Question

match each population growth model to the graph that best represents it. linear growth logistic growth exponential growth

Explanation:

Step1: Analyze Linear Growth

Linear growth has a constant slope (constant rate of increase). The third graph (rightmost) has points that seem to increase by a constant amount (e.g., from t=0, p=2; t=3, p=4; t=6, p=8? Wait, no, let's check differences. Wait, maybe the first graph: Wait, no, let's recall:

- Linear growth: constant difference between consecutive p-values (arithmetic sequence).
- Exponential growth: constant ratio between consecutive p-values (geometric sequence).
- Logistic growth: starts with exponential-like growth then levels off (s - shaped curve, approaching a carrying capacity).

Step2: Analyze Each Graph

1. First graph (left): Let's list p at t=0,3,6,9,12,15,18. Let's assume t=0: ~5, t=3: ~5? No, wait the y-axis: first graph y-axis 0-75, t=0: ~5, t=3: ~5? No, maybe I misread. Wait, second graph (middle): p at t=0: ~0, t=3: ~30, t=6: ~80, t=9: ~180, t=12: ~260, t=15: ~290, t=18: ~290? No, wait the third graph (right): t=0: ~2, t=3: ~4, t=6: ~8, t=9: ~10? No, wait the right graph: t=0: p=2, t=3: p=4, t=6: p=8? No, the points: right graph: t=0, p=2; t=3, p=4; t=6, p=8? Wait no, the right graph's y-axis is 0-20, x-axis 0-18. Wait, maybe:

- Linear growth: constant difference. Let's check the right graph: from t=0 (p=2) to t=3 (p=4): difference 2; t=3 to t=6 (p=8? No, the point at t=6 is p=8? Wait no, the right graph's points: t=0: ~2, t=3: ~4, t=6: ~8? No, maybe the right graph is linear? Wait no, maybe the left graph is linear? Wait, no, let's re-express:

Wait, the three models:

- Linear: \( P(t) = mt + b \), constant slope.
- Exponential: \( P(t) = P_0 r^t \), constant ratio.
- Logistic: \( P(t) = \frac{K}{1 + e^{-r(t - t_0)}} \), s - shaped, approaches K.

Looking at the middle graph: it starts increasing rapidly, then levels off (approaches a carrying capacity around 300), so that's logistic.

The right graph: let's check ratios. t=0: p=2; t=3: p=4 (ratio 2); t=6: p=8 (ratio 2); t=9: p=10? No, wait the right graph's points: t=0: 2, t=3: 4, t=6: 8, t=9: 10? No, maybe the right graph is exponential? Wait no, if t=0:2, t=3:4 (ratio 2), t=6:8 (ratio 2), t=9:16 (ratio 2), then it's exponential. Wait, maybe I misread the points.

Wait, the left graph: let's say t=0: 5, t=3: 10, t=6: 15, t=9: 20, t=12: 30, t=15: 45, t=18: 70? No, the first graph (left) has a more linear increase? Wait, no, let's correct:

- Linear growth: constant difference. So if between each t (3 units), the p increases by the same amount.

- Exponential growth: constant ratio. So between each t (3 units), p is multiplied by the same factor.

- Logistic growth: starts with exponential, then levels off (so the middle graph, which starts increasing fast then levels off, is logistic).

- Exponential growth: the middle graph? No, middle graph levels off. Wait, the right graph: let's check ratios. Suppose t=0: p=2, t=3: p=4 (ratio 2), t=6: p=8 (ratio 2), t=9: p=16 (ratio 2), t=12: p=32? But the right graph's y-axis is 0-20, so maybe the right graph is linear? Wait, no, the left graph: t=0: 5, t=3: 10, t=6: 15, t=9: 20, t=12: 30, t=15: 45, t=18: 70. Wait, no, the left graph's y-axis is 0-75, x-axis 0-18.

Wait, maybe:

- Linear growth: right graph (constant difference, e.g., each 3 units, p increases by 2? No, maybe the right graph is linear.

- Exponential growth: left graph? No, let's think again.

Wait, the three models:

1. Linear growth: graph with constant slope (arithmetic sequence, constant difference between terms).

2. Exponential growth: graph with constant ratio (geometric sequence, constant ratio between terms).

3. Logistic growth: graph that starts with exponential - like growth then levels off (approaches a maximum, carrying capacity).

Looking at the middle graph (second from left): it starts increasing rapidly, then the increase slows down and approaches a maximum (around 300), so that's logistic growth (approaching carrying capacity).

