Model Comparison

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Answer Key & Explanations

Model Comparison · Grade 11 · Worksheet 3

1. Compare f(x)=11x+6, g(x)=x²+16, h(x)=6^x for large x. Which function grows fastest? Answer: h(x)=6^x Solution: Analyze f(x)=11x+6 (linear function). As x increases, f(x) grows at a constant rate of 11 per unit increase in x. For large x, f(x) ≈ 11x.
   Full step-by-step solution

   Step 1: Analyze f(x)=11x+6 (linear function). As x increases, f(x) grows at a constant rate of 11 per unit increase in x. For large x, f(x) ≈ 11x. Step 2: Analyze g(x)=x²+16 (quadratic function). As x increases, g(x) grows proportionally to x squared. For large x, g(x) ≈ x². Quadratic growth is faster than linear growth because x² eventually exceeds 11x for sufficiently large x. Step 3: Analyze h(x)=6^x (exponential function with base 6 > 1). As x increases, h(x) multiplies by 6 for each unit increase in x. For large x, h(x) ≈ 6^x. Step 4: Compare growth rates for large x. Exponential functions with base > 1 always grow faster than any polynomial function (linear, quadratic, cubic, etc.) for sufficiently large x. Since 6^x grows much faster than x² and 11x, h(x) will eventually be the largest. Therefore, h(x)=6^x grows fastest for large x.

2. Hana is a city planner evaluating three models for the growth of a new urban district's population over time, where t is measured in years since the district opened. Model L predicts linear growth: P(t) = 2400 + 180t. Model Q predicts quadratic growth: P(t) = 12t^2 + 2400. Model E predicts exponential growth: P(t) = 2400(1.06)^t. Determine which model predicts the highest population after 20 years, and by how many people that model exceeds the second-highest prediction. Answer: Model E; 3102 people Solution: Evaluate Model L (linear) at t = 20. P(20) = 2400 + 180(20) = 2400 + 3600 = 6000 people. Evaluate Model Q (quadratic) at t = 20.
   Full step-by-step solution

   Step 1: Evaluate Model L (linear) at t = 20. P(20) = 2400 + 180(20) = 2400 + 3600 = 6000 people. Step 2: Evaluate Model Q (quadratic) at t = 20. P(20) = 12(20)^2 + 2400 = 12(400) + 2400 = 4800 + 2400 = 7200 people. Step 3: Evaluate Model E (exponential) at t = 20. P(20) = 2400(1.06)^20. First, compute (1.06)^20. Note that (1.06)^2 = 1.1236. (1.06)^4 = (1.1236)^2 = 1.2625 (rounded). (1.06)^8 = (1.2625)^2 = 1.5938 (rounded). (1.06)^16 = (1.5938)^2 = 2.5402 (rounded). Then (1.06)^20 = (1.06)^16 * (1.06)^4 = 2.5402 * 1.2625 = 3.2075 (rounded). Thus, P(20) = 2400 * 3.2075 = 7698 people (rounded to the nearest whole number). Step 4: Compare the three populations. Model L: 6000 Model Q: 7200 Model E: 7698 Model E predicts the highest population. Step 5: Find the difference between Model E and the second-highest (Model Q). Difference = 7698 - 7200 = 498 people. However, for a more precise calculation: (1.06)^20 = (1.06^10)^2. 1.06^5 = 1.3382, 1.06^10 = 1.3382^2 = 1.7908, 1.06^20 = 1.7908^2 = 3.2071. Then P(20) = 2400 * 3.2071 = 7697.04, which rounds to 7697 people. Difference = 7697 - 7200 = 497 people. Using a calculator for exact value: (1.06)^20 = 3.207135... P(20) = 2400 * 3.207135 = 7697.124, so 7697 people. Difference = 7697 - 7200 = 497 people. The answer is Model E exceeds Model Q by 497 people.

