Construct Linear Functions

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Answer Key & Explanations

Construct Linear Functions · Grade 9 · Worksheet 1

1. Aroha is tracking the cost of a gym membership. The membership has an initial registration fee and a fixed monthly charge. After 4 months, the total cost is $260. After 9 months, the total cost is $460. Assuming the total cost increases linearly with the number of months, write a linear function C(m) that gives the total cost after m months. Then use your function to find the total cost after 15 months. Answer: $700 Solution: Identify the two points: (4, 260) and (9, 460). Find the slope (monthly charge): m = (460 - 260) / (9 - 4) = 200 / 5 = 40. Use point-slope form: C(m) - 260 = 40(m - 4).
   Full step-by-step solution

   Step 1: Identify the two points: (4, 260) and (9, 460). Step 2: Find the slope (monthly charge): m = (460 - 260) / (9 - 4) = 200 / 5 = 40. Step 3: Use point-slope form: C(m) - 260 = 40(m - 4). Step 4: Simplify to slope-intercept form: C(m) = 40m - 160 + 260 = 40m + 100. Step 5: The linear function is C(m) = 40m + 100. Step 6: Find cost after 15 months: C(15) = 40(15) + 100 = 600 + 100 = 700. The answer is $700.

2. A local tech startup is analyzing their app's user growth. They found that the number of users follows the function f(x) = 1200(1.08)^x, where x represents months since launch. Meanwhile, their revenue is modeled by g(x) = 25x + 500. If they want to calculate the revenue per user after x months, what composite function would represent this relationship? Answer: g(x)/f(x) = (25x + 500)/(1200(1.08)^x) Solution: - f(x) = 1200(1.08)^x represents the number of users after x months. - g(x) = 25x + 500 represents the revenue after x months. Determine what "revenue per user" means.
   Full step-by-step solution

   Step 1: Understand the given functions. We have two functions: - f(x) = 1200(1.08)^x represents the number of users after x months. - g(x) = 25x + 500 represents the revenue after x months. Step 2: Determine what "revenue per user" means. Revenue per user is the total revenue divided by the total number of users. So, after x months, revenue per user = g(x) / f(x). Step 3: Write the composite function for revenue per user. Substitute the expressions for g(x) and f(x): Revenue per user = (25x + 500) / (1200(1.08)^x). Step 4: Final answer. The composite function representing revenue per user after x months is: g(x)/f(x) = (25x + 500) / (1200(1.08)^x).

3. Points (5, 13) and (9, 29). Find f(x) = mx + b. Answer: f(x) = 4x - 7 Solution: Find the slope m = (y2 - y1) / (x2 - x1) = (29 - 13) / (9 - 5) = 16 / 4 = 4. Use point-slope form: y - y1 = m(x - x1). Using (5, 13): y - 13 = 4(x - 5).
   Full step-by-step solution

   Step 1: Find the slope m = (y2 - y1) / (x2 - x1) = (29 - 13) / (9 - 5) = 16 / 4 = 4. Step 2: Use point-slope form: y - y1 = m(x - x1). Using (5, 13): y - 13 = 4(x - 5). Step 3: Simplify: y - 13 = 4x - 20. Step 4: Solve for y: y = 4x - 20 + 13 = 4x - 7. Step 5: Therefore, f(x) = 4x - 7.

4. Noah is tracking the cost of renting a delivery truck. The rental company charges a fixed daily fee plus a variable cost per kilometer driven. On Monday, Noah drove 80 km and paid $220. On Wednesday, he drove 150 km and paid $345. Assuming the cost is a linear function of the distance driven, determine the linear function C(d) that gives the total cost in dollars for driving d kilometers. Answer: C(d) = 1.25d + 120 Solution: Let C(d) = md + b, where m is the cost per km and b is the fixed daily fee.
   Full step-by-step solution

   Step 1: Let C(d) = md + b, where m is the cost per km and b is the fixed daily fee. Step 2: Using (80, 220): 220 = 80m + b Step 3: Using (150, 345): 345 = 150m + b Step 4: Subtract the first equation from the second: (345 - 220) = (150m - 80m) + (b - b) -> 125 = 70m Step 5: Solve for m: m = 125 / 70 = 1.25 (cost per km) Step 6: Substitute m = 1.25 into 220 = 80(1.25) + b: 220 = 100 + b -> b = 120 Step 7: The linear function is C(d) = 1.25d + 120. The answer is C(d) = 1.25d + 120.

5. Olivia is training for a charity bike ride. She records the distance she travels and the amount of money pledged per kilometer. On her first training ride of 7 kilometers, she raises $91. On her second ride of 13 kilometers, she raises $169. Assuming the relationship between distance traveled (d, in km) and money raised (R, in dollars) is linear, construct a linear function R(d) that models this relationship. Then use your function to determine how much money Olivia would raise if she completes a 21-kilometer ride. Answer: R(21) = 273 dollars Solution: Identify the two data points as (d1, R1) = (7, 91) and (d2, R2) = (13, 169). Calculate the slope m = (R2 - R1) / (d2 - d1) = (169 - 91) / (13 - 7) = 78 / 6 = 13. This means Olivia raises $13 per kilometer.
   Full step-by-step solution

   Step 1: Identify the two data points as (d1, R1) = (7, 91) and (d2, R2) = (13, 169). Step 2: Calculate the slope m = (R2 - R1) / (d2 - d1) = (169 - 91) / (13 - 7) = 78 / 6 = 13. This means Olivia raises $13 per kilometer. Step 3: Use the point-slope form: R - 91 = 13(d - 7). Step 4: Simplify to slope-intercept form: R = 13d - 91 + 91 = 13d. So the linear function is R(d) = 13d. Step 5: Substitute d = 21 into the function: R(21) = 13 * 21 = 273. The answer is 273 dollars.

6. A tech startup is modeling their user growth with the function f(x) = 2x² - 12x + 20, where x represents months since launch and f(x) represents thousands of users. The company wants to know the minimum number of users they'll have during this growth period. At what month will this minimum occur, and what is the minimum number of users? Answer: Month 3 with 2,000 users Solution: f(x) = 2x² - 12x + 20 where x = months since launch, f(x) = thousands of users. This is a quadratic function of the form ax² + bx + c, with a = 2, b = -12, c = 20.
   Full step-by-step solution

   Let's solve this step by step. We are given: f(x) = 2x² - 12x + 20 where x = months since launch, f(x) = thousands of users. --- **Step 1: Identify the type of function** This is a quadratic function of the form ax² + bx + c, with a = 2, b = -12, c = 20. Since a = 2 > 0, the parabola opens upwards, so the vertex is the minimum point. --- **Step 2: Find the x-coordinate of the vertex** For a quadratic ax² + bx + c, the x-coordinate of the vertex is: x = -b / (2a) Here, b = -12, a = 2 x = -(-12) / (2 * 2) = 12 / 4 = 3 So the minimum occurs at month 3. --- **Step 3: Find the minimum number of users** Substitute x = 3 into f(x): f(3) = 2*(3)² - 12*(3) + 20 = 2*9 - 36 + 20 = 18 - 36 + 20 = (18 + 20) - 36 = 38 - 36 = 2 Since f(x) is in thousands of users, f(3) = 2 means 2,000 users. --- **Step 4: Conclusion** The minimum occurs at month 3 with 2,000 users. --- **Final Answer:** Month 3 with 2,000 users