewrite the equation as $a + b = 1$ .

$a + b = 1$

$9 = 9 a + 3 b$

$4 = 4 a + 2 b$

$c = 0$

Step 2.3.3.2

Subtract $b$ from both sides of the equation.

$a = 1 - b$

$9 = 9 a + 3 b$

$4 = 4 a + 2 b$

$c = 0$

$a = 1 - b$

$9 = 9 a + 3 b$

$4 = 4 a + 2 b$

$c = 0$

Step 2.3.4

Replace all occurrences of $a$ with $1 - b$ in each equation.

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Step 2.3.4.1

Replace all occurrences of $a$ in $9 = 9 a + 3 b$ with $1 - b$ .

$9 = 9 (1 - b) + 3 b$

$a = 1 - b$

$4 = 4 a + 2 b$

$c = 0$

Step 2.3.4.2

Simplify the right side.

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Step 2.3.4.2.1

Simplify $9 (1 - b) + 3 b$ .

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Step 2.3.4.2.1.1

Simplify each term.

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Step 2.3.4.2.1.1.1

Apply the distributive property.

$9 = 9 \cdot 1 + 9 (- b) + 3 b$

$a = 1 - b$

$4 = 4 a + 2 b$

$c = 0$

Step 2.3.4.2.1.1.2

Multiply $9$ by $1$ .

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

$a = 1 - b$

$4 = 4 a + 2 b$

$c = 0$

Step 2.3.4.2.1.1.3

Multiply $- 1$ by $9$ .

$9 = 9 - 9 b + 3 b$

$a = 1 - b$

$4 = 4 a + 2 b$

$c = 0$

$9 = 9 - 9 b + 3 b$

$a = 1 - b$

$4 = 4 a + 2 b$

$c = 0$

Step 2.3.4.2.1.2

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

$9 = 9 - 6 b$

$a = 1 - b$

$4 = 4 a + 2 b$

$c = 0$

$9 = 9 - 6 b$

$a = 1 - b$

$4 = 4 a + 2 b$

$c = 0$

$9 = 9 - 6 b$

$a = 1 - b$

$4 = 4 a + 2 b$

$c = 0$

Step 2.3.4.3

Replace all occurrences of $a$ in $4 = 4 a + 2 b$ with $1 - b$ .

$4 = 4 (1 - b) + 2 b$

$9 = 9 - 6 b$

$a = 1 - b$

$c = 0$

Step 2.3.4.4

Simplify the right side.

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Step 2.3.4.4.1

Simplify $4 (1 - b) + 2 b$ .

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Step 2.3.4.4.1.1

Simplify each term.

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Step 2.3.4.4.1.1.1

Apply the distributive property.

$4 = 4 \cdot 1 + 4 (- b) + 2 b$

$9 = 9 - 6 b$

$a = 1 - b$

$c = 0$

Step 2.3.4.4.1.1.2

Multiply $4$ by $1$ .

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

$9 = 9 - 6 b$

$a = 1 - b$

$c = 0$

Step 2.3.4.4.1.1.3

Multiply $- 1$ by $4$ .

$4 = 4 - 4 b + 2 b$

$9 = 9 - 6 b$

$a = 1 - b$

$c = 0$

$4 = 4 - 4 b + 2 b$

$9 = 9 - 6 b$

$a = 1 - b$

$c = 0$

Step 2.3.4.4.1.2

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

$4 = 4 - 2 b$

$9 = 9 - 6 b$

$a = 1 - b$

$c = 0$

$4 = 4 - 2 b$

$9 = 9 - 6 b$

$a = 1 - b$

$c = 0$

$4 = 4 - 2 b$

$9 = 9 - 6 b$

$a = 1 - b$

$c = 0$

$4 = 4 - 2 b$

$9 = 9 - 6 b$

$a = 1 - b$

$c = 0$

Step 2.3.5

Solve for $b$ in $4 = 4 - 2 b$ .

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Step 2.3.5.1

Rewrite the equation as $4 - 2 b = 4$ .

$4 - 2 b = 4$

$9 = 9 - 6 b$

$a = 1 - b$

$c = 0$

Step 2.3.5.2

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

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Step 2.3.5.2.1

Subtract $4$ from both sides of the equation.

$- 2 b = 4 - 4$

$9 = 9 - 6 b$

$a = 1 - b$

$c = 0$

Step 2.3.5.2.2

Subtract $4$ from $4$ .

$- 2 b = 0$

$9 = 9 - 6 b$

$a = 1 - b$

$c = 0$

$- 2 b = 0$

$9 = 9 - 6 b$

$a = 1 - b$

$c = 0$

Step 2.3.5.3

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

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Step 2.3.5.3.1

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

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

$9 = 9 - 6 b$

$a = 1 - b$

$c = 0$

Step 2.3.5.3.2

Simplify the left side.

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Step 2.3.5.3.2.1

Cancel the common factor of $- 2$ .

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Step 2.3.5.3.2.1.1

Cancel the common factor.

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

$9 = 9 - 6 b$

$a = 1 - b$

$c = 0$

Step 2.3.5.3.2.1.2

Divide $b$ by $1$ .

