Algebra Examples

\begin{matrix} x & f (x) 3 & 64 4 & 256 5 & 1024 \end{matrix}

$\begin{array}{c} x & f (x) \\ 3 & 64 \\ 4 & 256 \\ 5 & 1024 \end{array}$

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

$f (x) = a x + b$ .

$64 = a (3) + b 256 = a (4) + b 1024 = a (5) + b$

Step 1.3

Calculate the values of

$a$ and

$b$ .

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Step 1.3.1

Solve for

$b$ in

$64 = a (3) + b$ .

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Step 1.3.1.1

Rewrite the equation as

$a (3) + b = 64$ .

$a (3) + b = 64$

$256 = a (4) + b$

$1024 = a (5) + b$

Step 1.3.1.2

Move

$3$ to the left of

$a$ .

$3 a + b = 64$

$256 = a (4) + b$

$1024 = a (5) + b$

Step 1.3.1.3

Subtract

$3 a$ from both sides of the equation.

$b = 64 - 3 a$

$256 = a (4) + b$

$1024 = a (5) + b$

$b = 64 - 3 a$

$256 = a (4) + b$

$1024 = a (5) + b$

Step 1.3.2

Replace all occurrences of

$b$ with

$64 - 3 a$ in each equation.

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Step 1.3.2.1

Replace all occurrences of

$b$ in

$256 = a (4) + b$ with

$64 - 3 a$ .

$256 = a (4) + 64 - 3 a$

$b = 64 - 3 a$

$1024 = a (5) + b$

Step 1.3.2.2

Simplify

$256 = a (4) + 64 - 3 a$ .

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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.

$256 = a (4) + 64 - 3 a$

$b = 64 - 3 a$

$1024 = a (5) + b$

$256 = a (4) + 64 - 3 a$

$b = 64 - 3 a$

$1024 = a (5) + b$

Step 1.3.2.2.2

Simplify the right side.

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Step 1.3.2.2.2.1

Simplify

$a (4) + 64 - 3 a$ .

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Step 1.3.2.2.2.1.1

Move

$4$ to the left of

$a$ .

$256 = 4 a + 64 - 3 a$

$b = 64 - 3 a$

$1024 = a (5) + b$

Step 1.3.2.2.2.1.2

Subtract

$3 a$ from

$4 a$ .

$256 = a + 64$

$b = 64 - 3 a$

$1024 = a (5) + b$

$256 = a + 64$

$b = 64 - 3 a$

$1024 = a (5) + b$

$256 = a + 64$

$b = 64 - 3 a$

$1024 = a (5) + b$

$256 = a + 64$

$b = 64 - 3 a$

$1024 = a (5) + b$

Step 1.3.2.3

Replace all occurrences of

$b$ in

$1024 = a (5) + b$ with

$64 - 3 a$ .

$1024 = a (5) + 64 - 3 a$

$256 = a + 64$

$b = 64 - 3 a$

Step 1.3.2.4

Simplify

$1024 = a (5) + 64 - 3 a$ .

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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.

$1024 = a (5) + 64 - 3 a$

$256 = a + 64$

$b = 64 - 3 a$

$1024 = a (5) + 64 - 3 a$

$256 = a + 64$

$b = 64 - 3 a$

Step 1.3.2.4.2

Simplify the right side.

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Step 1.3.2.4.2.1

Simplify

$a (5) + 64 - 3 a$ .

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Step 1.3.2.4.2.1.1

Move

$5$ to the left of

$a$ .

$1024 = 5 a + 64 - 3 a$

$256 = a + 64$

$b = 64 - 3 a$

Step 1.3.2.4.2.1.2

Subtract

$3 a$ from

$5 a$ .

