HaCkEd By KaSpEr511

Function Transformations

August 12th, 2009

We will discuss about how to obtain a graph of a function by transforming the graph of the standard functions. We will discuss problems from 6th Edition Stewart’s Calculus book, section 1.2.

Graph the function below by starting with the graph of one of the standard functions, and then applying the appropriate transformations.
10. $y=1-x^2$
12. $y=x^2-4x+3$
14. $y=4\sin 3x$
16. $y=\frac{1}{x-4}$
18. $y=(x+2)^4+3$
20. $y=1+\sqrt[3]{x-1}$
22. $y=\frac{1}{4}\tan(x-\frac{\pi}{4})$
24. $y=|x^2-2x|$

First, lets try problem 10. Start with the graph of the standard function $f(x)=x^2$ . We can perfom transformations

$\usepackage[all]{xy}\xymatrix{f(x)=x^2\ar[rr]^{\text{flip about}}_{\text{ the x- axis }}&&-f(x)=-x^2\ar[rr]^{\text{shift up}}_{\text{1 unit}}&&-f(x)+1=-x^2+1}$

Starting from the graph of $y=x^2$ , we flip the graph about the $x$ -axis to get the graph of $y=-x^2$ and then after that we shift the graph one unit upward to get the graph of $y=-x^2+1=1-x^2$ .

kuadrat2

For problem 14, we need to figure out what transformations that can bring $y=\sin x$ into $y=4\sin(3x)$ . One way to do that is by using the following series of transformations:

$\usepackage[all]{xy}\xymatrix{f(x)=\sin x\ar[rr]^{\text{shrink horiz.}}_{\text{ factor: 1/3}}&&f(3x)=\sin(3x)\ar[rr]^{\text{stretch vert.}}_{\text{factor: 4}}&&4f(3x)=4\sin(3x)}$

The first transformation shrink the graph of $y=\sin x$ with a factor of $1/3$ . In the graph of $y=\sin x$ the interval $[-\pi,\pi]$ only contains one full wave. But after stretching, this interval contains three full wave. After that, we stretch vertically so that the amplitude of $y=4\sin(3x)$ is four times of the amplitude of $y=\sin(3x)$ .