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Scientists measured the concentration of an exprerimental medication in blood samples over time. If a quadratic function is used to model the data, which of following best explains th meaning of the vertex ?

Options

A. The average maximum medicine concentration in the blood sample is 7.5 units.
B. The average maximum medicine concentration is more than 20 units.
C. The average maximum medicine concentration in the blood samples occurs around 2.5 minutes.
D. The average maximum medicine concentration in the blood sample occurs bgetween 5 and 10 minutes.

Explanation

To solve the question about the meaning of the vertex in the context of a quadratic function modeling the concentration of an experimental medication in blood samples over time, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding the Quadratic Function**:
- A quadratic function is typically represented in the form \( f(x) = ax^2 + bx + c \), which graphs as a parabola. The vertex of this parabola represents either a maximum or minimum point depending on the direction it opens (upwards or downwards).

**Hint**: Recall that the vertex of a parabola can indicate the maximum or minimum value of the function.

2. **Identifying the Context**:
- In this scenario, the scientists are measuring the concentration of a medication over time. The concentration of the medication is expected to rise to a peak (maximum) and then fall, which suggests that the parabola opens downwards.

**Hint**: Think about how medication concentration might behave over time—usually, it increases to a peak and then decreases.

3. **Finding the Vertex**:
- The vertex of the quadratic function in this context represents the point in time when the concentration of the medication in the blood is at its highest.

**Hint**: The vertex will give you the time at which the concentration is maximized.

4. **Analyzing the Given Options**:
- The options provided in the question need to be evaluated based on the information derived from the vertex.
- The vertex indicates the maximum concentration of the medication, which is approximately 2.6 units, occurring between 5 to 10 minutes.

**Hint**: Compare the options to the information about the vertex to determine which one accurately reflects the maximum concentration and the time it occurs.

5. **Selecting the Correct Answer**:
- The correct interpretation of the vertex is that it indicates the average maximum concentration of the medication in the blood occurs around 5 to 10 minutes. Therefore, option D is the most appropriate answer.

**Hint**: Look for the option that aligns with both the maximum concentration value and the time frame indicated by the vertex.

### Final Answer:
The average maximum medicine concentration in blood occurs around 5 to 10 minutes (Option D).