Sketch the graph of the following rational functions
`y=(7-2x)/(3x+5)`

Answer

To sketch the graph of the rational function \( y = \frac{7 - 2x}{3x + 5} \), we will follow these steps:

### Step 1: Find the y-intercept
To find the y-intercept, we set \( x = 0 \) in the equation:

\[ y = \frac{7 - 2(0)}{3(0) + 5} = \frac{7}{5} \]

So, the y-intercept is at the point \( (0, \frac{7}{5}) \).

### Step 2: Find the x-intercept
To find the x-intercept, we set \( y = 0 \):

\[ 0 = \frac{7 - 2x}{3x + 5} \]

This implies that the numerator must be zero:

\[ 7 - 2x = 0 \implies 2x = 7 \implies x = \frac{7}{2} \]

So, the x-intercept is at the point \( (\frac{7}{2}, 0) \).

### Step 3: Find the vertical asymptote
The vertical asymptote occurs where the denominator is zero:

\[ 3x + 5 = 0 \implies 3x = -5 \implies x = -\frac{5}{3} \]

So, the vertical asymptote is at \( x = -\frac{5}{3} \).

### Step 4: Find the horizontal asymptote
To find the horizontal asymptote, we look at the degrees of the numerator and denominator. Since both the numerator and denominator are linear (degree 1), we can find the horizontal asymptote by taking the ratio of the leading coefficients:

\[ y = \frac{-2}{3} \]

So, the horizontal asymptote is at \( y = -\frac{2}{3} \).

### Step 5: Sketch the graph
Now that we have the intercepts and asymptotes, we can sketch the graph:

1. Plot the y-intercept \( (0, \frac{7}{5}) \) which is approximately \( (0, 1.4) \).
2. Plot the x-intercept \( (\frac{7}{2}, 0) \) which is approximately \( (3.5, 0) \).
3. Draw a vertical dashed line at \( x = -\frac{5}{3} \) (approximately -1.67).
4. Draw a horizontal dashed line at \( y = -\frac{2}{3} \) (approximately -0.67).
5. The graph will approach the vertical asymptote but will not touch it. It will also approach the horizontal asymptote as \( x \) goes to positive or negative infinity.

### Final Graph
The graph will have two branches:
- One in the first quadrant approaching the horizontal asymptote as \( x \) increases.
- The other in the third quadrant approaching the horizontal asymptote as \( x \) decreases.