Question

advanced pre - calculus
mt 1 review - transformations
describe the transformations necessary to transform the graph of f(x) into that of g(x).

1. $f(x)=\frac{1}{x}$

$g(x)=\frac{1}{x - 1}+1$

1. $f(x)=sqrt{x}$

$g(x)=2sqrt{x - 1}$

1. $f(x)=\frac{1}{x}$

$g(x)=\frac{3}{x - 3}$

1. $f(x)=sqrt{x}$

$g(x)=\frac{1}{2}sqrt{x + 1}$

Explanation:

1.

Step1: Horizontal shift

For the function $y = \frac{1}{x - 1}+1$ compared to $y=\frac{1}{x}$, the $x$ in the denominator is replaced with $x - 1$. According to the rule of horizontal - shifts of functions $y = f(x - h)$, when $h>0$, the graph of $y = f(x)$ is shifted $h$ units to the right. Here $h = 1$, so the graph of $y=\frac{1}{x}$ is shifted 1 unit to the right.

Step2: Vertical shift

The $+ 1$ outside the fraction means that according to the rule of vertical - shifts of functions $y=f(x)+k$, when $k > 0$, the graph of $y = f(x)$ is shifted $k$ units up. Here $k = 1$, so the graph is shifted 1 unit up.

Step1: Horizontal shift

For the function $g(x)=2\sqrt{x - 1}$ compared to $f(x)=\sqrt{x}$, the $x$ inside the square - root is replaced with $x - 1$. According to the rule of horizontal - shifts of functions $y = f(x - h)$ with $h>0$, the graph of $y = f(x)$ is shifted $h$ units to the right. Here $h = 1$, so the graph of $y=\sqrt{x}$ is shifted 1 unit to the right.

Step2: Vertical stretch

The coefficient 2 in front of the square - root means that according to the rule of vertical stretches of functions $y = af(x)$ with $a>1$, the graph of $y = f(x)$ is vertically stretched by a factor of $a$. Here $a = 2$, so the graph is vertically stretched by a factor of 2.

Step1: Horizontal shift

For the function $g(x)=\frac{3}{x - 3}$ compared to $f(x)=\frac{1}{x}$, the $x$ in the denominator is replaced with $x - 3$. According to the rule of horizontal - shifts of functions $y = f(x - h)$ with $h>0$, the graph of $y = f(x)$ is shifted $h$ units to the right. Here $h = 3$, so the graph of $y=\frac{1}{x}$ is shifted 3 units to the right.

Step2: Vertical stretch

The coefficient 3 in the numerator means that according to the rule of vertical stretches of functions $y = af(x)$ (where $f(x)=\frac{1}{x}$), when $a = 3>1$, the graph of $y = f(x)$ is vertically stretched by a factor of 3.

Answer:

The graph of $f(x)=\frac{1}{x}$ is shifted 1 unit to the right and 1 unit up to get the graph of $g(x)=\frac{1}{x - 1}+1$.

2.