Section 12.3 Lecture

Section 12.3: Vertical and Horizontal Transformations of Function Graphs Worksheet 12.3

Recall: Transformations of prototype functions.

Example: Vertical Transformations. Prototype function: $$f(x)=x^2$$

$$y=f(x) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, y=f(x)$$ $$y=f(x)+2 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, y=f(x)-3$$

Example: Horizontal Transformations. Prototype function: $$f(x)=x^2$$

$$y=f(x) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, y=f(x)$$ $$y=f(x+2) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, y=f(x-3)$$

Generally Speaking...

Suppose you have a graph of $y=f(x).$ Then:

1) The graph of $y=f(x)+c$ is the graph of $y=f(x)$ shifted up $c$ units.

2) The graph of $y=f(x)-c$ is the graph of $y=f(x)$ shifted down $c$ units.

3) The graph of $y=f(x+c)$ is the graph of $y=f(x)$ shifted to the left $c$ units.

4) The graph of $y=f(x-c)$ is the graph of $y=f(x)$ shifted to the right $c$ units.

Example: Combining Transformations. Prototype function: $$f(x)=x^2$$

$$y=f(x)$$ $$y=f(x-2)+3$$

Example : Consider the basic (prototype) function $f(x)=|x|.$ Write the transformation of $y=f(x)$ which gives the graph below.

Example: Use transformations to graph the function $$f(x)=\sqrt{x+2}+3.$$

A List of Prototype Functions We Know $$\begin{array}{lr} f(x)=x & \mbox{linear}\\ f(x)=x^2 &\mbox{squaring}\\ f(x)=x^3 &\mbox{cubic}\\ f(x)=\sqrt{x} &\mbox{square root}\\ f(x)=\sqrt[3]{x} &\mbox{cube root}\\ f(x)=|x| &\mbox{absolute value}\\ f(x)=\frac{1}{x} &\mbox{reciprocal}\\ f(x)=b^x &\mbox{exponential}\\ f(x)=\log_b(x) &\mbox{logarithmic}\\ \end{array}$$