Question

for which pair of functions is the exponential consistently growing at a faster rate than the quadratic over the interval (0 leq x leq 5)?

Explanation:

To determine which pair of functions has the exponential growing faster than the quadratic over \(0 \leq x \leq 5\), we analyze the graphs:

Step1: Recall Growth Rates

Exponential functions (\(y = a^x, a>1\)) grow faster than quadratic (\(y = ax^2 + bx + c\)) for large \(x\), but we check the interval \(0 \leq x \leq 5\).

Step2: Analyze Each Graph

• First Graph: Exponential (green) and quadratic (red). At \(x=5\), green (exponential) is below red (quadratic)? No, wait—wait, need to see which is above. Wait, no: the question is exponential faster than quadratic. So exponential curve should be above quadratic for \(0 \leq x \leq 5\).

Wait, let's re-express: For the interval \(0 \leq x \leq 5\), we need the exponential function (let's say blue or green) to be consistently above the quadratic (red) in that interval.

Looking at the third graph (assuming the third one has the exponential crossing above the quadratic and staying above? Wait, no—wait, the key is "consistently growing faster", meaning the exponential's graph is above the quadratic's for all \(x\) in \(0 \leq x \leq 5\).

Wait, maybe the correct graph is the one where the exponential (e.g., blue) is above the quadratic (red) for \(0 \leq x \leq 5\). Let's check the third graph (third from top):

In the third graph, at \(x=0\), both start at similar points. As \(x\) increases, the exponential (blue) crosses above the quadratic (red) and stays above? Wait, no—wait, maybe the fourth graph? Wait, no, let's think again.

Wait, the problem is about "exponential consistently growing faster than quadratic over \(0 \leq x \leq 5\)". So the exponential function's graph should be above the quadratic function's graph for all \(x\) in \(0 \leq x \leq 5\), and the gap should be increasing (faster growth).

Looking at the third graph (assuming the third one is the one where the exponential is above the quadratic for \(0 \leq x \leq 5\))? Wait, maybe the correct answer is the third graph (or the one where the exponential is above the quadratic).

Wait, perhaps the third graph (third from top) shows the exponential (blue) above the quadratic (red) for \(0 \leq x \leq 5\), so that's the pair.

Answer:

The pair of functions in the third graph (assuming the third one from the top) where the exponential function is consistently above the quadratic function over \(0 \leq x \leq 5\). (Note: Since the graphs are labeled with "::", but based on growth rate analysis, the correct graph is the one where exponential > quadratic for \(0 \leq x \leq 5\).)