3. Compare f(x) = 13x + 20, g(x) = 3x² + 11, h(x) = 4^x for large x. Which function grows fastest? Answer: h(x) = 4^x Solution: Analyze f(x) = 13x + 20 (linear function). As x increases, f(x) grows at a constant rate of 13 per unit increase in x. For large x, f(x) behaves like 13x.
   Full step-by-step solution

   Step 1: Analyze f(x) = 13x + 20 (linear function). As x increases, f(x) grows at a constant rate of 13 per unit increase in x. For large x, f(x) behaves like 13x. Step 2: Analyze g(x) = 3x² + 11 (quadratic function). As x increases, g(x) grows proportionally to x². For large x, g(x) behaves like 3x². Quadratic growth is faster than linear growth because x² grows faster than x. Step 3: Analyze h(x) = 4^x (exponential function with base 4 > 1). As x increases, h(x) multiplies by 4 for each unit increase in x. For large x, h(x) behaves like 4^x. Step 4: Compare growth rates. For small x, linear or quadratic might be larger, but for large x: - Linear: f(x) ≈ 13x - Quadratic: g(x) ≈ 3x² - Exponential: h(x) ≈ 4^x Exponential functions with base > 1 always grow faster than any polynomial function (including quadratic) for sufficiently large x. Therefore, h(x) = 4^x grows fastest. The answer is h(x) = 4^x.

4. Compare f(x)=12x+15, g(x)=2x²+10, h(x)=4^x for large x. Which function grows fastest? Answer: h(x)=4^x Solution: Analyze f(x)=12x+15 (linear function). For large x, f(x) grows at a constant rate of 12 per unit increase in x. Step 2: Analyze g(x)=2x²+10 (quadratic function).
   Full step-by-step solution

   Step 1: Analyze f(x)=12x+15 (linear function). For large x, f(x) grows at a constant rate of 12 per unit increase in x. Step 2: Analyze g(x)=2x²+10 (quadratic function). For large x, g(x) grows proportionally to x², which is faster than linear growth. Step 3: Analyze h(x)=4^x (exponential function). For large x, h(x) grows by multiplying by 4 for each unit increase in x. Step 4: Compare growth rates. For large x, exponential functions with base > 1 always outpace any polynomial function. Therefore, h(x)=4^x grows fastest. The answer is h(x)=4^x.

5. Matiu is comparing three functions: f(x) = 13x + 21 (linear), g(x) = 4x² + 11 (quadratic), and h(x) = 6^x (exponential). For large x, which function grows fastest? Answer: h(x) = 6^x Solution: Analyze f(x) = 13x + 21 (linear). As x increases, f(x) grows by adding 13 each time. For x = 10, f(10) = 13(10) + 21 = 151.
   Full step-by-step solution

   Step 1: Analyze f(x) = 13x + 21 (linear). As x increases, f(x) grows by adding 13 each time. For x = 10, f(10) = 13(10) + 21 = 151. For x = 100, f(100) = 13(100) + 21 = 1321. Step 2: Analyze g(x) = 4x² + 11 (quadratic). As x increases, g(x) grows proportionally to x². For x = 10, g(10) = 4(100) + 11 = 411. For x = 100, g(100) = 4(10000) + 11 = 40011. Step 3: Analyze h(x) = 6^x (exponential). As x increases, h(x) multiplies by 6 each time. For x = 10, h(10) = 6^10 = 60466176. For x = 100, h(100) = 6^100, which is astronomically large (approximately 6.5 × 10^77). Step 4: Compare growth rates. For large x, exponential functions with base > 1 always outpace any polynomial function (linear or quadratic). Therefore, h(x) = 6^x grows fastest. The answer is h(x) = 6^x.

