# MUIC Math

Mathematics at MUIC

wiki:graph_tran

# Graph Transformations

Suppose we know what the graph of a particular function looks like. We can use graph transformations to quickly sketch graphs of functions that are transformed from the original function. We can divide transformations into two types:

1. Transformations outside the original function.
2. Transformations inside the original function.

## Transformations outside the original function These transformations apply to the output of the function. In other words, they apply to the value of $f(\ldots)$. For example, $f(x)+5,\quad f(x)-2,\quad 3f(x),\quad -f(x)$

Geometrically, any transformation outside the function will have an effect in the vertical direction on the original graph. For example, the graph of $y=f(x)+3$ is obtained by shifting the graph of $y=f(x)$ upward by 3 units.

In the following table, we assume that $d>0$ and $c>1$.

New function Effect on original graph
$f(x)+d$ Shift up by $d$
$f(x)-d$ Shift down by $d$
$cf(x)$ Stretch vertically by $c$
$\frac{1}{c}f(x)$ Shrink vertically by $\frac{1}{c}$
$-f(x)$ Flip over the $x$-axis

Example 1. To sketch the graph of $y=x^2+2$, we note that it is $f(x)+2$ where $f(x)=x^2$. Thus the graph of $y=x^2+2$ is obtained by shifting the graph of $y=x^2$ upward by 2 units. Similarly, the graph of $y=x^2-2$ is obtained by shifting the graph of $y=x^2$ downward by 2 units. ## Transformations inside the original function These transformations apply to the input of the function. In other words, they apply to $(\ldots)$ inside $f(\ldots)$. For example,

$f(x+5),\quad f(x-2),\quad f(3x),\quad f(-x)$

Geometrically, any transformation inside the function will have an effect in the horizontal direction. For example, the graph of $y=f(x-5)$ is obtained by shifting the graph of $y=f(x)$ to the right by 5 units.

In the following table, we assume that $d>0$ and $c>1$.

New function Effect on original graph
$f(x+d)$ Shift left by $d$
$f(x-d)$ Shift right by $d$
$f(cx)$ Shrink horizontally by $\frac{1}{c}$
$\frac{1}{c}f(x)$ Stretch horizontally by $c$
$f(-x)$ Flip over the $y$-axis

Example 2. To sketch the graph of $y=(x+2)^2$, we note that it is $f(x+2)$ where $f(x)=x^2$. Thus the graph of $y=(x+2)^2$ is obtained by shifting the graph of $y=x^2$ to the left by 2 units. Similarly, the graph of $y=(x-2)^2$ is obtained by shifting the graph of $y=x^2$ to the right by 2 units. ## Transformations inside and outside the original function

We can also have transformations inside and outside the original function.

Example 3. The graph of $y=\sqrt{x-3}+2$ is obtained from the graph of $y=\sqrt{x}$ by shifting it right by 3 units and up by 2 units.  