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.