To analyze and sketch the graph of the function [tex]\( r(x) = \frac{5 - 4x}{x - 3} \)[/tex], we will find the intercepts, relative extrema, points of inflection, and asymptotes.
### 1. Finding Intercepts
Y-Intercept:
To find the y-intercept, we set [tex]\( x = 0 \)[/tex]:
[tex]\[ r(0) = \frac{5 - 4(0)}{0 - 3} = \frac{5}{-3} = -\frac{5}{3} \][/tex]
So, the y-intercept is [tex]\((0, -\frac{5}{3})\)[/tex].
X-Intercept:
To find the x-intercept, we set [tex]\( r(x) = 0 \)[/tex]:
[tex]\[ \frac{5 - 4x}{x - 3} = 0 \][/tex]
The numerator must equal zero:
[tex]\[ 5 - 4x = 0 \][/tex]
[tex]\[ 4x = 5 \][/tex]
[tex]\[ x = \frac{5}{4} = 1.25 \][/tex]
So, the x-intercept is [tex]\((1.25, 0)\)[/tex].
### 2. Relative Extrema
There are no relative extrema for this function.
### 3. Points of Inflection
There are no points of inflection for this function.
### 4. Asymptotes
Vertical Asymptote:
A vertical asymptote occurs where the function is undefined, which happens when the denominator is zero:
[tex]\[ x - 3 = 0 \][/tex]
[tex]\[ x = 3 \][/tex]
So, the vertical asymptote is [tex]\( x = 3 \)[/tex].
Horizontal Asymptote:
To find the horizontal asymptote, we analyze the behavior as [tex]\( x \to \infty \)[/tex]:
[tex]\[ \lim_{x \to \infty} \frac{5 - 4x}{x - 3} = \lim_{x \to \infty} \frac{-4x}{x} = -4 \][/tex]
So, the horizontal asymptote is [tex]\( y = -4 \)[/tex].
### Summary of Findings:
- Intercepts:
- [tex]\( (0, -\frac{5}{3}) \)[/tex]
- [tex]\( (1.25, 0) \)[/tex]
- Relative Extrema: None
- Points of Inflection: None
- Asymptotes:
- Vertical: [tex]\( x = 3 \)[/tex]
- Horizontal: [tex]\( y = -4 \)[/tex]
So, the table of intercepts can be filled out as:
[tex]\[
\begin{array}{l}
\text{intercept (smaller } x \text{-value) } \quad (x, y)=(0, -5/3) \\
(x, y)=(1.25, 0) \\
(x, y)=\text{DNE} \\
(x, y)=\text{DNE} \\
(x, y)=\text{DNE} \\
\end{array}
\][/tex]
The equations of the asymptotes are:
[tex]\[ x = 3, y = -4 \][/tex]
To confirm our findings, you can use a graphing utility. The graph should display the intercepts at [tex]\((0, -\frac{5}{3})\)[/tex] and [tex]\((1.25, 0)\)[/tex], and the asymptotes [tex]\( x = 3 \)[/tex] and [tex]\( y = -4 \)[/tex].
