Symmetry behavior
The symmetry behavior provides information about whether the graph of a function is symmetric about an axis or a point.
Circular symmetry
!
Remember
For axis symmetry to the y-axis must apply:
$f(-x)=f(x)$
$f(-x)=f(x)$
Example
Check if $f(x)=x^4$ is axisymmetric to the y-axis.
$f(\color{red}{-x})=(\color{red}{-x})^4=x^4$$f(x)=x^4$
=> axisymmetric to the y-axis, because $f(-x)=f(x)$
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Extra
The graph of a function can also be axisymmetric to any axis if:
$f(c-x)=f(c+x)$
$c$ is the equation of the axis.
$f(c-x)=f(c+x)$
$c$ is the equation of the axis.
Point reflection
!
Remember
For point symmetry to the origin must apply:
$f(-x)=-f(x)$
$f(-x)=-f(x)$
Example
Check if $f(x)=x^3$ is point symmetric to the origin.
$f(\color{red}{-x})=(\color{red}{-x})^3=-x^3$$-f(x)=-x^3$
=> point symmetric to the origin because $f(-x)=-f(x)$
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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.
$f(x_0-x)-y_0=$ $-(f(x_0+x)+{y}_0)$
$x_0$ and $y_0$ are the coordinates of the point.