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

$9 = 9 - 6 b$

$a = 1 - b$

$c = 0$

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

$9 = 9 - 6 b$

$a = 1 - b$

$c = 0$

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

$9 = 9 - 6 b$

$a = 1 - b$

$c = 0$

Step 2.3.5.3.3

Simplify the right side.

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Step 2.3.5.3.3.1

Divide $0$ by $- 2$ .

$b = 0$

$9 = 9 - 6 b$

$a = 1 - b$

$c = 0$

$b = 0$

$9 = 9 - 6 b$

$a = 1 - b$

$c = 0$

$b = 0$

$9 = 9 - 6 b$

$a = 1 - b$

$c = 0$

$b = 0$

$9 = 9 - 6 b$

$a = 1 - b$

$c = 0$

Step 2.3.6

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

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Step 2.3.6.1

Replace all occurrences of $b$ in $9 = 9 - 6 b$ with $0$ .

$9 = 9 - 6 \cdot 0$

$b = 0$

$a = 1 - b$

$c = 0$

Step 2.3.6.2

Simplify the right side.

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Step 2.3.6.2.1

Simplify $9 - 6 \cdot 0$ .

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Step 2.3.6.2.1.1

Multiply $- 6$ by $0$ .

$9 = 9 + 0$

$b = 0$

$a = 1 - b$

$c = 0$

Step 2.3.6.2.1.2

Add $9$ and $0$ .

$9 = 9$

$b = 0$

$a = 1 - b$

$c = 0$

$9 = 9$

$b = 0$

$a = 1 - b$

$c = 0$

$9 = 9$

$b = 0$

$a = 1 - b$

$c = 0$

Step 2.3.6.3

Replace all occurrences of $b$ in $a = 1 - b$ with $0$ .

$a = 1 - (0)$

$9 = 9$

$b = 0$

$c = 0$

Step 2.3.6.4

Simplify the right side.

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Step 2.3.6.4.1

Subtract $0$ from $1$ .

$a = 1$

$9 = 9$

$b = 0$

$c = 0$

$a = 1$

$9 = 9$

$b = 0$

$c = 0$

$a = 1$

$9 = 9$

$b = 0$

$c = 0$

Step 2.3.7

Remove any equations from the system that are always true.

$a = 1$

$b = 0$

$c = 0$

Step 2.3.8

List all of the solutions.

$a = 1, b = 0, c = 0$

Step 2.4

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

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Step 2.4.1

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

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

Multiply ${(0)}^{2}$ by $1$ .

$y = {(0)}^{2} + (0) \cdot (0) + 0$

Step 2.4.1.1.2

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

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

Step 2.4.1.1.3

Multiply $0$ by $0$ .

$y = 0 + 0 + 0$

Step 2.4.1.2

Simplify by adding numbers.

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Step 2.4.1.2.1

Add $0$ and $0$ .

$y = 0 + 0$

Step 2.4.1.2.2

Add $0$ and $0$ .

$y = 0$

Step 2.4.2

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

$0 = 0$

Step 2.4.3

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

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Step 2.4.3.1

Simplify each term.

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Step 2.4.3.1.1

Multiply ${(1)}^{2}$ by $1$ .

$y = {(1)}^{2} + (0) \cdot (1) + 0$

Step 2.4.3.1.2

One to any power is one.

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

Step 2.4.3.1.3

Multiply $0$ by $1$ .

$y = 1 + 0 + 0$

Step 2.4.3.2

Simplify by adding numbers.

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Step 2.4.3.2.1

Add $1$ and $0$ .

$y = 1 + 0$

Step 2.4.3.2.2

Add $1$ and $0$ .

$y = 1$

Step 2.4.4

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

$1 = 1$

Step 2.4.5

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

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Step 2.4.5.1

Simplify each term.

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Step 2.4.5.1.1

Multiply ${(2)}^{2}$ by $1$ .

$y = {(2)}^{2} + (0) \cdot (2) + 0$

Step 2.4.5.1.2

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

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

Step 2.4.5.1.3

Multiply $0$ by $2$ .

$y = 4 + 0 + 0$

Step 2.4.5.2

Simplify by adding numbers.

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Step 2.4.5.2.1

Add $4$ and $0$ .

$y = 4 + 0$

Step 2.4.5.2.2

Add $4$ and $0$ .

$y = 4$

Step 2.4.6

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = 4$ and $y = 4$ .

$4 = 4$

Step 2.4.7

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 1$ , $b = 0$ , $c = 0$ , and $x = 3$ .

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Step 2.4.7.1

Simplify each term.

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Step 2.4.7.1.1

Multiply ${(3)}^{2}$ by $1$ .

$y = {(3)}^{2} + (0) \cdot (3) + 0$

Step 2.4.7.1.2

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

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

Step 2.4.7.1.3

Multiply $0$ by $3$ .

$y = 9 + 0 + 0$

Step 2.4.7.2

Simplify by adding numbers.

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Step 2.4.7.2.1

Add $9$ and $0$ .

$y = 9 + 0$

Step 2.4.7.2.2

Add $9$ and $0$ .

$y = 9$

Step 2.4.8

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 3$ . This check passes since $y = 9$ and $y = 9$ .

$9 = 9$

Step 2.4.9

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 $y = x^{2}$ .

$y = x^{2}$