Math Curve sketching Symmetry behavior

# Symmetry behavior

The symmetry behavior provides information about whether the graph of a function is symmetric about an axis or a point.

## Circular symmetry

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### Remember

For axis symmetry to the y-axis must apply:
$f(-x)=f(x)$

### Example

Check if $f(x)=x^4$ is axisymmetric to the y-axis.

$f(\color{red}{-x})=(\color{red}{-x})^4=x^4$
$f(x)=x^4$
=> axisymmetric to the y-axis, because $f(-x)=f(x)$
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### Extra

The graph of a function can also be axisymmetric to any axis if:
$f(c-x)=f(c+x)$

$c$ is the equation of the axis.

## Point reflection

!

### Remember

For point symmetry to the origin must apply:
$f(-x)=-f(x)$

### Example

Check if $f(x)=x^3$ is point symmetric to the origin.

$f(\color{red}{-x})=(\color{red}{-x})^3=-x^3$
$-f(x)=-x^3$
=> point symmetric to the origin because $f(-x)=-f(x)$
i

### Extra

The graph of a function can also be point-symmetric to any point if:
$f(x_0-x)-y_0=$ $-(f(x_0+x)+{y}_0)$

$x_0$ and $y_0$ are the coordinates of the point.