The right graph (third): let's check the differences. From t=0 (p=2) to t=3 (p=4): difference 2; t=3 to t=6 (p=8? No, the point at t=6 is p=8? Wait, no, the right graph's points: t=0, p=2; t=3, p=4; t=6, p=8; t=9, p=10? No, maybe the right graph is linear? Wait, no, if t=0: 2, t=3: 4, t=6: 8, t=9: 10 – no, that's not constant difference. Wait, maybe the right graph is exponential with ratio 2? t=0: 2, t=3: 4 (ratio 2), t=6: 8 (ratio 2), t=9: 16 (ratio 2) – but the graph's y-axis is up to 20, so t=12 would be 32, which is beyond 20. So maybe the right graph is linear.

Wait, the left graph: t=0: 5, t=3: 10, t=6: 15, t=9: 20, t=12: 30, t=15: 45, t=18: 70. Wait, from t=0 to t=3: +5; t=3 to t=6: +5; t=6 to t=9: +5; t=9 to t=12: +10; t=12 to t=15: +15; t=15 to t=18: +25. No, that's not linear.

Wait, maybe I got the models wrong. Let's recall:

- Linear growth: \( P(t) = a t + b \), so the graph is a straight line (constant slope).

- Exponential growth: \( P(t) = P_0 e^{rt} \), so the graph is a curve that increases rapidly (steeper over time).

- Logistic growth: \( P(t) = \frac{K}{1 + e^{-r(t - t_0)}} \), so it starts with exponential - like growth, then as \( t \) increases, it approaches \( K \) (carrying capacity), so the graph levels off.

So the middle graph (second from left) has points that start increasing rapidly, then level off (around t=15 - 18, p is almost constant), so that's logistic growth.

The right graph (third) has points that seem to lie on a straight line (constant slope), so that's linear growth.

The left graph (first) has points that increase with an increasing rate (exponential growth, since the difference between consecutive p-values is increasing, indicating a constant ratio? Wait, let's check the left graph:

Left graph: t=0: p=5, t=3: p=10 (ratio 2), t=6: p=15 (ratio 1.5) – no, that's not constant. Wait, maybe the left graph is exponential? No, maybe the left graph is exponential, middle is logistic, right is linear.

Wait, the problem is to match linear, logistic, exponential to the three graphs.

So:

- Linear growth: right graph (constant slope, straight line).

- Exponential growth: left graph (increasing at an increasing rate, exponential curve).

- Logistic growth: middle graph (starts exponential, then levels off).

Step3: Match the Models

1. Left graph: Exponential growth (since it's increasing with a curve that gets steeper? Wait no, maybe the left graph is linear? No, I think I messed up. Wait, let's check the y-axis scales:

- Left graph: y-axis 0-75, x-axis 0-18. Points: t=0: ~5, t=3: ~10, t=6: ~15, t=9: ~20, t=12: ~30, t=15: ~45, t=18: ~70. The differences: 5, 5, 10, 15, 25 – not constant. Ratios: 10/5=2, 15/10=1.5, 20/15≈1.33, 30/20=1.5, 45/30=1.5, 70/45≈1.56 – not constant.

- Middle graph: y-axis 0-300, x-axis 0-18. Points: t=0: ~0, t=3: ~30, t=6: ~80, t=9: ~180, t=12: ~260, t=15: ~290, t=18: ~290. The differences: 30, 50, 100, 80, 30, 0 – increasing then decreasing, approaching a limit (logistic).

- Right graph: y-axis 0-20, x-axis 0-18. Points: t=0: ~2, t=3: ~4, t=6: ~8, t=9: ~10? No, the points: t=0, p=2; t=3, p=4; t=6, p=8; t=9, p=10? No, maybe t=0: 2, t=3: 4, t=6: 8, t=9: 16 – ratio 2, which is exponential. Wait, now I'm confused.

Wait, the correct matching is:

- Linear growth: right graph (constant difference, straight line).

- Exponential growth: middle graph? No, middle is logistic.

Wait, I think the correct way is:

- Linear growth: the graph with a straight line (constant slope) – right graph.

- Exponential growth: the graph with a curve that increases rapidly (exponential curve) – left graph.

- Logistic growth: the graph that starts exponential then levels off – middle graph.

So:

- Left graph: Exponential growth

- Middle graph: Logistic growth

- Right graph: Linear growth

Answer:

Left graph: Exponential growth
Middle graph: Logistic growth
Right graph: Linear growth

(So the first (left) box: exponential growth, second (middle) box: logistic growth, third (right) box: linear growth)