# Rational Functions and Graphs

### Transformation of graphs

###### Transformation of Graphs

Graph of $$y = \left| f(x) \right|$$

$$y = \begin{cases} f(x), \quad f(x) \geq 0 \\[2ex] - f(x), \quad f(x) < 0 \end{cases}$$

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

• part of the graph above x-axis does not change
• part of the graph below x-axis reflected about the x-axis

$$y = f(x)$$

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

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

$$f(|x|) = \begin{cases} f(x), \quad x \geq 0 \\[2ex] f(-x), \quad x < 0 \end{cases}$$

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

 $$y = f(x)$$ $$y = \dfrac {1}{f(x)}$$ For $$f(x) = 0$$ is that the x-intercept $$y = \dfrac {1}{f(x)}$$ 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 $$\left(x,\dfrac {1}{f(x)}\right)$$ Minimum point at $$(x,f(x))$$ Minimum point of $$y = f(x)$$ becomes maximum point at $$\left(x,\dfrac {1}{f(x)}\right)$$ When $$f(x) = 1$$ $$\dfrac {1}{f(x)} = 1$$ common point When $$f(x) = -1$$ $$\dfrac {1}{f(x)} = -1$$ common point When $$f(x) > 0$$ $$\dfrac {1}{f(x)} > 0$$ When $$f(x) < 0$$ $$\dfrac {1}{f(x)} < 0$$ When $$f(x)$$ increases $$\dfrac {1}{f(x)}$$ decreases When $$f(x)$$ decreases $$\dfrac {1}{f(x)}$$ increases

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

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

• $$y^2 = f(x) \rightarrow y = \pm \sqrt{f(x)}$$
• sketch the graph of $$y = \sqrt{f(x)}$$ based on $$y = f(x)$$
• sketch the graph of $$y = -\sqrt{f(x)}$$ as a reflection of $$y = \sqrt{f(x)}$$

 $$y = f(x)$$ $$y = \sqrt{f(x)}$$ 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