Graph y=3x(x-1)(x-2)

$y = 3 x (x - 1) (x - 2)$

Step 1

Find the point at $x = - 1$ .

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = 3 {(- 1)}^{3} - 9 {(- 1)}^{2} + 6 (- 1)$

Simplify the result.

Simplify each term.

Raise $- 1$ to the power of $3$ .

$f (- 1) = 3 \cdot - 1 - 9 {(- 1)}^{2} + 6 (- 1)$

Multiply $3$ by $- 1$ .

$f (- 1) = - 3 - 9 {(- 1)}^{2} + 6 (- 1)$

Raise $- 1$ to the power of $2$ .

$f (- 1) = - 3 - 9 \cdot 1 + 6 (- 1)$

Multiply $- 9$ by $1$ .

$f (- 1) = - 3 - 9 + 6 (- 1)$

Multiply $6$ by $- 1$ .

$f (- 1) = - 3 - 9 - 6$

Simplify by subtracting numbers.

Subtract $9$ from $- 3$ .

$f (- 1) = - 12 - 6$

Subtract $6$ from $- 12$ .

$f (- 1) = - 18$

The final answer is $- 18$ .

$- 18$

Convert $- 18$ to decimal.

$y = - 18$

Step 2

Find the point at $x = 0$ .

Replace the variable $x$ with $0$ in the expression.

$f (0) = 3 {(0)}^{3} - 9 {(0)}^{2} + 6 (0)$

Simplify the result.

Simplify each term.

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

$f (0) = 3 \cdot 0 - 9 {(0)}^{2} + 6 (0)$

Multiply $3$ by $0$ .

$f (0) = 0 - 9 {(0)}^{2} + 6 (0)$

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

$f (0) = 0 - 9 \cdot 0 + 6 (0)$

Multiply $- 9$ by $0$ .

$f (0) = 0 + 0 + 6 (0)$

Multiply $6$ by $0$ .

$f (0) = 0 + 0 + 0$

Simplify by adding numbers.

Add $0$ and $0$ .

$f (0) = 0 + 0$

Add $0$ and $0$ .

$f (0) = 0$

The final answer is $0$ .

$0$

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 3$ .

Replace the variable $x$ with $3$ in the expression.

$f (3) = 3 {(3)}^{3} - 9 {(3)}^{2} + 6 (3)$

Simplify the result.

Simplify each term.

Multiply $3$ by ${(3)}^{3}$ by adding the exponents.

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

Raise $3$ to the power of $1$ .

$f (3) = 3 {(3)}^{3} - 9 {(3)}^{2} + 6 (3)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (3) = 3^{1 + 3} - 9 {(3)}^{2} + 6 (3)$

Add $1$ and $3$ .

$f (3) = 3^{4} - 9 {(3)}^{2} + 6 (3)$

Raise $3$ to the power of $4$ .

$f (3) = 81 - 9 {(3)}^{2} + 6 (3)$

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

$f (3) = 81 - 9 \cdot 9 + 6 (3)$

Multiply $- 9$ by $9$ .

$f (3) = 81 - 81 + 6 (3)$

Multiply $6$ by $3$ .

$f (3) = 81 - 81 + 18$

Simplify by adding and subtracting.

Subtract $81$ from $81$ .

$f (3) = 0 + 18$

Add $0$ and $18$ .

$f (3) = 18$

The final answer is $18$ .

$18$

Convert $18$ to decimal.

$y = 18$

Step 4

The cubic function can be graphed using the function behavior and the points.

$\begin{array}{c} x & y \\ - 1 & - 18 \\ 0 & 0 \\ 1 & 0 \\ 2 & 0 \\ 3 & 18 \end{array}$

Step 5

The cubic function can be graphed using the function behavior and the selected points.

Falls to the left and rises to the right

$\begin{array}{c} x & y \\ - 1 & - 18 \\ 0 & 0 \\ 1 & 0 \\ 2 & 0 \\ 3 & 18 \end{array}$

Step 6

Graph y=3x(x-1)(x-2)

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