Find the Symmetry y=(x-1)^3

$y = {(x - 1)}^{3}$

Simplify the right side.

Simplify ${(x - 1)}^{3}$ .

Use the Binomial Theorem.

$y = x^{3} + 3 x^{2} \cdot - 1 + 3 x {(- 1)}^{2} + {(- 1)}^{3}$

Simplify each term.

Multiply $- 1$ by $3$ .

$y = x^{3} - 3 x^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}$

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

$y = x^{3} - 3 x^{2} + 3 x \cdot 1 + {(- 1)}^{3}$

Multiply $3$ by $1$ .

$y = x^{3} - 3 x^{2} + 3 x + {(- 1)}^{3}$

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

$y = x^{3} - 3 x^{2} + 3 x - 1$

There are three types of symmetry:

1. X-Axis Symmetry

2. Y-Axis Symmetry

3. Origin Symmetry

If $(x, y)$ exists on the graph, then the graph is symmetric about the:

1. X-Axis if $(x, - y)$ exists on the graph

2. Y-Axis if $(- x, y)$ exists on the graph

3. Origin if $(- x, - y)$ exists on the graph

Check if the graph is symmetric about the x-axis by plugging in $- y$ for $y$ .

$- y = x^{3} - 3 x^{2} + 3 x - 1$

Simplify the left side.

Remove parentheses.

$- y = x^{3} - 3 x^{2} + 3 x - 1$

Since the equation is not identical to the original equation, it is not symmetric to the x-axis.

Not symmetric to the x-axis

Check if the graph is symmetric about the y-axis by plugging in $- x$ for $x$ .

$y = {(- x)}^{3} - 3 {(- x)}^{2} + 3 (- x) - 1$

Simplify the right side.

Simplify each term.

Apply the product rule to $- x$ .

$y = {(- 1)}^{3} x^{3} - 3 {(- x)}^{2} + 3 (- x) - 1$

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

$y = - x^{3} - 3 {(- x)}^{2} + 3 (- x) - 1$

Apply the product rule to $- x$ .

$y = - x^{3} - 3 ({(- 1)}^{2} x^{2}) + 3 (- x) - 1$

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

$y = - x^{3} - 3 (1 x^{2}) + 3 (- x) - 1$

Multiply $x^{2}$ by $1$ .

$y = - x^{3} - 3 x^{2} + 3 (- x) - 1$

Multiply $- 1$ by $3$ .

$y = - x^{3} - 3 x^{2} - 3 x - 1$

Since the equation is not identical to the original equation, it is not symmetric to the y-axis.

Not symmetric to the y-axis

Check if the graph is symmetric about the origin by plugging in $- x$ for $x$ and $- y$ for $y$ .

$- y = {(- x)}^{3} - 3 {(- x)}^{2} + 3 (- x) - 1$

Solve for $y$ .

Simplify the left side.

Remove parentheses.

$- y = {(- x)}^{3} - 3 {(- x)}^{2} + 3 (- x) - 1$

Simplify the right side.

Simplify each term.

Apply the product rule to $- x$ .

$- y = {(- 1)}^{3} x^{3} - 3 {(- x)}^{2} + 3 (- x) - 1$

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

$- y = - x^{3} - 3 {(- x)}^{2} + 3 (- x) - 1$

Apply the product rule to $- x$ .

$- y = - x^{3} - 3 ({(- 1)}^{2} x^{2}) + 3 (- x) - 1$

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

$- y = - x^{3} - 3 (1 x^{2}) + 3 (- x) - 1$

Multiply $x^{2}$ by $1$ .

$- y = - x^{3} - 3 x^{2} + 3 (- x) - 1$

Multiply $- 1$ by $3$ .

$- y = - x^{3} - 3 x^{2} - 3 x - 1$

Simplify the left side.

Multiply $- (- y)$ .

Multiply $- 1$ by $- 1$ .

$1 y = - ({(- x)}^{3} - 3 {(- x)}^{2} + 3 (- x) - 1)$

Multiply $y$ by $1$ .

$y = - ({(- x)}^{3} - 3 {(- x)}^{2} + 3 (- x) - 1)$

Simplify the right side.

Simplify $- ({(- x)}^{3} - 3 {(- x)}^{2} + 3 (- x) - 1)$ .

Simplify each term.

Apply the product rule to $- x$ .

$y = - ({(- 1)}^{3} x^{3} - 3 {(- x)}^{2} + 3 (- x) - 1)$

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

$y = - (- x^{3} - 3 {(- x)}^{2} + 3 (- x) - 1)$

Apply the product rule to $- x$ .

$y = - (- x^{3} - 3 ({(- 1)}^{2} x^{2}) + 3 (- x) - 1)$

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

$y = - (- x^{3} - 3 (1 x^{2}) + 3 (- x) - 1)$

Multiply $x^{2}$ by $1$ .

$y = - (- x^{3} - 3 x^{2} + 3 (- x) - 1)$

Multiply $- 1$ by $3$ .

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

Apply the distributive property.

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

Simplify.

Multiply $- - x^{3}$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $x^{3}$ by $1$ .

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

Multiply $- 3$ by $- 1$ .

$y = x^{3} + 3 x^{2} - (- 3 x) - - 1$

Multiply $- 3$ by $- 1$ .

$y = x^{3} + 3 x^{2} + 3 x - - 1$

Multiply $- 1$ by $- 1$ .

$y = x^{3} + 3 x^{2} + 3 x + 1$

Since the equation is not identical to the original equation, it is not symmetric to the origin.

Not symmetric to the origin

Determine the symmetry.

Not symmetric to the x-axis

Not symmetric to the y-axis

Not symmetric to the origin

Find the Symmetry y=(x-1)^3

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