Graph x^3+3x^2-40x-84

$x^{3} + 3 x^{2} - 40 x - 84$

Step 1

Find the point at $x = - 3$ .

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

$f (- 3) = {(- 3)}^{3} + 3 {(- 3)}^{2} - 40 \cdot - 3 - 84$

Simplify the result.

Simplify each term.

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

$f (- 3) = - 27 + 3 {(- 3)}^{2} - 40 \cdot - 3 - 84$

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

$f (- 3) = - 27 + 3 \cdot 9 - 40 \cdot - 3 - 84$

Multiply $3$ by $9$ .

$f (- 3) = - 27 + 27 - 40 \cdot - 3 - 84$

Multiply $- 40$ by $- 3$ .

$f (- 3) = - 27 + 27 + 120 - 84$

Simplify by adding and subtracting.

Add $- 27$ and $27$ .

$f (- 3) = 0 + 120 - 84$

Add $0$ and $120$ .

$f (- 3) = 120 - 84$

Subtract $84$ from $120$ .

$f (- 3) = 36$

The final answer is $36$ .

$36$

Convert $36$ to decimal.

$y = 36$

Step 2

Find the point at $x = - 2$ .

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

$f (- 2) = {(- 2)}^{3} + 3 {(- 2)}^{2} - 40 \cdot - 2 - 84$

Simplify the result.

Simplify each term.

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

$f (- 2) = - 8 + 3 {(- 2)}^{2} - 40 \cdot - 2 - 84$

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

$f (- 2) = - 8 + 3 \cdot 4 - 40 \cdot - 2 - 84$

Multiply $3$ by $4$ .

$f (- 2) = - 8 + 12 - 40 \cdot - 2 - 84$

Multiply $- 40$ by $- 2$ .

$f (- 2) = - 8 + 12 + 80 - 84$

Simplify by adding and subtracting.

Add $- 8$ and $12$ .

$f (- 2) = 4 + 80 - 84$

Add $4$ and $80$ .

$f (- 2) = 84 - 84$

Subtract $84$ from $84$ .

$f (- 2) = 0$

The final answer is $0$ .

$0$

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = - 1$ .

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

$f (- 1) = {(- 1)}^{3} + 3 {(- 1)}^{2} - 40 \cdot - 1 - 84$

Simplify the result.

Simplify each term.

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

$f (- 1) = - 1 + 3 {(- 1)}^{2} - 40 \cdot - 1 - 84$

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

$f (- 1) = - 1 + 3 \cdot 1 - 40 \cdot - 1 - 84$

Multiply $3$ by $1$ .

$f (- 1) = - 1 + 3 - 40 \cdot - 1 - 84$

Multiply $- 40$ by $- 1$ .

$f (- 1) = - 1 + 3 + 40 - 84$

Simplify by adding and subtracting.

Add $- 1$ and $3$ .

$f (- 1) = 2 + 40 - 84$

Add $2$ and $40$ .

$f (- 1) = 42 - 84$

Subtract $84$ from $42$ .

$f (- 1) = - 42$

The final answer is $- 42$ .

$- 42$

Convert $- 42$ to decimal.

$y = - 42$

Step 4

Find the point at $x = 0$ .

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

$f (0) = {(0)}^{3} + 3 {(0)}^{2} - 40 \cdot 0 - 84$

Simplify the result.

Simplify each term.

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

$f (0) = 0 + 3 {(0)}^{2} - 40 \cdot 0 - 84$

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

$f (0) = 0 + 3 \cdot 0 - 40 \cdot 0 - 84$

Multiply $3$ by $0$ .

$f (0) = 0 + 0 - 40 \cdot 0 - 84$

Multiply $- 40$ by $0$ .

$f (0) = 0 + 0 + 0 - 84$

Simplify by adding and subtracting.

Add $0$ and $0$ .

$f (0) = 0 + 0 - 84$

Add $0$ and $0$ .

$f (0) = 0 - 84$

Subtract $84$ from $0$ .

$f (0) = - 84$

The final answer is $- 84$ .

$- 84$

Convert $- 84$ to decimal.

$y = - 84$

Step 5

Find the point at $x = 1$ .

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

$f (1) = {(1)}^{3} + 3 {(1)}^{2} - 40 \cdot 1 - 84$

Simplify the result.

Simplify each term.

One to any power is one.

$f (1) = 1 + 3 {(1)}^{2} - 40 \cdot 1 - 84$

One to any power is one.

$f (1) = 1 + 3 \cdot 1 - 40 \cdot 1 - 84$

Multiply $3$ by $1$ .

$f (1) = 1 + 3 - 40 \cdot 1 - 84$

Multiply $- 40$ by $1$ .

$f (1) = 1 + 3 - 40 - 84$

Simplify by adding and subtracting.

Add $1$ and $3$ .

$f (1) = 4 - 40 - 84$

Subtract $40$ from $4$ .

$f (1) = - 36 - 84$

Subtract $84$ from $- 36$ .

$f (1) = - 120$

The final answer is $- 120$ .

$- 120$

Convert $- 120$ to decimal.

$y = - 120$

Step 6

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

$\begin{array}{c} x & y \\ - 3 & 36 \\ - 2 & 0 \\ - 1 & - 42 \\ 0 & - 84 \\ 1 & - 120 \end{array}$

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

$\begin{array}{c} x & y \\ - 3 & 36 \\ - 2 & 0 \\ - 1 & - 42 \\ 0 & - 84 \\ 1 & - 120 \end{array}$

Step 8

Graph x^3+3x^2-40x-84

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