In this notes, we study the remaining transformations of a function, namely horizontal shifting, vertical shifting, and dilation (stretching and shrinking). We also discuss how those transformations can be used to sketch the graph of a complicated function from its most basic form.

Horizontal Shifting (translation to the right or left):

Example. Let us plot $y=x^2$, $y=(x-3)^2$, and $y=(x+3)^2$ all together. As you can see clearly, the graph of $y=(x-3)^2$ is 3 units translation to the right of the graph of $y=x^2$, while the graph of $y=(x+3)^2$ is 3 units translations to the left of the graph of $y=x^2$.

Let $a>0$. Then:

The graph of $y=f(x-a)$ is $a$ units translation to the right of the graph of $y=f(x)$;

The graph of $y=f(x+a)$ is $a$ units translation to the left of the graph of $y=f(x)$.

Vertical Shifting (translation upward or downward):

Example. Let us plot $y=x^2$, $y=x^2+3$, and $y=x^2-3$ all together. As you can see clearly, the graph of $y=x^2+3$ is 3 units translation upward of the graph of $y=x^2$, while the graph of $y=x^2-3$ is 3 units translations downward of the graph of $y=x^2$.

Let $a>0$. Then:

The graph of $y=f(x)+a$ is $a$ units translation to upward of the graph of $y=f(x)$;

The graph of $y=f(x)-a$ is $a$ units translation to downward of the graph of $y=f(x)$.

Stretching and Shrinking:

Example. Let us compare the graphs of $y=x^2$, $y=2x^2$ and $y=\frac{1}{2}x^2$. The graph of $y=2x^2$ looks thinner than the graph of $y=x^2$, while the graph of $y=\frac{1}{2}x^2$ looks wider than the graph of $y=x^2$. The graph of $y=2x^2$ can be considered as a (vetical) stretching of the graph of $y=x^2$. Similarly, the graph of $=\frac{1}{2}x^2$ can be considered as a (vetical) shrinking of the graph of $y=x^2$.

If $a>0$, the graph of $y=af(x)$ is a (vertical) stretching of the graph of $y=f(x)$;

If $0<a<1$, the graph of $y=af(x)$ is a (vertical) shrinking of the graph of $y=f(x)$.

Graphing functions using transformations: Sometimes we can sketch graph of a function by analyzing transformations involved such as reflection, stretching/shrinking, horizontal/vertical shifting.

Example. Sketch the graph of $f(x)=\frac{1}{3}x^3+2$ using transformations.

Solution. The most basic form of the function is $y=x^3$. So we obtain the graph of $f(x)$ from $y=x^3$

1. first by shrinking $y=\frac{1}{3}x^3$
2. and then translate the resultign function 2 units upward $y=\frac{1}{3}x^3+2$.

Example. Sketch the graph of $g(x)=-(x-3)^2+5$ using transformations.

Solution. The most basic form of the function is $y=x^2$. We obtain the graph of $g(x)$ from $y=x^2$

1. first by reflection $y=-x^2$
2. next translate the resulting function 3 units to the right
3. and finally translate the resulting function 5 units upward.

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.

If you are beginning to study freshmen calculus, it would be definitely a good idea to review some of the important formulas from precalculus before you get into more serious stuff in calculus. My top recommendation of such formulas would be the following.

Expansion of Polynomials

1. \((a+b)^2=a^2+2ab+b^2\)
2. \((a-b)^2=a^2-2ab+b^2\)
3. \((a+b)^3=a^3+3a^2b+3ab^2+b^3\)
4. \((a-b)^3=a^3-3a^2b+3ab^2-b^3\)

Factorization of Polynomials

1. \(a^2-b^2=(a+b)(a-b)\)
2. \(a^3-b^3=(a-b)(a^2+ab+b^2)\)
3. \(a^3+b^3=(a+b)(a^2-ab+b^2)\)

Trigonometric Identities

1. \(\cos^2\theta+\sin^2\theta=1\)
2. \(\tan^2\theta+1=\sec^2\theta\)

Sine Sum and Difference Formulas

1. \(\sin(\theta_1+\theta_2)=\sin\theta_1\cos\theta_2+\cos\theta_1\sin\theta_2\)
2. \(\sin(\theta_1-\theta_2)=\sin\theta_1\cos\theta_2-\cos\theta_1\sin\theta_2\)

Sine Double Angle Formula \[\sin2\theta=2\sin\theta\cos\theta\]

Cosine Sum and Difference Formulas

1. \(\cos(\theta_1+\theta_2)=\cos\theta_1\cos\theta_2-\sin\theta_1\sin\theta_2\)
2. \(\cos(\theta_1-\theta_2)=\cos\theta_1\cos\theta_2+\sin\theta_1\sin\theta_2\)

Cosine Double Angle Formula \begin{eqnarray*}\cos2\theta&=&\cos^2\theta-\sin^2\theta\\&=&2\cos^2\theta-1\\&=&1-2\sin^2\theta\end{eqnarray*}

From this Cosine Double Angle Formula, we obtain Half Angle Formulas.

Half Angle Formulas

1. \(\cos^2\theta=\displaystyle\frac{1+\cos2\theta}{2}\) or equivalently \(\cos\theta=\pm\sqrt{\displaystyle\frac{1+\cos2\theta}{2}}\)
2. \(\sin^2\theta=\displaystyle\frac{1-\cos2\theta}{2}\) or equivalently \(\sin\theta=\pm\sqrt{\displaystyle\frac{1-\cos2\theta}{2}}\)

For the above formulas from trigonometry, there are actually only three formulas you need to remember. They are $\cos^2\theta+\sin^2\theta=1$ and sine and cosine sum formulas. The rest of the formulas from trigonometry that are listed above can be stemmed from these three formulas.