**Transformations of the graph of a function $y=f(x)$:**

The transformations we consider include reflection, horizontal shifting, vertical shifting, stretching and shrinking. Here we discuss reflection and related symmetries. In the following notes, we will discuss the rest of transformations.

**Reflection about the $x$-axis:**

*Example*. Consider $y=\sqrt{x}$. If you plug in $-y$ for $y$, we obtain $-y=\sqrt{x}$ or $y=-\sqrt{x}$. Let us plot them all togather.

If you fold the two plots along the $x$-axis, the graphs of $y=\sqrt{x}$ and $y=-\sqrt{x}$ would overlap. So we can say that $y=-\sqrt{x}$ is the reflection of $y=\sqrt{x}$ with respect to $x$-axis, and vice versa.

In general, the graph of $y=-f(x)$ is the reflection of the graph of $y=f(x)$ with respect to the $x$-axis.

**Reflection about the $y$-axis:**

*Example*: Now this time we consider the function $y=2x+1$. If you plug in $-x$ for $x$, we obtain $y=-2x+1$. Let us plot them both again.

You can clearly see that if you fold the two plots along teh $y$-axis, they would overlap. So we can say that the graph of $y=-2x+1$ is the reflection of the graph of $y=2x+1$.

In general, the graph of $y=f(-x)$ is the reflection of the graph of $y=f(x)$ with respect to the $y$-axis.

**Symmetry about the $y$-axis:**

*Example*. Consider the function $f(x)=x^2-1$. If you plug in $-x$ for $x$, $f(-x)=(-x)^2-1=x^2-1=f(x)$. So the function is unchanged by the reflection with respect to the $y$-axis.

In general, if the function $y=f(x)$ satisfies the property $$f(-x)=f(x),$$ we say that the graph of $f(x)$ is *symmetric about the $y$-axis*.

**Symmetry about the $x$-axis:**

*Example*: Consider $x=y^2$. This is a function of the form $x=f(y)$. If you plug in $-y$ for $y$, $(-y)^2=y^2=x$. So the function still does not change.

In general, if the function $x=f(y)$ satisfies the property $$f(-y)=f(y),$$ we say that the graph of $x=f(y)$ is *symmetric about the $x$-axis*.

**Symmetry about the origin:**

*Example*: Consider the function $y=x^3$. If you plug in $-x$ for $x$ and $-y$ for $y$, the equation does not change.

The effect of $x\mapsto -x$ and $y\mapsto -y$ in the equation $y=f(x)$ amounts to rotating the graph of $y=f(x)$ about the origin by $180^\circ$. If the graph of $y=f(x)$ does not change after the rotation of the graph about the origin by $180^\circ$ (or equivalently $-y=f(-x)$ is the same as the equation $y=f(x)$), then we say that the graph is *symmetric about the origin*.

**Even Functions and Odd Functions:**

*Definition*. A function $y=f(x)$ is a called an *even function* if it satisfies $f(-x)=f(x)$. The reason such function is called even is that one of the simplest examples of even functions is of the form $x^{\mbox{even integer}}$, for instance, $x^2$, $x^4$, $x^6$, etc.

*Definition*. A function $y=f(x)$ is a called an *odd function* if it satisfies $f(-x)=-f(x)$. The reason such function is called odd is that one of the simplest examples of odd functions is of the form $x^{\mbox{odd integer}}$, for instance, $x$, $x^3$, $x^5$, etc.

From the equations we can easily see that the graph of an even functions is symmetric about the $y$-axis, while the graph of an odd function is symmetric about the origin.

*Example*. $f(x)=5x^7-6x^3-2x$ is an odd function because \begin{align*}f(-x)&=5(-x)^7-6(-x)^3-2(-x)\\&=-5x^7+6x^3+2x\\&=-f(x).\end{align*}

*Remark.* Note that a constant multiple of an odd function is an odd function and the sum of an odd functions is an odd function. Since each term $x^7$, $x^3$, $x$ are odd functions, we can also see that $f(x)$ is an odd function.

*Example*. $g(x)=5x^6-3x^2-7$ is an even function because \begin{align*}

g(-x)&=5(-x)^6-3(-x)^2-7\\&=5x^6-3x^2-7\\&=g(x).

\end{align*}

*Remark*. Note that a constant function (number) is an even function, a constant multiple of an even function is an even function and the sum of an even functions is an even function. Since each term $x^6$, $x^2$, $-7$ are even functions, we can also see that $g(x)$ is an even function.

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