12 If the function f has five distinct zeros, which of the following could represent the complete graph of f in the xy‑plane?

Options

A.
B.
C.
D.

Answer

To solve the problem of identifying which graph could represent a function \( f \) with five distinct zeros, we need to analyze the characteristics of the graphs provided in the options. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Concept of Zeros
A zero of a function is a value of \( x \) for which \( f(x) = 0 \). If a function has five distinct zeros, it means that the graph of the function intersects the x-axis at five different points.

**Hint:** Remember that each intersection with the x-axis represents a zero of the function.

### Step 2: Analyze Each Option
We will examine each option to count the number of x-intercepts (where the graph crosses the x-axis).

- **Option A:** Count the x-intercepts.
- The graph intersects the x-axis at 1, 2, 3, and 4 points.
- Therefore, it has 4 distinct zeros.

**Hint:** Look for how many times the graph crosses the x-axis.

- **Option B:** Count the x-intercepts.
- The graph intersects the x-axis at 1, 2, 3, and 4 points.
- Therefore, it also has 4 distinct zeros.

**Hint:** Compare the number of x-intercepts with the required number of distinct zeros.

- **Option C:** Count the x-intercepts.
- The graph intersects the x-axis at 1, 2, 3, 4, 5, and 6 points.
- Therefore, it has 6 distinct zeros.

**Hint:** Ensure that the number of intercepts matches the required number of distinct zeros.

- **Option D:** Count the x-intercepts.
- The graph intersects the x-axis at 1, 2, 3, 4, and 5 points.
- Therefore, it has exactly 5 distinct zeros.

**Hint:** This is the critical step; look for the option that has exactly 5 distinct x-intercepts.

### Step 3: Conclusion
After analyzing all the options, we find that only **Option D** has exactly 5 distinct zeros. Thus, the complete graph of the function \( f \) with five distinct zeros is represented by Option D.

**Final Answer:** The answer is **Option D**.