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Construct Linear Functions

Grade 9 · Algebra · Worksheet 3

  1. Mere is tracking the growth of a native tree species for a science project. She measures the tree's height at two different times. At the start of her observation (month 0), the tree is 12 centimeters tall. After 5 months, the tree has grown to 42 centimeters tall. Assuming the tree grows at a constant rate, construct a linear function H(t) that models the height of the tree in centimeters after t months. Then, use this function to predict the height of the tree after 8 months. Answer: ______________
  2. Mason is tracking the cost of renting a bicycle from a local shop. The shop charges a fixed initial fee plus a constant rate per hour. After 4 hours, the total cost is $26. After 9 hours, the total cost is $46. Determine the linear function C(h) that models the total cost in dollars as a function of the number of hours h rented. Then use this function to find the total cost for renting the bicycle for 12 hours. Answer: ______________
  3. Isabella is monitoring the water level in a reservoir during a drought. At 8:00 AM, the water level is 87 centimeters above the critical low mark. At 2:00 PM (6 hours later), the water level has dropped to 69 centimeters above the critical low mark. Assuming the water level decreases at a constant rate, write a linear function L(t) that models the water level in centimeters above the critical mark t hours after 8:00 AM. Then, use your function to predict how many hours after 8:00 AM the water level will reach the critical low mark (0 cm). Answer: ______________
  4. Aroha is tracking the cost of renting a hall for her community event. The rental company charges a fixed setup fee plus an hourly rate. For a 3-hour event, the total cost is $260. For a 7-hour event, the total cost is $460. Write a linear function C(h) that represents the total cost in dollars for renting the hall for h hours. Then, use your function to determine the cost for a 10-hour event. Answer: ______________
  5. f(x) = 4x³ - 2x² + 7x - 5, f(-2) = ? Answer: ______________
  6. f(x) = 2x + 5, g(x) = x² - 3, find f(g(2)) = ? Answer: ______________
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Answer Key & Explanations

Construct Linear Functions · Grade 9 · Worksheet 3

  1. Mere is tracking the growth of a native tree species for a science project. She measures the tree's height at two different times. At the start of her observation (month 0), the tree is 12 centimeters tall. After 5 months, the tree has grown to 42 centimeters tall. Assuming the tree grows at a constant rate, construct a linear function H(t) that models the height of the tree in centimeters after t months. Then, use this function to predict the height of the tree after 8 months. Answer: 60 centimeters Solution: Identify the two data points. At t = 0 months, height = 12 cm gives point (0, 12). At t = 5 months, height = 42 cm gives point (5, 42).
    Full step-by-step solution

    Step 1: Identify the two data points. At t = 0 months, height = 12 cm gives point (0, 12). At t = 5 months, height = 42 cm gives point (5, 42). Step 2: Find the slope m (growth rate per month). m = (42 - 12) / (5 - 0) = 30 / 5 = 6 cm per month. Step 3: The y-intercept b is the initial height at t = 0, so b = 12. Step 4: Write the linear function: H(t) = 6t + 12. Step 5: Predict height after 8 months by substituting t = 8: H(8) = 6(8) + 12 = 48 + 12 = 60. The answer is 60 centimeters.

  2. Mason is tracking the cost of renting a bicycle from a local shop. The shop charges a fixed initial fee plus a constant rate per hour. After 4 hours, the total cost is $26. After 9 hours, the total cost is $46. Determine the linear function C(h) that models the total cost in dollars as a function of the number of hours h rented. Then use this function to find the total cost for renting the bicycle for 12 hours. Answer: 58 Solution: Identify the two points from the problem. The cost depends on hours, so hours are the independent variable (h) and cost is the dependent variable (C). The points are (4, 26) and (9, 46).
    Full step-by-step solution

    Step 1: Identify the two points from the problem. The cost depends on hours, so hours are the independent variable (h) and cost is the dependent variable (C). The points are (4, 26) and (9, 46). Step 2: Find the slope m using the formula m = (y2 - y1) / (x2 - x1). m = (46 - 26) / (9 - 4) = 20 / 5 = 4. So the hourly rate is $4 per hour. Step 3: Use the point-slope form to find the equation. Use point (4, 26): C - 26 = 4(h - 4) C - 26 = 4h - 16 C = 4h + 10 So the linear function is C(h) = 4h + 10. Step 4: Find the cost for 12 hours by substituting h = 12: C(12) = 4(12) + 10 = 48 + 10 = 58. The answer is 58 dollars.

