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 $y=x^3$

- first by shrinking $y=\frac{1}{3}x^3$
- 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 $y=x^2$

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