Transformation of graphs

Graph of $y = | f (x) |$

$y = {\begin{cases} f (x), f (x) \geq 0 \\ - f (x), f (x) < 0 \end{cases}$

Graph of $y = | f (x) |$ based on given $y = f (x)$ :

$y = f (x)$

Rendered by QuickLaTeX.com

$y = | f (x) |$

Rendered by QuickLaTeX.com

Graph of $y = f (| x |)$

$f (| x |) = {\begin{cases} f (x), x \geq 0 \\ f (- x), x < 0 \end{cases}$

Rendered by QuickLaTeX.com

Graph of $y = \frac{1}{f (x)}$

Rendered by QuickLaTeX.com

$y = f (x)$	$y = \frac{1}{f (x)}$
For $f (x) = 0$ is that the x-intercept	$y = \frac{1}{f (x)}$ does not exist x-intercept of $y = f (x)$ becomes the vertical asymtotes
Vertical asymtotes of $f (x)$	Vertical asymtotes of $y = f (x)$ becomes x-intercept, but does not touch x-axis (indicate by hollow circles)
Maximum point at $(x, f (x))$	Maximum point of $y = f (x)$ becomes minimum point at $(x, \frac{1}{f (x)})$
Minimum point at $(x, f (x))$	Minimum point of $y = f (x)$ becomes maximum point at $(x, \frac{1}{f (x)})$
When $f (x) = 1$	$\frac{1}{f (x)} = 1$ common point
When $f (x) = - 1$	$\frac{1}{f (x)} = - 1$ common point
When $f (x) > 0$	$\frac{1}{f (x)} > 0$
When $f (x) < 0$	$\frac{1}{f (x)} < 0$
When $f (x)$ increases	$\frac{1}{f (x)}$ decreases
When $f (x)$ decreases	$\frac{1}{f (x)}$ increases

Graph of $y^{2} = f (x)$

Graph of $y^{2} = f (x)$ based on given $y = f (x)$ :

Rendered by QuickLaTeX.com

$y = f (x)$	$y = \sqrt{f (x)}$
For $f (x) < 0$	$\sqrt{f (x)}$ is not defined, no graph
For any point $(h, k)$ where $k > 0$	The point will become $(h, \sqrt{k})$
Horizontal asymptote $y = c$	Horizontal asymtote will become $y = \sqrt{c}$
Vertical asymptote $x = a$	Vertical asymptote $x = a$ , no change
When $f (x) = 0$	$\sqrt{f (x)} = 0$
When $f (x) = 1$	$\sqrt{f (x)} = 1$
When $0 < f (x) < 1$	$\sqrt{f (x)} > f (x)$ , that means the graph will be higher
When $f (x) > 1$	$\sqrt{f (x)} < f (x)$ , that means the graph will be lower