Sketch the graph of the following rational functions
`y=(2x+1)/(x-3)`

Answer

To sketch the graph of the rational function \( y = \frac{2x + 1}{x - 3} \), we will follow these steps:

### Step 1: Identify the asymptotes
1. **Vertical Asymptote**: Set the denominator equal to zero to find the vertical asymptote.
\[ x - 3 = 0 \implies x = 3 \]
This means there is a vertical asymptote at \( x = 3 \).

2. **Horizontal Asymptote**: For rational functions where the degrees of the numerator and denominator are the same, the horizontal asymptote can be found by taking the ratio of the leading coefficients.
\[ \text{Horizontal Asymptote} = \frac{2}{1} = 2 \]
So, there is a horizontal asymptote at \( y = 2 \).

### Step 2: Find the intercepts
1. **Y-intercept**: Set \( x = 0 \) to find the y-intercept.
\[ y = \frac{2(0) + 1}{0 - 3} = \frac{1}{-3} = -\frac{1}{3} \]
Thus, the y-intercept is at \( (0, -\frac{1}{3}) \).

2. **X-intercept**: Set \( y = 0 \) to find the x-intercept.
\[ 2x + 1 = 0 \implies 2x = -1 \implies x = -\frac{1}{2} \]
Thus, the x-intercept is at \( (-\frac{1}{2}, 0) \).

### Step 3: Analyze the behavior around the asymptotes
- As \( x \) approaches \( 3 \) from the left (\( x \to 3^- \)), \( y \to -\infty \) (since the denominator approaches zero negatively).
- As \( x \) approaches \( 3 \) from the right (\( x \to 3^+ \)), \( y \to +\infty \) (since the denominator approaches zero positively).
- As \( x \to \infty \), \( y \) approaches the horizontal asymptote \( y = 2 \).
- As \( x \to -\infty \), \( y \) also approaches the horizontal asymptote \( y = 2 \).

### Step 4: Sketch the graph
- Plot the vertical asymptote at \( x = 3 \).
- Plot the horizontal asymptote at \( y = 2 \).
- Plot the intercepts \( (0, -\frac{1}{3}) \) and \( (-\frac{1}{2}, 0) \).
- Draw the curve approaching the asymptotes and passing through the intercepts.

### Final Graph
The graph will have two branches:
- One in the first quadrant (approaching the horizontal asymptote \( y = 2 \) as \( x \to \infty \) and going to \( +\infty \) as \( x \to 3^+ \)).
- Another in the third quadrant (approaching the horizontal asymptote \( y = 2 \) as \( x \to -\infty \) and going to \( -\infty \) as \( x \to 3^- \)).