https://file.aiquickdraw.com/mathbot/file/1727287348634gf046o6l.jpg

To solve this problem, we need to analyze the graph shown in the image to identify the tangent and secant lines and determine the slope at \( x = 0 \).

### Step 1: Identify the function and the lines
- **Function (Green curve)**: This is the given function in the graph, which resembles a parabola. The key task is to find the slope of this function at \( x = 0 \).
- **Tangent Line**: A tangent line touches the curve at exactly one point and represents the slope of the function at that point. It should touch the function only at \( x = 0 \) in this case.
- **Secant Line**: A secant line intersects the function at two or more points. In this case, the secant line likely intersects the curve at two distinct points near \( x = 0 \).

### Step 2: Analyze the Lines
- **Blue Line**: This line intersects the curve at two points, one at \( x = -1 \) and another at \( x = 1 \). Therefore, this line is the **secant line**.
- **Red Line**: This line touches the green curve at only one point at \( x = 0 \), making it the **tangent line**.

### Step 3: Find the Slope of the Function at \( x = 0 \)
- The **slope of the tangent line** at a point represents the derivative of the function at that point. 
- Visually from the graph, the red line (the tangent) has a slope of **0**, because it is horizontal at \( x = 0 \). This implies the function has a derivative (or slope) of **0** at \( x = 0 \), meaning the graph has a local minimum at this point.

### Conclusion:
- The **blue line** is the **secant line**.
- The **red line** is the **tangent line**, and the slope of the function at \( x = 0 \) is **0**.

Would you like further clarification on any part of this? 

Here are some related questions for deeper understanding:
1. What is the geometric significance of a tangent line's slope?
2. How can we calculate the exact slope of a secant line?
3. What does the slope of 0 at a point say about the nature of that point (e.g., minimum, maximum)?
4. How would the secant line change if we considered different points of intersection with the function?
5. How does the derivative relate to the graphical representation of the function at different points?

**Tip:** The slope of the tangent line at a point is equivalent to the derivative of the function at that point, which measures the rate of change of the function's value.

Use the graph below to find the slope of the function at x = 0. Which line is the secant line? Which is the tangent line?

Learn how to find the slope of a function at x = 0 by analyzing the tangent and secant lines in the graph. Understand the relationship between the tangent line's slope and the derivative. Suitable for students in Grades 11-12.

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