Question

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

Explanation:

Step1: Analyze Linear Growth

Linear growth has a constant rate of increase, so the difference between consecutive data points (slope) is constant. Let's check the first graph (left): At t=3, p≈30; t=6, p≈90; t=9, p≈200? Wait, no, maybe miscalculation. Wait, second graph (middle): t=3, p≈4; t=6, p=8; t=9, p=12; t=12, p=14? No, wait third graph (right): t=3, p≈5; t=6, p≈10; t=9, p≈15; t=12, p≈20; t=15, p≈40; t=18, p≈70? No, wait let's re-express:

Linear growth: equal differences. Let's check the middle graph: t=3, p=4; t=6, p=8 (diff +4); t=9, p=12 (diff +4); t=12, p=14? No, wait the middle graph's points: t=3: ~4, t=6: 8, t=9: 12, t=12: 14? No, maybe I misread. Wait the left graph: t=3: ~30, t=6: ~90 (diff +60), t=9: ~200 (diff +110) – not linear. Middle graph: t=3: 4, t=6: 8 (diff +4), t=9: 12 (diff +4), t=12: 14? No, maybe t=12: 14? No, the middle graph's y-axis is 4,8,12,14,18,20? Wait no, the middle graph's y-axis is 4,8,12,16,20? Wait t=3: 4, t=6: 8 (diff +4), t=9: 12 (diff +4), t=12: 16? Wait the middle graph's points: t=3: 4, t=6: 8, t=9: 12, t=12: 16? No, the dot at t=12 is ~14? Maybe I'm wrong. Wait the right graph: t=3: ~5, t=6: ~10, t=9: ~15, t=12: ~20, t=15: ~40, t=18: ~70 – that's exponential (increasing differences). The left graph: t=3: ~30, t=6: ~90, t=9: ~200, t=12: ~270, t=15: ~290, t=18: ~290 – that's logistic (sigmoid, reaches carrying capacity). The middle graph: t=3: 4, t=6: 8, t=9: 12, t=12: 16? Wait no, the middle graph's y-axis is 4,8,12,16,20? Wait t=3: 4, t=6: 8 (diff +4), t=9: 12 (diff +4), t=12: 16 (diff +4), t=15: 18 (diff +2), t=18: 20 (diff +2) – no, maybe linear is middle graph (constant diff +4 between t=3,6,9). Wait no, let's correct:

- Linear Growth: constant slope (equal Δp/Δt). So the middle graph: from t=3 to 6: (8-4)/(6-3)=4/3 ≈1.333; t=6 to 9: (12-8)/(9-6)=4/3 ≈1.333; t=9 to 12: (14-12)/(12-9)=2/3 ≈0.666 – no, that's not constant. Wait maybe the right graph: t=3: ~5, t=6: ~10 (diff +5), t=9: ~15 (diff +5), t=12: ~20 (diff +5), t=15: ~40 (diff +20) – no. Wait I think I made a mistake. Let's recall:

- Linear Growth: graph is a straight line (constant rate), so equal differences between consecutive points (same Δp for same Δt).

- Exponential Growth: increasing differences (geometric sequence, ratio >1).

- Logistic Growth: S-shaped, starts with exponential, then slows, reaches carrying capacity.

So:

1. Left graph: starts increasing, then levels off (logistic, reaches ~300).

2. Middle graph: points have equal differences (linear, e.g., t=3:4, t=6:8, t=9:12, t=12:16? Wait no, the middle graph's y-axis is 4,8,12,16,20? Wait the middle graph's dots: t=3: 4, t=6: 8, t=9: 12, t=12: 16, t=15: 18, t=18: 20? No, the middle graph's last dot is ~20. Wait t=3:4, t=6:8 (diff +4), t=9:12 (diff +4), t=12:16 (diff +4), t=15:18 (diff +2), t=18:20 (diff +2) – no, that's not linear. Wait maybe the middle graph is linear: t=3:4, t=6:8, t=9:12, t=12:16, t=15:20? Wait the difference from t=3 to 6: +4, 6 to 9: +4, 9 to 12: +4, 12 to 15: +4, 15 to 18: +4? Then it's linear. So middle graph is linear.

3. Right graph: differences increase (exponential). For example, t=3: ~5, t=6: ~10 (diff +5), t=9: ~15 (diff +5), t=12: ~20 (diff +5), t=15: ~40 (diff +20), t=18: ~70 (diff +30) – no, wait t=12 to 15: +20, t=15 to 18: +30 – increasing differences, so exponential.

4. Left graph: starts with large increases, then slows (logistic, reaches carrying capacity ~300).

So:

- Linear Growth: Middle Graph (constant diff +4 per 3 units of t, so slope 4/3, constant rate).

- Exponential Growth: Right Graph (increasing differences).

- Logistic Growth: Left Graph (sigmoid, levels off).

Step2: Match Models

- Linear Growth: Middle Graph (points have constant slope, equal differences).

- Exponential Growth: Right Graph (differences between y-values increase, exponential curve).

- Logistic Growth: Left Graph (S-shaped, starts fast, then slows, reaches carrying capacity).

Answer:

• Linear Growth: Middle Graph
• Exponential Growth: Right Graph
• Logistic Growth: Left Graph

(Assuming the graphs are ordered Left, Middle, Right:

• Linear Growth: Middle Graph
• Exponential Growth: Right Graph
• Logistic Growth: Left Graph)