Find the Symmetry x=(y-2)^2-8

$x = {(y - 2)}^{2} - 8$

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

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

$x = {((- y) - 2)}^{2} - 8$

Step 4

Simplify the right side.

Simplify ${((- y) - 2)}^{2} - 8$ .

Simplify each term.

Rewrite ${(- y - 2)}^{2}$ as $(- y - 2) (- y - 2)$ .

$x = (- y - 2) (- y - 2) - 8$

Expand $(- y - 2) (- y - 2)$ using the FOIL Method.

Apply the distributive property.

$x = - y (- y - 2) - 2 (- y - 2) - 8$

Apply the distributive property.

$x = - y (- y) - y \cdot - 2 - 2 (- y - 2) - 8$

Apply the distributive property.

$x = - y (- y) - y \cdot - 2 - 2 (- y) - 2 \cdot - 2 - 8$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$x = - 1 \cdot - 1 y \cdot y - y \cdot - 2 - 2 (- y) - 2 \cdot - 2 - 8$

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$x = - 1 \cdot - 1 (y \cdot y) - y \cdot - 2 - 2 (- y) - 2 \cdot - 2 - 8$

Multiply $y$ by $y$ .

$x = - 1 \cdot - 1 y^{2} - y \cdot - 2 - 2 (- y) - 2 \cdot - 2 - 8$

Multiply $- 1$ by $- 1$ .

$x = 1 y^{2} - y \cdot - 2 - 2 (- y) - 2 \cdot - 2 - 8$

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

$x = y^{2} - y \cdot - 2 - 2 (- y) - 2 \cdot - 2 - 8$

Multiply $- 2$ by $- 1$ .

$x = y^{2} + 2 y - 2 (- y) - 2 \cdot - 2 - 8$

Multiply $- 1$ by $- 2$ .

$x = y^{2} + 2 y + 2 y - 2 \cdot - 2 - 8$

Multiply $- 2$ by $- 2$ .

$x = y^{2} + 2 y + 2 y + 4 - 8$

Add $2 y$ and $2 y$ .

$x = y^{2} + 4 y + 4 - 8$

Subtract $8$ from $4$ .

$x = y^{2} + 4 y - 4$

Step 5

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 6

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

$- x = {(y - 2)}^{2} - 8$

Step 7

Simplify.

Simplify the left side.

Remove parentheses.

$- x = {(y - 2)}^{2} - 8$

Simplify the right side.

Simplify ${(y - 2)}^{2} - 8$ .

Simplify each term.

Rewrite ${(y - 2)}^{2}$ as $(y - 2) (y - 2)$ .

$- x = (y - 2) (y - 2) - 8$

Expand $(y - 2) (y - 2)$ using the FOIL Method.

Apply the distributive property.

$- x = y (y - 2) - 2 (y - 2) - 8$

Apply the distributive property.

$- x = y \cdot y + y \cdot - 2 - 2 (y - 2) - 8$

Apply the distributive property.

$- x = y \cdot y + y \cdot - 2 - 2 y - 2 \cdot - 2 - 8$

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

$- x = y^{2} + y \cdot - 2 - 2 y - 2 \cdot - 2 - 8$

Move $- 2$ to the left of $y$ .

$- x = y^{2} - 2 \cdot y - 2 y - 2 \cdot - 2 - 8$

Multiply $- 2$ by $- 2$ .

$- x = y^{2} - 2 y - 2 y + 4 - 8$

Subtract $2 y$ from $- 2 y$ .

$- x = y^{2} - 4 y + 4 - 8$

Subtract $8$ from $4$ .

$- x = y^{2} - 4 y - 4$

Step 8

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 9

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

$- x = {((- y) - 2)}^{2} - 8$

Step 10

Solve for $x$ .

Simplify the left side.

Remove parentheses.

$- x = {((- y) - 2)}^{2} - 8$

Simplify the right side.

Simplify ${((- y) - 2)}^{2} - 8$ .

Simplify each term.

Rewrite ${(- y - 2)}^{2}$ as $(- y - 2) (- y - 2)$ .

$- x = (- y - 2) (- y - 2) - 8$

Expand $(- y - 2) (- y - 2)$ using the FOIL Method.

Apply the distributive property.

$- x = - y (- y - 2) - 2 (- y - 2) - 8$

Apply the distributive property.

$- x = - y (- y) - y \cdot - 2 - 2 (- y - 2) - 8$

Apply the distributive property.

