f'(x) = 3 graph .

Then select three possible graphs of f.

To graph the derivative \( f'(x) = 3 \), note that this is a constant function. This means that the slope of the tangent line to \( f(x) \) is constant and equal to 3 everywhere.

1. **Understanding \( f'(x) = 3 \):**
   - Since the derivative is constant and positive, the original function \( f(x) \) must be a straight line with a positive slope.
   - Specifically, if \( f'(x) = 3 \), the slope of \( f(x) \) is always 3, meaning \( f(x) \) has the form:
     \[
     f(x) = 3x + C
     \]
     where \( C \) is a constant representing the y-intercept of the function.

2. **Graph of \( f'(x) = 3 \):**
   - The graph of \( f'(x) = 3 \) is simply a horizontal line at \( y = 3 \), showing the constant slope.

3. **Possible graphs of \( f(x) \):**
   Since \( f(x) = 3x + C \), the graph of \( f(x) \) is a straight line with slope 3. Three possible variations of \( f(x) \) can be:
   - \( f(x) = 3x \) (a line passing through the origin),
   - \( f(x) = 3x + 1 \) (a line with slope 3 and y-intercept 1),
   - \( f(x) = 3x - 2 \) (a line with slope 3 and y-intercept -2).

Each graph would look like a straight line, but shifted vertically depending on the value of \( C \).

---

Would you like more details on how these functions behave? Here are five related questions:

1. What does it mean for a function's derivative to be constant?
2. How do we find the derivative of more complex functions?
3. How would the graph of \( f(x) \) change if the slope was different?
4. Can the derivative of a function be negative, and how would that affect the graph?
5. How do we interpret the constant \( C \) in the function \( f(x) = 3x + C \)?

**Tip:** If you know the derivative, integrating it helps find the original function!

f'(x) = 3 graph. Then select three possible graphs of f.

Understand how to graph the derivative f'(x) = 3 and identify three possible graphs of the original function f(x). This solution covers the concept of constant derivatives and linear functions, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation