Graph 2-(x-1)^3

$2 - {(x - 1)}^{3}$

Step 1

Find the point at $x = - 1$ .

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

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

Simplify the result.

Simplify each term.

Subtract $1$ from $- 1$ .

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

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

$f (- 1) = 2 + 8$

Multiply $- 1$ by $- 8$ .

$f (- 1) = 2 + 8$

Add $2$ and $8$ .

$f (- 1) = 10$

The final answer is $10$ .

$10$

Convert $10$ to decimal.

$y = 10$

Step 2

Find the point at $x = 0$ .

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

$f (0) = 2 - {((0) - 1)}^{3}$

Simplify the result.

Simplify each term.

Subtract $1$ from $0$ .

$f (0) = 2 - {(- 1)}^{3}$

Multiply $- 1$ by ${(- 1)}^{3}$ by adding the exponents.

Multiply $- 1$ by ${(- 1)}^{3}$ .

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

$f (0) = 2 + (- 1) {(- 1)}^{3}$

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

$f (0) = 2 + {(- 1)}^{1 + 3}$

Add $1$ and $3$ .

$f (0) = 2 + {(- 1)}^{4}$

Raise $- 1$ to the power of $4$ .

$f (0) = 2 + 1$

Add $2$ and $1$ .

$f (0) = 3$

The final answer is $3$ .

$3$

Convert $3$ to decimal.

$y = 3$

Step 3

Find the point at $x = 1$ .

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

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

Simplify the result.

Simplify each term.

Subtract $1$ from $1$ .

$f (1) = 2 - 0^{3}$

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

$f (1) = 2 - 0$

Multiply $- 1$ by $0$ .

$f (1) = 2 + 0$

Add $2$ and $0$ .

$f (1) = 2$

The final answer is $2$ .

$2$

Convert $2$ to decimal.

$y = 2$

Step 4

Find the point at $x = 2$ .

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

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

Simplify the result.

Simplify each term.

Subtract $1$ from $2$ .

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

One to any power is one.

$f (2) = 2 - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$f (2) = 2 - 1$

Subtract $1$ from $2$ .

$f (2) = 1$

The final answer is $1$ .

$1$

Convert $1$ to decimal.

$y = 1$

Step 5

Find the point at $x = 3$ .

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

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

Simplify the result.

Simplify each term.

Subtract $1$ from $3$ .

$f (3) = 2 - 2^{3}$

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

$f (3) = 2 - 1 \cdot 8$

Multiply $- 1$ by $8$ .

$f (3) = 2 - 8$

Subtract $8$ from $2$ .

$f (3) = - 6$

The final answer is $- 6$ .

$- 6$

Convert $- 6$ to decimal.

$y = - 6$

Step 6

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

$\begin{array}{c} x & y \\ - 1 & 10 \\ 0 & 3 \\ 1 & 2 \\ 2 & 1 \\ 3 & - 6 \end{array}$

Step 7

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

Rises to the left and falls to the right

$\begin{array}{c} x & y \\ - 1 & 10 \\ 0 & 3 \\ 1 & 2 \\ 2 & 1 \\ 3 & - 6 \end{array}$

Step 8

Graph 2-(x-1)^3

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