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

Answer

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

### Step 1: Identify the intercepts
1. **Y-intercept**: Set \( x = 0 \) in the function.
\[ y = \frac{0 + 3}{0 - 2} = \frac{3}{-2} = -\frac{3}{2} \]
So, the y-intercept is at the point \( (0, -\frac{3}{2}) \).

2. **X-intercept**: Set \( y = 0 \) in the function.
\[ 0 = \frac{x + 3}{x - 2} \implies x + 3 = 0 \implies x = -3 \]
So, the x-intercept is at the point \( (-3, 0) \).

### Step 2: Determine the vertical asymptote
The vertical asymptote occurs where the denominator is zero. Set the denominator equal to zero:
\[ x - 2 = 0 \implies x = 2 \]
Thus, there is a vertical asymptote at \( x = 2 \).

### Step 3: Determine the horizontal asymptote
For large values of \( x \), the function behaves like:
\[ y \approx \frac{x}{x} = 1 \]
Thus, the horizontal asymptote is \( y = 1 \).

### Step 4: Sketch the graph
1. Plot the intercepts: \( (0, -\frac{3}{2}) \) and \( (-3, 0) \).
2. Draw the vertical asymptote at \( x = 2 \) (the graph will approach this line but never touch it).
3. Draw the horizontal asymptote at \( y = 1 \).
4. As \( x \) approaches \( 2 \) from the left, \( y \) will approach \( -\infty \) and from the right, \( y \) will approach \( +\infty \).
5. The graph will be in two parts: one part in the left of the vertical asymptote and one part in the right.

### Final Graph
You should have a graph that shows:
- A curve approaching the vertical asymptote \( x = 2 \) from the left and going to \( -\infty \).
- A curve approaching the vertical asymptote \( x = 2 \) from the right and going to \( +\infty \).
- The horizontal asymptote at \( y = 1 \) where the graph flattens out as \( x \) goes to \( \pm \infty \).