Section 12.4 Lecture

Section 12.4: Reflecting, Stretching, and Shrinking of Function Graphs Worksheet 12.4

Recall: Transformations of prototype functions. (Round 2.)

Example: Stretching and Shrinking. Prototype function: $$f(x)=x^2$$

$$y=f(x) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, y=f(x)$$ $$y=4f(x) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, y=0.25f(x)$$

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

$$y=f(x) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, y=f(x)$$ $$y=-f(x) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, y=-4f(x)$$

Generally Speaking...

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

1) The graph of $y=cf(x)$ stretched $c$ units when $c \gt 1.$

2) The graph of $y=cf(x)$ is the graph of $y=f(x)$ shrunk down (or compressed) by $c$ when $0 \lt c \lt 1$.

3) The graph of $y=-f(x)$ is the graph of $y=f(x)$ reflected over the $x$ axis.

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

$$y=f(x)$$ $$y=-0.25f(x-2)+3$$

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 $$f(x)=0.5(x+2)^3+3.$$

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