Algebra Examples

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

Step 1

There are three types of symmetry:

1. X-Axis Symmetry

2. Y-Axis Symmetry

3. Origin Symmetry

Step 2

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

Step 3

Use the Binomial Theorem.

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

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

Multiply $- 1$ by $3$ .

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

Step 4.2

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

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

Step 4.3

Multiply $3$ by $1$ .

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

Step 4.4

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

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

Step 5

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

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

Step 6

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

Not symmetric to the x-axis

Step 7

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

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

Step 8

Simplify each term.

Tap for more steps...

Step 8.1

Apply the product rule to $- x$ .

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

Step 8.2

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

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

Step 8.3

Apply the product rule to $- x$ .

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

Step 8.4

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

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

Step 8.5

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

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

Step 8.6

Multiply $- 1$ by $3$ .

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

Step 9

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

Not symmetric to the y-axis

Step 10

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

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

Step 11

Simplify each term.

Tap for more steps...

Step 11.1

Apply the product rule to $- x$ .

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

Step 11.2

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

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

Step 11.3

Apply the product rule to $- x$ .

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

Step 11.4

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

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

Step 11.5

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

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

Step 11.6

Multiply $- 1$ by $3$ .

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

Step 12

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

Not symmetric to the origin

Step 13

Determine the symmetry.

Not symmetric to the x-axis

Not symmetric to the y-axis

Not symmetric to the origin

Step 14