$- x = - y (- y) - y \cdot - 2 - 2 (- y) - 2 \cdot - 2 - 8$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$- x = - 1 \cdot - 1 y \cdot y - y \cdot - 2 - 2 (- y) - 2 \cdot - 2 - 8$

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$- x = - 1 \cdot - 1 (y \cdot y) - y \cdot - 2 - 2 (- y) - 2 \cdot - 2 - 8$

Multiply $y$ by $y$ .

$- x = - 1 \cdot - 1 y^{2} - y \cdot - 2 - 2 (- y) - 2 \cdot - 2 - 8$

Multiply $- 1$ by $- 1$ .

$- x = 1 y^{2} - y \cdot - 2 - 2 (- y) - 2 \cdot - 2 - 8$

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

$- x = y^{2} - y \cdot - 2 - 2 (- y) - 2 \cdot - 2 - 8$

Multiply $- 2$ by $- 1$ .

$- x = y^{2} + 2 y - 2 (- y) - 2 \cdot - 2 - 8$

Multiply $- 1$ by $- 2$ .

$- x = y^{2} + 2 y + 2 y - 2 \cdot - 2 - 8$

Multiply $- 2$ by $- 2$ .

$- x = y^{2} + 2 y + 2 y + 4 - 8$

Add $2 y$ and $2 y$ .

$- x = y^{2} + 4 y + 4 - 8$

Subtract $8$ from $4$ .

$- x = y^{2} + 4 y - 4$

Simplify the left side.

Multiply $- (- x)$ .

Multiply $- 1$ by $- 1$ .

$1 x = - ({((- y) - 2)}^{2} - 8)$

Multiply $x$ by $1$ .

$x = - ({((- y) - 2)}^{2} - 8)$

Simplify the right side.

Simplify $- ({((- y) - 2)}^{2} - 8)$ .

Simplify each term.

Rewrite ${(- y - 2)}^{2}$ as $(- y - 2) (- y - 2)$ .

$x = - ((- y - 2) (- y - 2) - 8)$

Expand $(- y - 2) (- y - 2)$ using the FOIL Method.

Apply the distributive property.

$x = - (- y (- y - 2) - 2 (- y - 2) - 8)$

Apply the distributive property.

$x = - (- y (- y) - y \cdot - 2 - 2 (- y - 2) - 8)$

Apply the distributive property.

$x = - (- y (- y) - y \cdot - 2 - 2 (- y) - 2 \cdot - 2 - 8)$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$x = - (- 1 \cdot - 1 y \cdot y - y \cdot - 2 - 2 (- y) - 2 \cdot - 2 - 8)$

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$x = - (- 1 \cdot - 1 (y \cdot y) - y \cdot - 2 - 2 (- y) - 2 \cdot - 2 - 8)$

Multiply $y$ by $y$ .

$x = - (- 1 \cdot - 1 y^{2} - y \cdot - 2 - 2 (- y) - 2 \cdot - 2 - 8)$

Multiply $- 1$ by $- 1$ .

$x = - (1 y^{2} - y \cdot - 2 - 2 (- y) - 2 \cdot - 2 - 8)$

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

$x = - (y^{2} - y \cdot - 2 - 2 (- y) - 2 \cdot - 2 - 8)$

Multiply $- 2$ by $- 1$ .

$x = - (y^{2} + 2 y - 2 (- y) - 2 \cdot - 2 - 8)$

Multiply $- 1$ by $- 2$ .

$x = - (y^{2} + 2 y + 2 y - 2 \cdot - 2 - 8)$

Multiply $- 2$ by $- 2$ .

$x = - (y^{2} + 2 y + 2 y + 4 - 8)$

Add $2 y$ and $2 y$ .

$x = - (y^{2} + 4 y + 4 - 8)$

Simplify by multiplying through.

Subtract $8$ from $4$ .

$x = - (y^{2} + 4 y - 4)$

Apply the distributive property.

$x = - y^{2} - (4 y) - - 4$

Simplify.

Multiply $4$ by $- 1$ .

$x = - y^{2} - 4 y - - 4$

Multiply $- 1$ by $- 4$ .

$x = - y^{2} - 4 y + 4$

Step 11

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

Not symmetric to the origin

Step 12

Determine the symmetry.

Not symmetric to the x-axis

Not symmetric to the y-axis

Not symmetric to the origin

Step 13

Find the Symmetry x=(y-2)^2-8

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