To graph the rational function $g(x) = \frac{-3x + 15}{x^2 - 7x + 10}$, we will follow these steps:

1. Find the vertical asymptotes:

Vertical asymptotes occur where the denominator equals zero (since division by zero is undefined).

Factor the denominator: $x^2 - 7x + 10 = (x - 5)(x - 2)$ So, the vertical asymptotes are at $x = 5$ and $x = 2$.

2. Find the horizontal asymptote:

The degree of the numerator (1) is less than the degree of the denominator (2), meaning the horizontal asymptote is at $y = 0$.

3. Find any holes in the graph:

Holes occur when a factor cancels out in both the numerator and denominator. Since the numerator cannot be factored further, there are no common factors with the denominator. Therefore, there are no holes in this function.

4. Plot key points:

To ensure the graph is accurate, we can plot points by evaluating $g(x)$ at specific values of $x$:

• Point 1: Let $x = 1$ $g(1) = \frac{-3(1) + 15}{(1-5)(1-2)} = \frac{-3 + 15}{(-4)(-1)} = \frac{12}{4} = 3$ So, $(1, 3)$ is on the graph.

• Point 2: Let $x = 0$ $g(0) = \frac{-3(0) + 15}{(0-5)(0-2)} = \frac{15}{(-5)(-2)} = \frac{15}{10} = 1.5$ So, $(0, 1.5)$ is on the graph.

• Point 3: Let $x = 3$ $g(3) = \frac{-3(3) + 15}{(3-5)(3-2)} = \frac{-9 + 15}{(-2)(1)} = \frac{6}{-2} = -3$ So, $(3, -3)$ is on the graph.

5. Graph the function:

• Plot the vertical asymptotes at $x = 5$ and $x = 2$.
• Draw the horizontal asymptote at $y = 0$.
• Plot the points $(1, 3)$, $(0, 1.5)$, and $(3, -3)$.

Once all this is done, you can complete the graph by drawing the curve through these points, ensuring it approaches the asymptotes.

Would you like me to provide a detailed sketch for this function?

5 Follow-up Questions:

1. What is the significance of vertical asymptotes in rational functions?
2. How do you determine if a rational function has a hole in its graph?
3. What is the behavior of a rational function near its vertical asymptotes?
4. Why does this particular function have a horizontal asymptote at $y = 0$?
5. How would the graph change if the numerator's degree were equal to or greater than the denominator’s degree?

Tip: Always check for common factors between the numerator and denominator to identify holes in the graph.