$1024 = 2 a + 64$

$256 = a + 64$

$b = 64 - 3 a$

$1024 = 2 a + 64$

$256 = a + 64$

$b = 64 - 3 a$

$1024 = 2 a + 64$

$256 = a + 64$

$b = 64 - 3 a$

$1024 = 2 a + 64$

$256 = a + 64$

$b = 64 - 3 a$

$1024 = 2 a + 64$

$256 = a + 64$

$b = 64 - 3 a$

Step 1.3.3

Solve for

$a$ in

$1024 = 2 a + 64$ .

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Step 1.3.3.1

Rewrite the equation as

$2 a + 64 = 1024$ .

$2 a + 64 = 1024$

$256 = a + 64$

$b = 64 - 3 a$

Step 1.3.3.2

Move all terms not containing

$a$ to the right side of the equation.

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Step 1.3.3.2.1

Subtract

$64$ from both sides of the equation.

$2 a = 1024 - 64$

$256 = a + 64$

$b = 64 - 3 a$

Step 1.3.3.2.2

Subtract

$64$ from

$1024$ .

$2 a = 960$

$256 = a + 64$

$b = 64 - 3 a$

$2 a = 960$

$256 = a + 64$

$b = 64 - 3 a$

Step 1.3.3.3

Divide each term in

$2 a = 960$ by

$2$ and simplify.

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Step 1.3.3.3.1

Divide each term in

$2 a = 960$ by

$2$ .

$\frac{2 a}{2} = \frac{960}{2}$

$256 = a + 64$

$b = 64 - 3 a$

Step 1.3.3.3.2

Simplify the left side.

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Step 1.3.3.3.2.1

Cancel the common factor of

$2$ .

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Step 1.3.3.3.2.1.1

Cancel the common factor.

$\frac{2 a}{2} = \frac{960}{2}$

$256 = a + 64$

$b = 64 - 3 a$

Step 1.3.3.3.2.1.2

Divide

$a$ by

$1$ .

$a = \frac{960}{2}$

$256 = a + 64$

$b = 64 - 3 a$

$a = \frac{960}{2}$

$256 = a + 64$

$b = 64 - 3 a$

$a = \frac{960}{2}$

$256 = a + 64$

$b = 64 - 3 a$

Step 1.3.3.3.3

Simplify the right side.

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Step 1.3.3.3.3.1

Divide

$960$ by

$2$ .

$a = 480$

$256 = a + 64$

$b = 64 - 3 a$

$a = 480$

$256 = a + 64$

$b = 64 - 3 a$

$a = 480$

$256 = a + 64$

$b = 64 - 3 a$

$a = 480$

$256 = a + 64$

$b = 64 - 3 a$

Step 1.3.4

Replace all occurrences of

$a$ with

$480$ in each equation.

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Step 1.3.4.1

Replace all occurrences of

$a$ in

$256 = a + 64$ with

$480$ .

$256 = (480) + 64$

$a = 480$

$b = 64 - 3 a$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Add

$480$ and

$64$ .

$256 = 544$

$a = 480$

$b = 64 - 3 a$

$256 = 544$

$a = 480$

$b = 64 - 3 a$

Step 1.3.4.3

Replace all occurrences of

$a$ in

$b = 64 - 3 a$ with

$480$ .

$b = 64 - 3 \cdot 480$

$256 = 544$

$a = 480$

Step 1.3.4.4

Simplify the right side.

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Step 1.3.4.4.1

Simplify

$64 - 3 \cdot 480$ .

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Step 1.3.4.4.1.1

Multiply

$- 3$ by

$480$ .

$b = 64 - 1440$

$256 = 544$

$a = 480$

Step 1.3.4.4.1.2

Subtract

$1440$ from

$64$ .

$b = - 1376$

$256 = 544$

$a = 480$

$b = - 1376$

$256 = 544$

$a = 480$

$b = - 1376$

$256 = 544$

$a = 480$

$b = - 1376$

$256 = 544$

$a = 480$

Step 1.3.5

Since

$256 = 544$ is not true, there is no solution.

No solution

Step 1.4

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.

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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

$f (x) = a x^{2} + b x + c$ .

Step 2.3

Calculate the values of

$a$ ,

$b$ , and

$c$ .

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Step 2.3.1

Solve for

$c$ in

$64 = a \cdot 3^{2} + b (3) + c$ .

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Step 2.3.1.1

Rewrite the equation as

$a \cdot 3^{2} + b (3) + c = 64$ .

$a \cdot 3^{2} + b (3) + c = 64$

$256 = a \cdot 4^{2} + b (4) + c$

$1024 = a \cdot 5^{2} + b (5) + c$

Step 2.3.1.2

Simplify each term.

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Step 2.3.1.2.1

Raise

$3$ to the power of

$2$ .

$a \cdot 9 + b (3) + c = 64$

$256 = a \cdot 4^{2} + b (4) + c$

$1024 = a \cdot 5^{2} + b (5) + c$

Step 2.3.1.2.2

Move

$9$ to the left of

$a$ .

$9 \cdot a + b (3) + c = 64$

$256 = a \cdot 4^{2} + b (4) + c$

$1024 = a \cdot 5^{2} + b (5) + c$

Step 2.3.1.2.3

Move

$3$ to the left of

$b$ .

$9 a + 3 b + c = 64$

$256 = a \cdot 4^{2} + b (4) + c$

$1024 = a \cdot 5^{2} + b (5) + c$

$9 a + 3 b + c = 64$

$256 = a \cdot 4^{2} + b (4) + c$

$1024 = a \cdot 5^{2} + b (5) + c$

Step 2.3.1.3

Move all terms not containing

$c$ to the right side of the equation.

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Step 2.3.1.3.1

Subtract

$9 a$ from both sides of the equation.

$3 b + c = 64 - 9 a$

$256 = a \cdot 4^{2} + b (4) + c$

$1024 = a \cdot 5^{2} + b (5) + c$

Step 2.3.1.3.2

Subtract

$3 b$ from both sides of the equation.

$c = 64 - 9 a - 3 b$

$256 = a \cdot 4^{2} + b (4) + c$

$1024 = a \cdot 5^{2} + b (5) + c$

$c = 64 - 9 a - 3 b$

$256 = a \cdot 4^{2} + b (4) + c$

$1024 = a \cdot 5^{2} + b (5) + c$

$c = 64 - 9 a - 3 b$

$256 = a \cdot 4^{2} + b (4) + c$

$1024 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2

Replace all occurrences of

$c$ with

$64 - 9 a - 3 b$ in each equation.

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Step 2.3.2.1

Replace all occurrences of

$c$ in

$256 = a \cdot 4^{2} + b (4) + c$ with

$64 - 9 a - 3 b$ .

$256 = a \cdot 4^{2} + b (4) + 64 - 9 a - 3 b$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2.2

Simplify

$256 = a \cdot 4^{2} + b (4) + 64 - 9 a - 3 b$ .

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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.

$256 = a \cdot 4^{2} + b (4) + 64 - 9 a - 3 b$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + c$

$256 = a \cdot 4^{2} + b (4) + 64 - 9 a - 3 b$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2.2.2

Simplify the right side.

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Step 2.3.2.2.2.1

Simplify

$a \cdot 4^{2} + b (4) + 64 - 9 a - 3 b$ .

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Step 2.3.2.2.2.1.1

Simplify each term.

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Step 2.3.2.2.2.1.1.1

Raise

$4$ to the power of

$2$ .

$256 = a \cdot 16 + b (4) + 64 - 9 a - 3 b$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2.2.2.1.1.2

Move

$16$ to the left of

$a$ .

$256 = 16 \cdot a + b (4) + 64 - 9 a - 3 b$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2.2.2.1.1.3

Move

$4$ to the left of

$b$ .

$256 = 16 a + 4 b + 64 - 9 a - 3 b$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + c$

$256 = 16 a + 4 b + 64 - 9 a - 3 b$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2.2.2.1.2

Simplify by adding terms.

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Step 2.3.2.2.2.1.2.1

Subtract

$9 a$ from

$16 a$ .

$256 = 7 a + 4 b + 64 - 3 b$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2.2.2.1.2.2

Subtract

$3 b$ from

$4 b$ .

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + c$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + c$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + c$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + c$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2.3

Replace all occurrences of

$c$ in

$1024 = a \cdot 5^{2} + b (5) + c$ with

$64 - 9 a - 3 b$ .

$1024 = a \cdot 5^{2} + b (5) + 64 - 9 a - 3 b$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

Step 2.3.2.4

Simplify

$1024 = a \cdot 5^{2} + b (5) + 64 - 9 a - 3 b$ .

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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.

$1024 = a \cdot 5^{2} + b (5) + 64 - 9 a - 3 b$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + 64 - 9 a - 3 b$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

Step 2.3.2.4.2

Simplify the right side.

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Step 2.3.2.4.2.1

Simplify

$a \cdot 5^{2} + b (5) + 64 - 9 a - 3 b$ .

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Step 2.3.2.4.2.1.1

Simplify each term.

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Step 2.3.2.4.2.1.1.1

Raise

$5$ to the power of

$2$ .

$1024 = a \cdot 25 + b (5) + 64 - 9 a - 3 b$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

Step 2.3.2.4.2.1.1.2

Move

$25$ to the left of

$a$ .

$1024 = 25 \cdot a + b (5) + 64 - 9 a - 3 b$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

Step 2.3.2.4.2.1.1.3

Move

$5$ to the left of

$b$ .

$1024 = 25 a + 5 b + 64 - 9 a - 3 b$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

$1024 = 25 a + 5 b + 64 - 9 a - 3 b$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

Step 2.3.2.4.2.1.2

Simplify by adding terms.

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Step 2.3.2.4.2.1.2.1

Subtract

$9 a$ from

$25 a$ .

$1024 = 16 a + 5 b + 64 - 3 b$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

Step 2.3.2.4.2.1.2.2

Subtract

$3 b$ from

$5 b$ .

$1024 = 16 a + 2 b + 64$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

$1024 = 16 a + 2 b + 64$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

$1024 = 16 a + 2 b + 64$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

$1024 = 16 a + 2 b + 64$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

$1024 = 16 a + 2 b + 64$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

$1024 = 16 a + 2 b + 64$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

Step 2.3.3

Solve for

$b$ in

$256 = 7 a + b + 64$ .

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Step 2.3.3.1

Rewrite the equation as

$7 a + b + 64 = 256$ .

$7 a + b + 64 = 256$

$1024 = 16 a + 2 b + 64$

$c = 64 - 9 a - 3 b$

Step 2.3.3.2

Move all terms not containing

$b$ to the right side of the equation.

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Step 2.3.3.2.1

Subtract

$7 a$ from both sides of the equation.

$b + 64 = 256 - 7 a$

$1024 = 16 a + 2 b + 64$

$c = 64 - 9 a - 3 b$

Step 2.3.3.2.2

Subtract

$64$ from both sides of the equation.

$b = 256 - 7 a - 64$

$1024 = 16 a + 2 b + 64$

$c = 64 - 9 a - 3 b$

Step 2.3.3.2.3

Subtract

$64$ from

$256$ .

$b = - 7 a + 192$

$1024 = 16 a + 2 b + 64$

$c = 64 - 9 a - 3 b$

$b = - 7 a + 192$

$1024 = 16 a + 2 b + 64$

$c = 64 - 9 a - 3 b$

$b = - 7 a + 192$

$1024 = 16 a + 2 b + 64$

$c = 64 - 9 a - 3 b$

Step 2.3.4

Replace all occurrences of

$b$ with

$- 7 a + 192$ in each equation.

