https://file.aiquickdraw.com/mathbot/file/1723780537733g3cjh6yy.jpg

To solve the problem, let's analyze the graphs provided and the choices given.

### Analyzing the Graphs:
1. **Graph a**: This is a linear function. A linear function has no turning points or roots (other than at most one root depending on its slope and y-intercept).
   
2. **Graph b**: This is a quadratic function, which is a parabola. It has one turning point (the vertex) and it can have either zero, one, or two real roots depending on its position relative to the x-axis.

3. **Graph c**: This is a cubic function. Cubic functions typically have up to two turning points and can have up to three real roots.

4. **Graph d**: This appears to be a quartic function. A quartic function can have up to three turning points and up to four real roots.

### Possible Answers:
The question asks us to identify the correct relationships between the graphs and the number of roots \( r \) and the number of turning points \( n \).

Now let's correlate the number of roots \( r \) and the number of turning points \( n \) for each graph:

- **Graph a** (linear function): 
  - \( r = 1 \) (intersects the x-axis once)
  - \( n = 0 \) (no turning points)

- **Graph b** (quadratic function):
  - \( r = 2 \) (intersects the x-axis twice)
  - \( n = 1 \) (one turning point)

- **Graph c** (cubic function):
  - \( r = 3 \) (intersects the x-axis three times)
  - \( n = 2 \) (two turning points)

- **Graph d** (quartic function):
  - \( r = 4 \) (intersects the x-axis four times)
  - \( n = 3 \) (three turning points)

### Correct Answer:
Given the analysis, the correct matching should be:

- Graph a: \( r = 1 \), \( n = 0 \)
- Graph b: \( r = 2 \), \( n = 1 \)
- Graph c: \( r = 3 \), \( n = 2 \)
- Graph d: \( r = 4 \), \( n = 3 \)

This matches with the following option:

**Option D: \( a: n=0 \) and \( r=1 \); \( b: n=1 \) and \( r=2 \); \( c: n=2 \) and \( r=3 \); \( d: n=3 \) and \( r=4 \).**

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Would you like further details or explanations? Here are some related questions:

1. How do you determine the number of roots of a polynomial function?
2. What is the general form of a quadratic function and its properties?
3. How can you identify a cubic function by its graph?
4. What is the significance of turning points in polynomial functions?
5. How do polynomial degrees relate to the number of roots and turning points?
6. What is the relationship between the derivative of a function and its turning points?
7. Can a cubic function have only one real root? Explain.
8. How does the discriminant help in identifying the number of roots for a quadratic equation?

**Tip:** When analyzing polynomial graphs, remember that the degree of the polynomial gives the maximum number of roots, and the degree minus one gives the maximum number of turning points.

Explore how to analyze polynomial graphs to determine the number of roots and turning points. This detailed analysis covers linear, quadratic, cubic, and quartic functions, their characteristics, and how to identify them based on graph properties. Ideal for high school students studying algebra and graphing functions.

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