  3. Isabella is monitoring the water level in a reservoir during a drought. At 8:00 AM, the water level is 87 centimeters above the critical low mark. At 2:00 PM (6 hours later), the water level has dropped to 69 centimeters above the critical low mark. Assuming the water level decreases at a constant rate, write a linear function L(t) that models the water level in centimeters above the critical mark t hours after 8:00 AM. Then, use your function to predict how many hours after 8:00 AM the water level will reach the critical low mark (0 cm). Answer: 29 hours Solution: Identify the data points. At t = 0 hours (8:00 AM), level = 87 cm. At t = 6 hours (2:00 PM), level = 69 cm.
    Full step-by-step solution

    Step 1: Identify the data points. At t = 0 hours (8:00 AM), level = 87 cm. At t = 6 hours (2:00 PM), level = 69 cm. Points: (0, 87) and (6, 69). Step 2: Find the slope m = (69 - 87) / (6 - 0) = (-18) / 6 = -3. So the water level drops 3 cm per hour. Step 3: Use the point (0, 87) as the y-intercept b. The linear function is L(t) = -3t + 87. Step 4: To find when the level reaches 0, set L(t) = 0: 0 = -3t + 87. Step 5: Solve for t: 3t = 87, so t = 87 / 3 = 29. The answer is 29 hours after 8:00 AM.

  4. Aroha is tracking the cost of renting a hall for her community event. The rental company charges a fixed setup fee plus an hourly rate. For a 3-hour event, the total cost is $260. For a 7-hour event, the total cost is $460. Write a linear function C(h) that represents the total cost in dollars for renting the hall for h hours. Then, use your function to determine the cost for a 10-hour event. Answer: $610 Solution: Identify the two data points from the problem: (3, 260) and (7, 460). Let h represent hours and C represent total cost. Calculate the slope (hourly rate) using the formula m = (C2 - C1) / (h2 - h1).
    Full step-by-step solution

    Step 1: Identify the two data points from the problem: (3, 260) and (7, 460). Let h represent hours and C represent total cost. Step 2: Calculate the slope (hourly rate) using the formula m = (C2 - C1) / (h2 - h1). So, m = (460 - 260) / (7 - 3) = 200 / 4 = 50. Step 3: Use point-slope form to find the equation. Using point (3, 260): C - 260 = 50(h - 3). Step 4: Simplify to slope-intercept form: C - 260 = 50h - 150. Then, C = 50h + 110. Step 5: The linear function is C(h) = 50h + 110. Step 6: To find the cost for 10 hours, substitute h = 10: C(10) = 50(10) + 110 = 500 + 110 = 610. The answer is $610.

  5. f(x) = 4x³ - 2x² + 7x - 5, f(-2) = ? Answer: -59 Solution: Substitute x = -2 into the function: f(-2) = 4(-2)³ - 2(-2)² + 7(-2) - 5 Evaluate (-2)³ = -8, so 4(-8) = -32 Evaluate (-2)² = 4, so -2(4) = -8 Evaluate 7(-2) = -14 Combine all terms: -32 - 8 - 14 - 5 -32 - 8 = -40 -40 - 14 = -54 -54 - 5 = -59 The answer is -59.
    Full step-by-step solution

    Step 1: Substitute x = -2 into the function: f(-2) = 4(-2)³ - 2(-2)² + 7(-2) - 5 Step 2: Evaluate (-2)³ = -8, so 4(-8) = -32 Step 3: Evaluate (-2)² = 4, so -2(4) = -8 Step 4: Evaluate 7(-2) = -14 Step 5: Combine all terms: -32 - 8 - 14 - 5 Step 6: -32 - 8 = -40 Step 7: -40 - 14 = -54 Step 8: -54 - 5 = -59 The answer is -59.

  6. f(x) = 2x + 5, g(x) = x² - 3, find f(g(2)) = ? Answer: 7 Solution: f(x) = 2x + 5 g(x) = x² - 3 We need to find f(g(2)). Find g(2). g(x) = x² - 3 Substitute x = 2: g(2) = (2)² - 3 g(2) = 4 - 3 g(2) = 1 Now find f(g(2)).
    Full step-by-step solution

    We are given: f(x) = 2x + 5 g(x) = x² - 3 We need to find f(g(2)). Step 1: Find g(2). g(x) = x² - 3 Substitute x = 2: g(2) = (2)² - 3 g(2) = 4 - 3 g(2) = 1 Step 2: Now find f(g(2)). Since g(2) = 1, f(g(2)) = f(1) Step 3: Evaluate f(1). f(x) = 2x + 5 Substitute x = 1: f(1) = 2(1) + 5 f(1) = 2 + 5 f(1) = 7 Step 4: Conclusion. f(g(2)) = 7 Final answer: 7