https://file.aiquickdraw.com/mathbot/file/1727187623973exd5lx2d.jpg

The problem is asking to match the graphs of functions (labeled A through D) with the graphs of their respective derivatives (labeled 1 through 4).

Here's how to approach the problem:

### Key observations when matching graphs:
1. **Critical points of the original function**:
   - At points where the original function (A–D) has a local maximum or minimum, the derivative should cross the x-axis (because the slope is zero at those points).
   
2. **Slope of the original function**:
   - When the original function is increasing, the derivative is positive (above the x-axis).
   - When the original function is decreasing, the derivative is negative (below the x-axis).

3. **Shape of the derivative**:
   - If the original function is linear (a straight line), the derivative will be a constant (a horizontal line).
   - If the original function is quadratic, the derivative will be linear.

Now, let’s match them based on these rules:

### Step-by-step matching:
1. **Graph A**: 
   - This is a cubic-like function with a local minimum and a local maximum, which means its derivative will have two roots (crosses the x-axis twice).
   - **Matches Derivative 2**: This derivative has two x-intercepts, corresponding to the two critical points in graph A.

2. **Graph B**: 
   - This is a linear function, meaning its derivative should be a constant value.
   - **Matches Derivative 4**: The derivative of graph B is a constant, which corresponds to the horizontal line in graph 4.

3. **Graph C**:
   - This function has a point of inflection where the slope flattens and then changes. It looks cubic in nature, but smoother.
   - **Matches Derivative 3**: The graph has a more complex behavior near the origin, showing rapid changes that match the behavior of graph C.

4. **Graph D**:
   - This graph has a pattern of oscillations with three critical points (two maxima and one minimum).
   - **Matches Derivative 1**: The derivative graph has three x-intercepts, which corresponds to the critical points in graph D.

### Final matches:
1. A → 2  
2. B → 4  
3. C → 3  
4. D → 1

Would you like further explanations on how the derivatives were identified, or do you have any other questions?

### Relative Questions:
1. What is the general relationship between the function and its derivative regarding increasing and decreasing behavior?
2. How does the second derivative of a function relate to the concavity of the original function?
3. How can critical points help in identifying maxima, minima, or inflection points of a function?
4. Why does a constant function have a derivative of zero, and what does this signify in terms of the slope?
5. How can you determine the number of turning points in a function based on its derivative?

### Tip:
Always examine the behavior of the slope at the critical points (maxima, minima, or inflection points) to understand how the graph of the derivative behaves.

Match the graphs of each function in A through D with the graph of its derivative 1 through 4 below.

Learn how to match graphs of functions with their corresponding derivatives by analyzing critical points and the behavior of slopes. This solution demonstrates how the first derivative reveals the slope of the original function and identifies maxima, minima, and inflection points.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation