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 change). Let's check the first graph: at t=3, p≈5; t=6, p≈10? Wait, no, first graph: t=0, p=0; t=3, p≈5; t=6, p≈10? Wait, no, looking at the first graph (left): the points seem to have a constant difference? Wait, maybe the middle graph? Wait, no, let's recall:

- Linear growth: constant increase per unit time (straight line, constant slope).
- Exponential growth: increasing by a constant ratio (curving upwards, increasing rate).
- Logistic growth: S - shaped, starts with exponential - like, then levels off.

First graph (left): Let's check the differences. At t=0, p=0; t=3, p≈5; t=6, p≈10; t=9, p≈15; t=12, p≈20; t=15, p≈30; t=18, p≈40? Wait, no, the left graph's y - axis: 0,15,30,45,60,75. At t=0, p=0; t=3, p≈5? No, maybe the left graph is linear? Wait, no, the middle graph: t=0, p=0; t=3, p≈30; t=6, p≈80; t=9, p≈190; t=12, p≈260; t=15, p≈290; t=18, p≈290. Wait, no, the right graph: t=0, p=2; t=3, p=4; t=6, p=8; t=9, p=11? No, t=0, p=2; t=3, p=4 (double); t=6, p=8 (double again); t=9, p=11? No, wait t=9: p=11? No, the right graph's y - axis: 0,4,8,12,16,20. t=0, p=2; t=3, p=4 (ratio 2); t=6, p=8 (ratio 2); t=9, p=11? No, maybe t=9, p=11? No, maybe I misread. Wait, the right graph: t=0, p=2; t=3, p=4; t=6, p=8; t=9, p=11? No, that's not exponential. Wait, maybe the middle graph is exponential? Wait, no, let's correct:

- Linear growth: constant difference. So if the left graph has points with constant difference (e.g., each time t increases by 3, p increases by a constant amount). Let's check left graph:

t: 0, 3, 6, 9, 12, 15, 18

p: 0, ~5, ~10, ~15, ~20, ~30, ~40? Wait, no, the left graph's y - axis is 0,15,30,45,60,75. So at t=0, p=0; t=3, p≈5? No, maybe the left graph is linear (constant slope), middle is logistic (levels off at the end), right is exponential.

Wait, let's re - identify:

1. Linear growth: The graph with constant slope (straight line, constant increase). So the left graph: let's check the slope. From t=0 to t=18, p goes from 0 to 70? No, maybe the left graph is linear (each time t increases by 3, p increases by a constant amount).

2. Exponential growth: The graph with increasing slope (curving upwards, like the right graph? Wait, the right graph: t=0, p=2; t=3, p=4; t=6, p=8; t=9, p=11? No, t=9, p=11 is not double of 8. Wait, maybe the middle graph is exponential? No, the middle graph at t=15 and t=18 is flat, so that's logistic (levels off).

3. Logistic growth: S - shaped, starts with exponential, then levels off. So middle graph: starts increasing rapidly, then levels off (t=15 and t=18 same), so logistic.

4. Exponential growth: right graph? Wait, t=0, p=2; t=3, p=4 (ratio 2); t=6, p=8 (ratio 2); t=9, p=11? No, t=9, p=11 is not 8*1.5. Wait, maybe the right graph is exponential (since it's doubling: t=0:2, t=3:4, t=6:8, t=9:11? No, maybe my mistake. Wait, the left graph: t=0, p=0; t=3, p=5; t=6, p=10; t=9, p=15; t=12, p=20; t=15, p=30; t=18, p=40? No, the y - axis for left is 0,15,30,45,60,75. So t=0:0, t=3:5 (no), maybe the left graph is linear (constant slope), middle is logistic (levels off), right is exponential.

Wait, let's recall the definitions:

- Linear growth: \( P(t)=mt + b \), constant slope \( m \).
- Exponential growth: \( P(t)=P_0e^{rt} \) or \( P(t)=P_0a^t \), increasing by a constant ratio.
- Logistic growth: \( P(t)=\frac{K}{1 + e^{-r(t - t_0)}} \), S - shaped, approaches carrying capacity \( K \).

So:

- Left graph: The points seem to have a constant rate of increase (linear), so linear growth.
- Middle graph: Starts increasing, then levels off (approaches a carrying capacity), so logistic growth.
- Right graph: The points seem to be doubling (t=0:2, t=3:4, t=6:8, t=9:11? No, t=9: p=11? Wait, no, t=9: p=11? No, the right graph's y - axis: 0,4,8,12,16,20. t=0, p=2; t=3, p=4 (ratio 2); t=6, p=8 (ratio 2); t=9, p=11? No, maybe t=9, p=11 is a mistake, but likely the right graph is exponential (since it's growing by a constant ratio, doubling every 3 units of time).

So matching:

- Linear growth: Left graph (first graph)
- Logistic growth: Middle graph (second graph)
- Exponential growth: Right graph (third graph)

Wait, let's check the middle graph: at t=15 and t=18, the population is almost the same, so it's reached carrying capacity, so logistic. The right graph: each time t increases by 3, the population doubles (t=0:2, t=3:4, t=6:8), so exponential. The left graph: each time t increases by 3, the population increases by a constant amount (e.g., t=0:0, t=3:5, t=6:10, t=9:15, etc.), so linear.

Step2: Match the Models

- Linear growth: The left - most graph (first graph) because it has a constant rate of increase (straight - line - like, constant slope).
- Logistic growth: The middle graph (second graph) because it starts with growth and then levels off (S - shaped, approaching carrying capacity).
- Exponential growth: The right - most graph (third graph) because it grows by a constant ratio (doubling every 3 units of time, curving upwards with increasing rate).

Answer:

• Linear growth: The Left Graph (first graph)
• Logistic growth: The Middle Graph (second graph)
• Exponential growth: The Right Graph (third graph)