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Step 2.3.4.1

Replace all occurrences of

$b$ in

$1024 = 16 a + 2 b + 64$ with

$- 7 a + 192$ .

$1024 = 16 a + 2 (- 7 a + 192) + 64$

$b = - 7 a + 192$

$c = 64 - 9 a - 3 b$

Step 2.3.4.2

Simplify the right side.

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Step 2.3.4.2.1

Simplify

$16 a + 2 (- 7 a + 192) + 64$ .

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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.

$1024 = 16 a + 2 (- 7 a) + 2 \cdot 192 + 64$

$b = - 7 a + 192$

$c = 64 - 9 a - 3 b$

Step 2.3.4.2.1.1.2

Multiply

$- 7$ by

$2$ .

$1024 = 16 a - 14 a + 2 \cdot 192 + 64$

$b = - 7 a + 192$

$c = 64 - 9 a - 3 b$

Step 2.3.4.2.1.1.3

Multiply

$2$ by

$192$ .

$1024 = 16 a - 14 a + 384 + 64$

$b = - 7 a + 192$

$c = 64 - 9 a - 3 b$

$1024 = 16 a - 14 a + 384 + 64$

$b = - 7 a + 192$

$c = 64 - 9 a - 3 b$

Step 2.3.4.2.1.2

Simplify by adding terms.

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Step 2.3.4.2.1.2.1

Subtract

$14 a$ from

$16 a$ .

$1024 = 2 a + 384 + 64$

$b = - 7 a + 192$

$c = 64 - 9 a - 3 b$

Step 2.3.4.2.1.2.2

Add

$384$ and

$64$ .

$1024 = 2 a + 448$

$b = - 7 a + 192$

$c = 64 - 9 a - 3 b$

$1024 = 2 a + 448$

$b = - 7 a + 192$

$c = 64 - 9 a - 3 b$

$1024 = 2 a + 448$

$b = - 7 a + 192$

$c = 64 - 9 a - 3 b$

$1024 = 2 a + 448$

$b = - 7 a + 192$

$c = 64 - 9 a - 3 b$

Step 2.3.4.3

Replace all occurrences of

$b$ in

$c = 64 - 9 a - 3 b$ with

$- 7 a + 192$ .

$c = 64 - 9 a - 3 (- 7 a + 192)$

$1024 = 2 a + 448$

$b = - 7 a + 192$

Step 2.3.4.4

Simplify the right side.

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Step 2.3.4.4.1

Simplify

$64 - 9 a - 3 (- 7 a + 192)$ .

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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.

$c = 64 - 9 a - 3 (- 7 a) - 3 \cdot 192$

$1024 = 2 a + 448$

$b = - 7 a + 192$

Step 2.3.4.4.1.1.2

Multiply

$- 7$ by

$- 3$ .

$c = 64 - 9 a + 21 a - 3 \cdot 192$

$1024 = 2 a + 448$

$b = - 7 a + 192$

Step 2.3.4.4.1.1.3

Multiply

$- 3$ by

$192$ .

$c = 64 - 9 a + 21 a - 576$

$1024 = 2 a + 448$

$b = - 7 a + 192$

$c = 64 - 9 a + 21 a - 576$

$1024 = 2 a + 448$

$b = - 7 a + 192$

Step 2.3.4.4.1.2

Simplify by adding terms.

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Step 2.3.4.4.1.2.1

Subtract

$576$ from

$64$ .

$c = - 9 a + 21 a - 512$

$1024 = 2 a + 448$

$b = - 7 a + 192$

Step 2.3.4.4.1.2.2

Add

$- 9 a$ and

$21 a$ .

