The image shows a graph of the function $y = \frac{x^2 + x - 6}{x - 2}$. The task is to sketch the graph of this function and identify any important features like intercepts, asymptotes, and any points of interest.

Step-by-Step Analysis

1. Factor the numerator: $x^2 + x - 6 = (x - 2)(x + 3)$ So the function simplifies to: $y = \frac{(x - 2)(x + 3)}{x - 2}$

2. Simplify the function: The function simplifies to: $y = x + 3$ except at $x = 2$, where the function is undefined because the denominator is zero.

3. Identify the vertical asymptote:

   • The function has a hole at $x = 2$, not a vertical asymptote. A vertical asymptote would occur if the denominator could not be canceled out, but here, it does get canceled. Therefore, the only discontinuity is a hole at $x = 2$.

4. Determine the intercepts:

   • Y-intercept: Substitute $x = 0$: $y = 0 + 3 = 3$ So, the y-intercept is at $(0, 3)$.

   • X-intercept: Set $y = 0$ and solve for $x$: $x + 3 = 0 \quad \Rightarrow \quad x = -3$ So, the x-intercept is at $(-3, 0)$.

5. Horizontal Asymptote:

   • The function $y = x + 3$ is linear, so it does not have a horizontal asymptote.

6. Graph behavior:

   • The graph is a straight line $y = x + 3$ with a hole at $x = 2$.

Key Features on the Graph:

• Hole at $x = 2$: This corresponds to the point $(2, 5)$, but the point is not included in the graph.
• X-intercept: $(-3, 0)$
• Y-intercept: $(0, 3)$
• No vertical asymptote: Only a removable discontinuity (hole) at $x = 2$.

Would you like more details or have any questions?

Related Questions:

1. What happens to the graph if the function had a denominator that doesn't cancel out?
2. How do you find the vertical and horizontal asymptotes for rational functions?
3. What are the differences between holes and vertical asymptotes in graphing rational functions?
4. How do you graph a rational function with both vertical and horizontal asymptotes?
5. How does the degree of the numerator and denominator influence the shape of the graph?

Tip: When graphing rational functions, always factor the numerator and denominator first to identify any removable discontinuities like holes.