Graph f(t)=3t(t-3)(t+4)

$f (t) = 3 t (t - 3) (t + 4)$

Step 1

Find the point at $x = - 2$ .

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = 3 {(- 2)}^{3} + 3 {(- 2)}^{2} - 36 \cdot - 2$

Simplify the result.

Simplify each term.

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

$f (- 2) = 3 \cdot - 8 + 3 {(- 2)}^{2} - 36 \cdot - 2$

Multiply $3$ by $- 8$ .

$f (- 2) = - 24 + 3 {(- 2)}^{2} - 36 \cdot - 2$

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

$f (- 2) = - 24 + 3 \cdot 4 - 36 \cdot - 2$

Multiply $3$ by $4$ .

$f (- 2) = - 24 + 12 - 36 \cdot - 2$

Multiply $- 36$ by $- 2$ .

$f (- 2) = - 24 + 12 + 72$

Simplify by adding numbers.

Add $- 24$ and $12$ .

$f (- 2) = - 12 + 72$

Add $- 12$ and $72$ .

$f (- 2) = 60$

The final answer is $60$ .

$60$

Convert $60$ to decimal.

$y = 60$

Step 2

Find the point at $x = - 1$ .

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

$f (- 1) = 3 {(- 1)}^{3} + 3 {(- 1)}^{2} - 36 \cdot - 1$

Simplify the result.

Simplify each term.

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

$f (- 1) = 3 \cdot - 1 + 3 {(- 1)}^{2} - 36 \cdot - 1$

Multiply $3$ by $- 1$ .

$f (- 1) = - 3 + 3 {(- 1)}^{2} - 36 \cdot - 1$

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

$f (- 1) = - 3 + 3 \cdot 1 - 36 \cdot - 1$

Multiply $3$ by $1$ .

$f (- 1) = - 3 + 3 - 36 \cdot - 1$

Multiply $- 36$ by $- 1$ .

$f (- 1) = - 3 + 3 + 36$

Simplify by adding numbers.

Add $- 3$ and $3$ .

$f (- 1) = 0 + 36$

Add $0$ and $36$ .

$f (- 1) = 36$

The final answer is $36$ .

$36$

Convert $36$ to decimal.

$y = 36$

Step 3

Find the point at $x = 0$ .

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

$f (0) = 3 {(0)}^{3} + 3 {(0)}^{2} - 36 \cdot 0$

Simplify the result.

Simplify each term.

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

$f (0) = 3 \cdot 0 + 3 {(0)}^{2} - 36 \cdot 0$

Multiply $3$ by $0$ .

$f (0) = 0 + 3 {(0)}^{2} - 36 \cdot 0$

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

$f (0) = 0 + 3 \cdot 0 - 36 \cdot 0$

Multiply $3$ by $0$ .

$f (0) = 0 + 0 - 36 \cdot 0$

Multiply $- 36$ 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 4

Find the point at $x = 1$ .

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

$f (1) = 3 {(1)}^{3} + 3 {(1)}^{2} - 36 \cdot 1$

Simplify the result.

Simplify each term.

One to any power is one.

$f (1) = 3 \cdot 1 + 3 {(1)}^{2} - 36 \cdot 1$

Multiply $3$ by $1$ .

$f (1) = 3 + 3 {(1)}^{2} - 36 \cdot 1$

One to any power is one.

$f (1) = 3 + 3 \cdot 1 - 36 \cdot 1$

Multiply $3$ by $1$ .

$f (1) = 3 + 3 - 36 \cdot 1$

Multiply $- 36$ by $1$ .

$f (1) = 3 + 3 - 36$

Simplify by adding and subtracting.

Add $3$ and $3$ .

$f (1) = 6 - 36$

Subtract $36$ from $6$ .

$f (1) = - 30$

The final answer is $- 30$ .

$- 30$

Convert $- 30$ to decimal.

$y = - 30$

Step 5

Find the point at $x = 2$ .

Replace the variable $x$ with $2$ in the expression.

$f (2) = 3 {(2)}^{3} + 3 {(2)}^{2} - 36 \cdot 2$

Simplify the result.

Simplify each term.

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

$f (2) = 3 \cdot 8 + 3 {(2)}^{2} - 36 \cdot 2$

Multiply $3$ by $8$ .

$f (2) = 24 + 3 {(2)}^{2} - 36 \cdot 2$

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

$f (2) = 24 + 3 \cdot 4 - 36 \cdot 2$

Multiply $3$ by $4$ .

$f (2) = 24 + 12 - 36 \cdot 2$

Multiply $- 36$ by $2$ .

$f (2) = 24 + 12 - 72$

Simplify by adding and subtracting.

Add $24$ and $12$ .

$f (2) = 36 - 72$

Subtract $72$ from $36$ .

$f (2) = - 36$

The final answer is $- 36$ .

$- 36$

Convert $- 36$ to decimal.

$y = - 36$

Step 6

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

$\begin{array}{c} x & y \\ - 2 & 60 \\ - 1 & 36 \\ 0 & 0 \\ 1 & - 30 \\ 2 & - 36 \end{array}$

Step 7

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

Falls to the left and rises to the right

$\begin{array}{c} x & y \\ - 2 & 60 \\ - 1 & 36 \\ 0 & 0 \\ 1 & - 30 \\ 2 & - 36 \end{array}$

Step 8

Graph f(t)=3t(t-3)(t+4)

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