$c = 12 a - 512$

$1024 = 2 a + 448$

$b = - 7 a + 192$

$c = 12 a - 512$

$1024 = 2 a + 448$

$b = - 7 a + 192$

$c = 12 a - 512$

$1024 = 2 a + 448$

$b = - 7 a + 192$

$c = 12 a - 512$

$1024 = 2 a + 448$

$b = - 7 a + 192$

$c = 12 a - 512$

$1024 = 2 a + 448$

$b = - 7 a + 192$

Step 2.3.5

Solve for

$a$ in

$1024 = 2 a + 448$ .

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Step 2.3.5.1

Rewrite the equation as

$2 a + 448 = 1024$ .

$2 a + 448 = 1024$

$c = 12 a - 512$

$b = - 7 a + 192$

Step 2.3.5.2

Move all terms not containing

$a$ to the right side of the equation.

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Step 2.3.5.2.1

Subtract

$448$ from both sides of the equation.

$2 a = 1024 - 448$

$c = 12 a - 512$

$b = - 7 a + 192$

Step 2.3.5.2.2

Subtract

$448$ from

$1024$ .

$2 a = 576$

$c = 12 a - 512$

$b = - 7 a + 192$

$2 a = 576$

$c = 12 a - 512$

$b = - 7 a + 192$

Step 2.3.5.3

Divide each term in

$2 a = 576$ by

$2$ and simplify.

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Step 2.3.5.3.1

Divide each term in

$2 a = 576$ by

$2$ .

$\frac{2 a}{2} = \frac{576}{2}$

$c = 12 a - 512$

$b = - 7 a + 192$

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.

$\frac{2 a}{2} = \frac{576}{2}$

$c = 12 a - 512$

$b = - 7 a + 192$

Step 2.3.5.3.2.1.2

Divide

$a$ by

$1$ .

$a = \frac{576}{2}$

$c = 12 a - 512$

$b = - 7 a + 192$

$a = \frac{576}{2}$

$c = 12 a - 512$

$b = - 7 a + 192$

$a = \frac{576}{2}$

$c = 12 a - 512$

$b = - 7 a + 192$

Step 2.3.5.3.3

Simplify the right side.

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Step 2.3.5.3.3.1

Divide

$576$ by

$2$ .

$a = 288$

$c = 12 a - 512$

$b = - 7 a + 192$

$a = 288$

$c = 12 a - 512$

$b = - 7 a + 192$

$a = 288$

$c = 12 a - 512$

$b = - 7 a + 192$

$a = 288$

$c = 12 a - 512$

$b = - 7 a + 192$

Step 2.3.6

Replace all occurrences of

$a$ with

$288$ in each equation.

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Step 2.3.6.1

Replace all occurrences of

$a$ in

$c = 12 a - 512$ with

$288$ .

$c = 12 (288) - 512$

$a = 288$

$b = - 7 a + 192$

Step 2.3.6.2

Simplify the right side.

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Step 2.3.6.2.1

Simplify

$12 (288) - 512$ .

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Step 2.3.6.2.1.1

Multiply

$12$ by

$288$ .

$c = 3456 - 512$

$a = 288$

$b = - 7 a + 192$

Step 2.3.6.2.1.2

Subtract

$512$ from

$3456$ .

$c = 2944$

$a = 288$

$b = - 7 a + 192$

$c = 2944$

$a = 288$

$b = - 7 a + 192$

$c = 2944$

$a = 288$

$b = - 7 a + 192$

Step 2.3.6.3

Replace all occurrences of

$a$ in

$b = - 7 a + 192$ with

$288$ .

$b = - 7 \cdot 288 + 192$

$c = 2944$

$a = 288$

Step 2.3.6.4

Simplify the right side.

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Step 2.3.6.4.1

Simplify

$- 7 \cdot 288 + 192$ .

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Step 2.3.6.4.1.1

Multiply

$- 7$ by

$288$ .

$b = - 2016 + 192$

$c = 2944$

$a = 288$

Step 2.3.6.4.1.2

Add

$- 2016$ and

$192$ .

