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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.