6. A company is analyzing three different investment growth models. The linear model is L(t) = 800 + 120t, the quadratic model is Q(t) = 800 + 40t², and the exponential model is E(t) = 800 × 1.15^t, where t is time in years and all outputs are in dollars. At what time t (in years) will the exponential model predict exactly the same value as the quadratic model? Round your answer to the nearest tenth of a year. Answer: 4.8 Solution: Set E(t) = Q(t) 800 × 1.15^t = 800 + 40t² Divide both sides by 40 to simplify 20 × 1.15^t = 20 + t² t² = 20 × 1.15^t - 20 Try values of t to find where both sides are equal At t = 4: 4² = 16 and 20 × 1.15^4 - 20 = 20 × 1.749 - 20 = 34.98 - 20 = 14.98 At t = 5: 5² = 25 and 20 × 1.15^5 - 20 = 20 ×…
   Full step-by-step solution

   Step 1: Set E(t) = Q(t) 800 × 1.15^t = 800 + 40t² Step 2: Divide both sides by 40 to simplify 20 × 1.15^t = 20 + t² Step 3: Rearrange the equation t² = 20 × 1.15^t - 20 Step 4: Try values of t to find where both sides are equal At t = 4: 4² = 16 and 20 × 1.15^4 - 20 = 20 × 1.749 - 20 = 34.98 - 20 = 14.98 At t = 5: 5² = 25 and 20 × 1.15^5 - 20 = 20 × 2.011 - 20 = 40.22 - 20 = 20.22 Step 5: The solution is between t = 4 and t = 5 At t = 4.8: 4.8² = 23.04 and 20 × 1.15^4.8 - 20 = 20 × 2.152 - 20 = 43.04 - 20 = 23.04 Step 6: Both sides equal 23.04 at t = 4.8 The answer is 4.8.

7. Compare f(x) = 9x + 14, g(x) = 3x² + 11, and h(x) = 7^x for large x. Which function grows fastest? Answer: h(x) = 7^x Solution: Identify the function types. f(x) = 9x + 14 is linear (degree 1). g(x) = 3x² + 11 is quadratic (degree 2).
   Full step-by-step solution

   Step 1: Identify the function types. f(x) = 9x + 14 is linear (degree 1). g(x) = 3x² + 11 is quadratic (degree 2). h(x) = 7^x is exponential (base > 1). Step 2: Compare linear and quadratic. For large x, the quadratic term 3x² grows much faster than the linear term 9x, so g(x) > f(x) for sufficiently large x. Step 3: Compare quadratic and exponential. For large x, an exponential function with base > 1 (here 7) grows faster than any polynomial function. For example, at x = 10: g(10) = 3(100) + 11 = 311, while h(10) = 7^10 = 282475249. The exponential is already far larger. Step 4: Conclusion. For large x, the order of growth is: f(x) < g(x) < h(x). Therefore, h(x) = 7^x grows fastest.

8. Compare f(x)=6x+11, g(x)=x²+16, h(x)=2^x for large x. Which function grows fastest? Answer: h(x)=2^x Solution: Analyze f(x)=6x+11 (linear function). As x increases, f(x) grows by a constant 6 for each unit increase in x. For large x, f(x) behaves like 6x.
   Full step-by-step solution

   Step 1: Analyze f(x)=6x+11 (linear function). As x increases, f(x) grows by a constant 6 for each unit increase in x. For large x, f(x) behaves like 6x. Step 2: Analyze g(x)=x²+16 (quadratic function). As x increases, g(x) grows proportionally to x². For large x, the +16 becomes negligible, so g(x) behaves like x². Step 3: Analyze h(x)=2^x (exponential function). As x increases, h(x) doubles for each unit increase in x. For large x, this growth far exceeds any polynomial growth. Step 4: Compare growth rates for large x. For any exponential function with base greater than 1, it will eventually outgrow any polynomial function. Since 2^x grows faster than x², and x² grows faster than 6x, the order from slowest to fastest is: f(x) (linear), g(x) (quadratic), h(x) (exponential). Therefore, h(x)=2^x grows fastest for large x.