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

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

Step 1

Find the point at $x = - 1$ .

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

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

Simplify the result.

Simplify each term.

Multiply $6$ by $- 1$ .

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

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

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

Multiply $- 9$ by $1$ .

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

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

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

Multiply $3$ by $- 1$ .

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

Simplify by subtracting numbers.

Subtract $9$ from $- 6$ .

$f (- 1) = - 15 - 3$

Subtract $3$ from $- 15$ .

$f (- 1) = - 18$

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.

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

Simplify the result.

Simplify each term.

Multiply $6$ by $0$ .

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

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

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

Multiply $- 9$ by $0$ .

$f (0) = 0 + 0 + 3 {(0)}^{3}$

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

$f (0) = 0 + 0 + 3 \cdot 0$

Multiply $3$ 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.

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

Simplify the result.

Simplify each term.

Multiply $6$ by $3$ .

$f (3) = 18 - 9 {(3)}^{2} + 3 {(3)}^{3}$

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

$f (3) = 18 - 9 \cdot 9 + 3 {(3)}^{3}$

Multiply $- 9$ by $9$ .

$f (3) = 18 - 81 + 3 {(3)}^{3}$

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

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

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

$f (3) = 18 - 81 + 3 {(3)}^{3}$

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

$f (3) = 18 - 81 + 3^{1 + 3}$

Add $1$ and $3$ .

$f (3) = 18 - 81 + 3^{4}$

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

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

Simplify by adding and subtracting.

Subtract $81$ from $18$ .

$f (3) = - 63 + 81$

Add $- 63$ and $81$ .

$f (3) = 18$

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(1-x)(2-x)

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