https://file.aiquickdraw.com/mathbot/file/1729629116899afpdtvdg.jpg

The exponential function decays at one-half the rate of the quadratic function.

The exponential function decays at the same rate as the quadratic function.

The exponential function decays at two-thirds the rate of the quadratic function.

The exponential function decays at three-fourths the rate of the quadratic function.


To analyze the decay rates of the functions over the interval \(-2 \leq x \leq 0\), let’s break down what the graph shows.

- **Blue curve**: This appears to represent the **quadratic function**.
- **Pink curve**: This is likely the **exponential function**.

In general:
1. **Quadratic functions** (e.g., \(y = ax^2\)) decrease and then increase symmetrically as \(x\) moves from negative to positive values.
2. **Exponential functions** (e.g., \(y = a \cdot b^x\), with \(0 < b < 1\) for decay) decrease much more rapidly than polynomials at first, but their decay slows as they approach the x-axis.

### Observing Decay in the Interval \(-2 \leq x \leq 0\)
- Both functions are decreasing in this interval, but their rates are different.
- **Exponential decay** tends to happen more quickly than quadratic decay for small negative values of \(x\).
- The slope (rate of decrease) of the **pink curve (exponential)** is steeper than the slope of the **blue curve (quadratic)** initially near \(x = -2\), but their decay seems to stabilize towards 0 at different rates.

### Selecting the Best Answer:
From the provided options:
1. **"The exponential function decays at one-half the rate of the quadratic function"**: This seems unlikely because exponential decay is typically stronger than quadratic decay at the start.
2. **"The exponential function decays at the same rate as the quadratic function"**: Not correct; their rates differ noticeably.
3. **"The exponential function decays at two-thirds the rate of the quadratic function"**: This might fit if the exponential function were only slightly slower, but still not the best match.
4. **"The exponential function decays at three-fourths the rate of the quadratic function"**: This option aligns better if the quadratic function decays only slightly faster over the interval \(-2 \leq x \leq 0\).

### Conclusion:
Based on the behavior observed from the graph, **the most reasonable answer is likely: "The exponential function decays at three-fourths the rate of the quadratic function."** The graph suggests that the quadratic function decays a bit faster over the interval in question.

---

Do you want more details on how I identified the decay rates or further explanation on exponential and quadratic behavior?  

Here are five related questions you might explore:
1. How do you calculate decay rates mathematically for exponential and quadratic functions?
2. What happens to exponential functions as \(x \rightarrow -\infty\)?
3. How does the rate of change differ between linear, quadratic, and exponential functions?
4. What are some real-world applications where exponential decay is more relevant than quadratic decay?
5. How do slopes and tangents explain decay behavior?

**Tip:** To compare decay rates quantitatively, you can calculate the derivative (slope) of both functions over the given interval.

A quadratic function and an exponential function are graphed below. How do the decay rates of the functions compare over the interval -2 ≤ x ≤ 0?

Explore how the decay rates of a quadratic function and an exponential function compare over the interval -2 ≤ x ≤ 0. This problem involves understanding the difference in the rate of change between exponential and quadratic functions, and interpreting their graphs.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation