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 differences between consecutive points. For the first graph: At t=3, P≈5; t=6, P≈7.5; t=9, P≈10; t=12, P≈22.5? Wait, no, maybe the third graph. Wait, third graph: t=0, P=4; t=3, P=5; t=6, P=8; t=9, P=10; t=12, P=14; t=15, P=16; t=18, P=20. Wait, no, let's recalculate differences. Wait, linear growth: the difference between P values (ΔP) should be constant for equal Δt (here Δt=3).

First graph (left): t=0 (P=5), t=3 (P≈5), t=6 (P≈7.5), t=9 (P≈10), t=12 (P≈22.5), t=15 (P≈40), t=18 (P≈70). Wait, no, maybe I misread. Wait, the y-axis: left graph y: 15, 30, 45, 60, 75. So t=0: P=5 (below 15), t=3: P=5, t=6: P=10, t=9: P=12.5, t=12: P=22.5, t=15: P=40, t=18: P=70. No, that's not linear.

Second graph (middle): t=0: P=0, t=3: P=50, t=6: P=100, t=9: P=200, t=12: P=275, t=15: P=300, t=18: P=300. Wait, t=3 to 6: ΔP=50, 6 to 9: ΔP=100, 9 to 12: ΔP=75, 12 to 15: ΔP=25, 15 to 18: ΔP=0. That's logistic (sigmoid, reaches carrying capacity).

Third graph (right): t=0: P=4, t=3: P=5 (ΔP=1), t=6: P=8 (ΔP=3), t=9: P=10 (ΔP=2), t=12: P=14 (ΔP=4), t=15: P=16 (ΔP=2), t=18: P=20 (ΔP=4). No, wait, maybe linear is the one with constant ΔP. Wait, no, let's check the three models:

- Linear growth: straight line, constant slope (ΔP/Δt constant).
- Exponential growth: curve, increasing slope (ΔP proportional to current P).
- Logistic growth: S-shaped, starts with exponential, then slows to carrying capacity.

Third graph (right): Let's check ratios. P(0)=4, P(3)=5 (5/4=1.25), P(6)=8 (8/5=1.6), P(9)=10 (10/8=1.25), P(12)=14 (14/10=1.4), P(15)=16 (16/14≈1.14), P(18)=20 (20/16=1.25). Not exponential. Wait, maybe I messed up.

Wait, correct approach:

1. Linear growth: The graph with a constant rate (straight line, equal ΔP for equal Δt). Let's check the right graph: t=0 (4), t=3 (5) → ΔP=1, Δt=3. t=3 to 6: 8-5=3 (ΔP=3, Δt=3 → slope 1). t=6 to 9: 10-8=2 (slope 2/3). No, that's not linear. Wait, maybe the left graph: t=0 (5), t=3 (5), t=6 (10), t=9 (12.5), t=12 (22.5), t=15 (40), t=18 (70). No. Wait, maybe the middle graph: no, middle is logistic. Wait, perhaps the right graph is linear? No, let's re-express:

Wait, the three models:

- Linear: y = mt + b, constant m.
- Exponential: y = ab^t, b > 1, increasing.
- Logistic: y = K / (1 + e^(-rt)), S-shaped, reaches K.

So:

- Left graph: starts low, increases, then maybe? No, left graph's points: t=0 (P=5), t=3 (5), t=6 (10), t=9 (12.5), t=12 (22.5), t=15 (40), t=18 (70). The increase accelerates, so exponential?

- Middle graph: t=0 (0), t=3 (50), t=6 (100), t=9 (200), t=12 (275), t=15 (300), t=18 (300). So it increases, then levels off (carrying capacity at 300), so logistic.

- Right graph: t=0 (4), t=3 (5), t=6 (8), t=9 (10), t=12 (14), t=15 (16), t=18 (20). Let's check the differences: t=0-3: +1, 3-6: +3, 6-9: +2, 9-12: +4, 12-15: +2, 15-18: +4. No, but maybe linear is the one with the most constant slope. Wait, no, maybe I made a mistake. Let's try again:

Wait, the three graphs:

1. Left graph (first): The points seem to follow a curve that starts slow, then grows faster (exponential? Or logistic? No, logistic has a sigmoid, but this is increasing with increasing slope).

2. Middle graph (second): Grows, then slows down and stabilizes (logistic, reaches carrying capacity).

3. Right graph (third): The points: t=0 (4), t=3 (5), t=6 (8), t=9 (10), t=12 (14), t=15 (16), t=18 (20). Let's check the slope between t=0-6: (8-4)/(6-0)=4/6=2/3. t=6-12: (14-8)/(12-6)=6/6=1. t=12-18: (20-14)/(18-12)=6/6=1. No, not constant. Wait, maybe the left graph is linear? No, left graph's t=0 (5), t=6 (10), t=12 (22.5), t=18 (70). The slope from 0-6: 5/6, 6-12: 12.5/6≈2.08, 12-18: 47.5/6≈7.92. Increasing slope (exponential).

Middle graph: t=0 (0), t=6 (100), t=12 (275), t=18 (300). So it's logistic (S-curve, reaches K=300).

Right graph: Let's check the ratios. P(0)=4, P(3)=5 (5/4=1.25), P(6)=8 (8/5=1.6), P(9)=10 (10/8=1.25), P(12)=14 (14/10=1.4), P(15)=16 (16/14≈1.14), P(18)=20 (20/16=1.25). Not exponential. Wait, maybe linear is the right graph? No, maybe I messed up the models.

Wait, correct matching:

- Linear growth: straight line, constant rate. So the graph where the change in P per unit t is constant. Let's check the right graph: t=0 (4), t=3 (5) (ΔP=1), t=6 (8) (ΔP=3), t=9 (10) (ΔP=2), t=12 (14) (ΔP=4), t=15 (16) (ΔP=2), t=18 (20) (ΔP=4). No, that's not constant. Wait, the left graph: t=0 (5), t=3 (5) (ΔP=0), t=6 (10) (ΔP=5), t=9 (12.5) (ΔP=2.5), t=12 (22.5) (ΔP=10), t=15 (40) (ΔP=17.5), t=18 (70) (ΔP=30). No. Middle graph: t=0 (0), t=3 (50) (ΔP=50), t=6 (100) (ΔP=50), t=9 (200) (ΔP=100), t=12 (275) (ΔP=75), t=15 (300) (ΔP=25), t=18 (300) (ΔP=0). Ah, t=0-3: 50, 3-6: 50 (constant), then 6-9: 100, etc. No, that's not linear. Wait, maybe the right graph is linear? No, I think I made a mistake. Let's recall:

- Linear: y = mx + b, so the graph is a straight line. So which graph looks like a straight line?

Wait, the right graph: let's plot the points (t, P):

t=0: 4

t=3: 5

t=6: 8

t=9: 10

t=12: 14

t=15: 16

t=18: 20

If we draw a line, does it fit? Let's check the slope between t=0 and t=18: (20-4)/(18-0)=16/18=8/9≈0.89. Between t=0 and t=6: (8-4)/6=4/6=2/3≈0.67. Not the same.

Left graph:

t=0: 5

t=3: 5

t=6: 10

t=9: 12.5

t=12: 22.5

t=15: 40

t=18: 70

Slope from 0-18: (70-5)/18=65/18≈3.61. From 0-6: (10-5)/6=5/6≈0.83. Not same.

Middle graph:

t=0: 0

t=3: 50

t=6: 100

t=9: 200

t=12: 275

t=15: 300

t=18: 300

Slope from 0-6: 100/6≈16.67. From 6-12: (275-100)/6=175/6≈29.17. Not same.

Wait, maybe the models are:

- Linear: constant rate (straight line) → right graph? No, maybe the left graph is exponential, middle is logistic, right is linear? No, I think I need to re-express:

Wait, the three growth models:

1. Linear growth: population increases by a constant amount each time period. So ΔP is constant.

2. Exponential growth: population increases by a constant percentage each time period. So ΔP is proportional to current P.

3. Logistic growth: population grows exponentially at first, then slows as it approaches carrying capacity (S-curve).

So:

- Linear: ΔP constant. Let's check the right graph: t=3-6: ΔP=8-5=3; t=6-9: 10-8=2; t=9-12:14-10=4; t=12-15:16-14=2; t=15-18:20-16=4. Not constant.

Left graph: t=3-6: 10-5=5; t=6-9:12.5-10=2.5; t=9-12:22.5-12.5=10; t=12-15:40-22.5=17.5; t=15-18:70-40=30. Not constant.

Middle graph: t=3-6:100-50=50; t=6-9:200-100=100; t=9-12:275-200=75; t=12-15:300-275=25; t=15-18:300-300=0. So it starts with ΔP=50 (constant for first two intervals), then increases, then decreases, then stops. So this is logistic (reaches carrying capacity at 300).

Exponential growth: ΔP proportional to P. So for left graph, P at t=6:10, t=9:12.5. 12.5/10=1.25. t=12:22.5, 22.5/12.5=1.8. No, not constant. Wait, t=0:5, t=3:5 (no growth), t=6:10 (10/5=2), t=9:12.5 (12.5/10=1.25), t=12:22.5 (22.5/12.5=1.8), t=15:40 (40/22.5≈1.78), t=18:70 (70/40=1.75). Not exponential.

Right graph: P at t=0:4, t=3:5 (5/4=1.25), t=6:8 (8/5=1.6), t=9:10 (10/8=1.25), t=12:14 (14/10=1.4), t=15:16 (16/14≈1.14), t=18:20 (20/16=1.25). The ratio is around 1.25-1.6, not constant. Wait, maybe the left graph is linear? No, I'm confused. Let's try to match:

- Linear growth: the graph with a straight line (constant slope). Let's assume the right graph is linear (maybe I misread the points). Wait, the right graph's y-axis: 4, 8, 12, 16, 20. So t=0:4, t=6:8 (Δt=6, ΔP=4 → slope 4/6=2/3), t=12:14 (Δt=6, ΔP=6 → slope 1), t=18:20 (Δt=6, ΔP=6 → slope 1). No, not constant.

Wait, maybe the middle graph is exponential? No, middle graph reaches a plateau (logistic). So:

- Logistic growth: middle graph (reaches carrying capacity at 300).

- Exponential growth: left graph (increases rapidly, no plateau).

- Linear growth: right graph (even though the slope isn't perfectly constant, maybe it's the closest).

So:

Left graph: exponential growth

Middle graph: logistic growth

Right graph: linear growth

Wait, but let's confirm:

- Linear: constant rate (straight line) → right graph.

- Exponential: increasing rate (curve) → left graph.

- Logistic: S-curve (increases, then plateaus) → middle graph.

Yes, that makes sense. So:

Left graph (first) → exponential growth

Middle graph (second) → logistic growth

Right graph (third) → linear growth

Step2: Match the Models

Now, match each model to the graph:

- Linear growth: right graph (third)

- Logistic growth: middle graph (second)

- Exponential growth: left graph (first)

Answer:

Left graph (first) → exponential growth
Middle graph (second) → logistic growth
Right graph (third) → linear growth

(So the boxes below the graphs should be filled as:
First box (left graph): exponential growth
Second box (middle graph): logistic growth
Third box (right graph): linear growth)