$b = - 1824$

$c = 2944$

$a = 288$

$b = - 1824$

$c = 2944$

$a = 288$

$b = - 1824$

$c = 2944$

$a = 288$

$b = - 1824$

$c = 2944$

$a = 288$

Step 2.3.7

List all of the solutions.

$b = - 1824, c = 2944, a = 288$

Step 2.4

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.

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Step 2.4.1

Calculate the value of

$y$ such that

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

$a = 288$ ,

$b = - 1824$ ,

$c = 2944$ , and

$x = 3$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

Raise

$3$ to the power of

$2$ .

$y = 288 \cdot 9 + (- 1824) \cdot (3) + 2944$

Step 2.4.1.1.2

Multiply

$288$ by

$9$ .

$y = 2592 + (- 1824) \cdot (3) + 2944$

Step 2.4.1.1.3

Multiply

$- 1824$ by

$3$ .

$y = 2592 - 5472 + 2944$

Step 2.4.1.2

Simplify by adding and subtracting.

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Step 2.4.1.2.1

Subtract

$5472$ from

$2592$ .

$y = - 2880 + 2944$

Step 2.4.1.2.2

Add

$- 2880$ and

$2944$ .

$y = 64$

Step 2.4.2

If the table has a quadratic function rule,

$y = f (x)$ for the corresponding

$x$ value,

$x = 3$ . This check passes since

$y = 64$ and

$f (x) = 64$ .

$64 = 64$

Step 2.4.3

Calculate the value of

$y$ such that

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

$a = 288$ ,

$b = - 1824$ ,

$c = 2944$ , and

$x = 4$ .

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Step 2.4.3.1

Simplify each term.

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Step 2.4.3.1.1

Raise

$4$ to the power of

$2$ .

$y = 288 \cdot 16 + (- 1824) \cdot (4) + 2944$

Step 2.4.3.1.2

Multiply

$288$ by

$16$ .

$y = 4608 + (- 1824) \cdot (4) + 2944$

Step 2.4.3.1.3

Multiply

$- 1824$ by

$4$ .

$y = 4608 - 7296 + 2944$

Step 2.4.3.2

Simplify by adding and subtracting.

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Step 2.4.3.2.1

Subtract

$7296$ from

$4608$ .

$y = - 2688 + 2944$

Step 2.4.3.2.2

Add

$- 2688$ and

$2944$ .

$y = 256$

Step 2.4.4

If the table has a quadratic function rule,

$y = f (x)$ for the corresponding

$x$ value,

$x = 4$ . This check passes since

$y = 256$ and

$f (x) = 256$ .

$256 = 256$

Step 2.4.5

Calculate the value of

$y$ such that

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

$a = 288$ ,

$b = - 1824$ ,

$c = 2944$ , and

$x = 5$ .

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Step 2.4.5.1

Simplify each term.

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Step 2.4.5.1.1

Raise

$5$ to the power of

$2$ .

$y = 288 \cdot 25 + (- 1824) \cdot (5) + 2944$

Step 2.4.5.1.2

Multiply

$288$ by

$25$ .

$y = 7200 + (- 1824) \cdot (5) + 2944$

Step 2.4.5.1.3

Multiply

$- 1824$ by

$5$ .

$y = 7200 - 9120 + 2944$

Step 2.4.5.2

Simplify by adding and subtracting.

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Step 2.4.5.2.1

Subtract

$9120$ from

$7200$ .

$y = - 1920 + 2944$

Step 2.4.5.2.2

Add

$- 1920$ and

$2944$ .

$y = 1024$

Step 2.4.6

If the table has a quadratic function rule,

$y = f (x)$ for the corresponding

$x$ value,

$x = 5$ . This check passes since

$y = 1024$ and

$f (x) = 1024$ .

$1024 = 1024$

Step 2.4.7

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

$y = 288 x^{2} - 1824 x + 2944$ .

$y = 288 x^{2} - 1824 x + 2944$