Use graphical methods to find any turning points if the graph of the function : f(x)= x^2 / x-6

To find the turning points of the function \( f(x) = \frac{x^2}{x - 6} \) using graphical methods, follow these steps:

### Step 1: Analyze the Function
The function \( f(x) = \frac{x^2}{x - 6} \) is a rational function, and it has a vertical asymptote where the denominator equals zero, i.e., at \( x = 6 \).

### Step 2: Identify Critical Points
To find critical points (where turning points may occur), we need to first find the derivative \( f'(x) \) and set it equal to zero. Critical points occur where the derivative is zero or undefined, but since we are focusing on a graphical method, we'll proceed by plotting the function.

### Step 3: Graph the Function
Graph the function \( f(x) = \frac{x^2}{x - 6} \). This can be done using graphing software, a graphing calculator, or by plotting points manually. The graph should clearly show the behavior of the function around the critical points.

### Step 4: Identify Turning Points on the Graph
A turning point occurs where the graph changes direction, i.e., where the slope of the tangent to the curve changes from positive to negative or vice versa.

### Analysis:
- **Asymptote at \( x = 6 \):** The function will have a vertical asymptote at \( x = 6 \).
- **Turning Points:** Based on the graphical representation, the turning points are where the function changes from increasing to decreasing or vice versa.

Given the form of the function, it’s expected that there will be a turning point before and after the asymptote. 

### Conclusion:
- The turning point before the asymptote (for \( x < 6 \)) is a local minimum.
- The turning point after the asymptote (for \( x > 6 \)) is a local maximum.

To accurately determine the \( x \)-coordinates of the turning points and their nature, the derivative can be used. However, graphically, these points can be identified as where the graph shifts direction.

Would you like the detailed steps for finding the derivative and verifying these points analytically? Or do you want to explore the graph more?

### Follow-Up Questions:
1. What is the derivative of \( f(x) \) for further analysis?
2. How does the vertical asymptote affect the function's behavior near \( x = 6 \)?
3. Can you explain why the function is undefined at \( x = 6 \)?
4. What happens to \( f(x) \) as \( x \) approaches \( 6 \) from the left and right?
5. How would you describe the end behavior of the function as \( x \) approaches infinity?
6. Are there any horizontal asymptotes for this function? If so, where?
7. Can we identify the exact coordinates of the turning points using algebraic methods?
8. How does the function behave for very large positive and negative values of \( x \)?

### Tip:
When graphing a rational function, always start by identifying the asymptotes and the intercepts to understand the overall shape of the graph.

Learn how to analyze the function f(x) = x^2 / (x - 6) graphically to find turning points. This problem covers rational functions, asymptotes, and graphical methods for analyzing functions. Suitable for high school students.

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