Find the Symmetry y=4-x^2

$y = 4 - x^{2}$

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 $y$ .

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

Simplify the left side.

Remove parentheses.

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

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 $x$ .

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

Simplify the right side.

Simplify each term.

Apply the product rule to $- x$ .

$y = 4 - ({(- 1)}^{2} x^{2})$

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

Move ${(- 1)}^{2}$ .

$y = 4 + {(- 1)}^{2} \cdot - 1 x^{2}$

Multiply ${(- 1)}^{2}$ by $- 1$ .

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

$y = 4 + {(- 1)}^{2} \cdot {(- 1)}^{1} x^{2}$

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

$y = 4 + {(- 1)}^{2 + 1} x^{2}$

Add $2$ and $1$ .

$y = 4 + {(- 1)}^{3} x^{2}$

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

$y = 4 - x^{2}$

Since the equation is identical to the original equation, it is symmetric to the y-axis.

Symmetric with respect to the y-axis

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

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

Solve for $y$ .

Simplify the left side.

Remove parentheses.

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

Simplify the right side.

Simplify each term.

Apply the product rule to $- x$ .

$- y = 4 - ({(- 1)}^{2} x^{2})$

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

Move ${(- 1)}^{2}$ .

$- y = 4 + {(- 1)}^{2} \cdot - 1 x^{2}$

Multiply ${(- 1)}^{2}$ by $- 1$ .

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

$- y = 4 + {(- 1)}^{2} \cdot {(- 1)}^{1} x^{2}$

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

$- y = 4 + {(- 1)}^{2 + 1} x^{2}$

Add $2$ and $1$ .

$- y = 4 + {(- 1)}^{3} x^{2}$

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

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

Simplify the left side.

Multiply $- (- y)$ .

Multiply $- 1$ by $- 1$ .

$1 y = - (4 - {(- x)}^{2})$

Multiply $y$ by $1$ .

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

Simplify the right side.

Simplify $- (4 - {(- x)}^{2})$ .

Simplify each term.

Apply the product rule to $- x$ .

$y = - (4 - ({(- 1)}^{2} x^{2}))$

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

Move ${(- 1)}^{2}$ .

$y = - (4 + {(- 1)}^{2} \cdot - 1 x^{2})$

Multiply ${(- 1)}^{2}$ by $- 1$ .

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

$y = - (4 + {(- 1)}^{2} \cdot {(- 1)}^{1} x^{2})$

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

$y = - (4 + {(- 1)}^{2 + 1} x^{2})$

Add $2$ and $1$ .

$y = - (4 + {(- 1)}^{3} x^{2})$

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

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

Simplify by multiplying through.

Apply the distributive property.

$y = - 1 \cdot 4 - - x^{2}$

Multiply $- 1$ by $4$ .

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

Multiply $- - x^{2}$ .

Multiply $- 1$ by $- 1$ .

$y = - 4 + 1 x^{2}$

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

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

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.

Symmetric with respect to the y-axis

Find the Symmetry y=4-x^2

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