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Construct Linear Functions

Grade 9 · Algebra · Worksheet 2

  1. 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: ______________
  2. Aroha is tracking the growth of a rare fern in her backyard. She measures the height of the fern at two different times. After 3 weeks, the fern is 15 cm tall. After 7 weeks, the fern is 27 cm tall. Assuming the fern grows at a constant rate, write a linear function h(t) that gives the height of the fern in centimeters after t weeks. Use your function to predict the height of the fern after 11 weeks. Answer: ______________
  3. Emma is analyzing the fuel efficiency of a delivery truck. The graph shows the relationship between the number of gallons of fuel used (x) and the distance traveled in miles (y). The line on the graph passes through the points (0, 0) and (25, 400). What is the linear function f(x) that models this relationship, representing the distance traveled for a given number of gallons of fuel used? Answer: ______________
  4. Mason is analyzing the distance a car travels over time. On a coordinate plane, a line representing the linear function f(x) passes through the points (9, 45) and (18, 90). What is the equation of this linear function in slope-intercept form? Answer: ______________
  5. Aroha is tracking the cost of renting a party hall. The total cost C(d) in dollars for renting the hall for d days is a linear function. She knows that renting the hall for 9 days costs $1,020, and renting it for 14 days costs $1,470. Determine the linear function C(d) that models the total cost, and use it to find the cost of renting the hall for 20 days. Answer: ______________
  6. A tech startup is modeling their app's 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 determine the minimum number of users they'll have and when this minimum occurs. Find the vertex of this quadratic function. Answer: ______________
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Answer Key & Explanations

Construct Linear Functions · Grade 9 · Worksheet 2

  1. 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.

  2. Aroha is tracking the growth of a rare fern in her backyard. She measures the height of the fern at two different times. After 3 weeks, the fern is 15 cm tall. After 7 weeks, the fern is 27 cm tall. Assuming the fern grows at a constant rate, write a linear function h(t) that gives the height of the fern in centimeters after t weeks. Use your function to predict the height of the fern after 11 weeks. Answer: 39 cm Solution: Identify the two points from the data: (3, 15) and (7, 27), where t is time in weeks and h is height in cm. Calculate the slope m = (27 - 15) / (7 - 3) = 12 / 4 = 3.
    Full step-by-step solution

    Step 1: Identify the two points from the data: (3, 15) and (7, 27), where t is time in weeks and h is height in cm. Step 2: Calculate the slope m = (27 - 15) / (7 - 3) = 12 / 4 = 3. Step 3: Use the point-slope form with one point, say (3, 15): h - 15 = 3(t - 3). Step 4: Simplify to slope-intercept form: h - 15 = 3t - 9, so h = 3t + 6. Step 5: The linear function is h(t) = 3t + 6. Step 6: Predict the height after 11 weeks: h(11) = 3(11) + 6 = 33 + 6 = 39. The answer is 39 cm.

  3. Emma is analyzing the fuel efficiency of a delivery truck. The graph shows the relationship between the number of gallons of fuel used (x) and the distance traveled in miles (y). The line on the graph passes through the points (0, 0) and (25, 400). What is the linear function f(x) that models this relationship, representing the distance traveled for a given number of gallons of fuel used? Answer: f(x) = 16x Solution: Identify the two points from the graph: (0, 0) and (25, 400). Calculate the slope (m) using the formula m = (y2 - y1) / (x2 - x1). m = (400 - 0) / (25 - 0) = 400 / 25.
    Full step-by-step solution

    Step 1: Identify the two points from the graph: (0, 0) and (25, 400). Step 2: Calculate the slope (m) using the formula m = (y2 - y1) / (x2 - x1). Step 3: m = (400 - 0) / (25 - 0) = 400 / 25. Step 4: Simplify the fraction: 400 / 25 = 16. Step 5: The slope m = 16 means the truck travels 16 miles per gallon of fuel. Step 6: Since the line passes through (0, 0), the y-intercept b = 0. Step 7: Write the equation in slope-intercept form: f(x) = mx + b. Step 8: Substitute the values: f(x) = 16x + 0. Step 9: The linear function is f(x) = 16x. The answer is f(x) = 16x.

  4. Mason is analyzing the distance a car travels over time. On a coordinate plane, a line representing the linear function f(x) passes through the points (9, 45) and (18, 90). What is the equation of this linear function in slope-intercept form? Answer: f(x) = 5x Solution: Identify the two points: (9, 45) and (18, 90). Calculate the slope using m = (y2 - y1) / (x2 - x1). m = (90 - 45) / (18 - 9) = 45 / 9 = 5.
    Full step-by-step solution

    Step 1: Identify the two points: (9, 45) and (18, 90). Step 2: Calculate the slope using m = (y2 - y1) / (x2 - x1). Step 3: m = (90 - 45) / (18 - 9) = 45 / 9 = 5. Step 4: Use slope-intercept form f(x) = mx + b. Substitute one point, (9, 45), and m = 5: 45 = 5(9) + b. Step 5: 45 = 45 + b, so b = 0. Step 6: The equation is f(x) = 5x + 0, or f(x) = 5x. The answer is f(x) = 5x.

  5. Aroha is tracking the cost of renting a party hall. The total cost C(d) in dollars for renting the hall for d days is a linear function. She knows that renting the hall for 9 days costs $1,020, and renting it for 14 days costs $1,470. Determine the linear function C(d) that models the total cost, and use it to find the cost of renting the hall for 20 days. Answer: C(d) = 90d + 210, cost for 20 days = $2,010 Solution: Identify the two data points as (9, 1020) and (14, 1470). Calculate the slope m = (1470 - 1020) / (14 - 9) = 450 / 5 = 90. Use the point-slope form: C(d) - 1020 = 90(d - 9).
    Full step-by-step solution

    Step 1: Identify the two data points as (9, 1020) and (14, 1470). Step 2: Calculate the slope m = (1470 - 1020) / (14 - 9) = 450 / 5 = 90. Step 3: Use the point-slope form: C(d) - 1020 = 90(d - 9). Step 4: Simplify: C(d) - 1020 = 90d - 810, so C(d) = 90d + 210. Step 5: For d = 20, C(20) = 90(20) + 210 = 1800 + 210 = 2010. The linear function is C(d) = 90d + 210, and the cost for 20 days is $2,010.

  6. A tech startup is modeling their app's 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 determine the minimum number of users they'll have and when this minimum occurs. Find the vertex of this quadratic function. Answer: (3, 2) Solution: To find the vertex of the quadratic function f(x) = 2x² - 12x + 20, we can use the vertex formula. For a quadratic in the form f(x) = ax² + bx + c, the x-coordinate of the vertex is given by x = -b / (2a).
    Full step-by-step solution

    To find the vertex of the quadratic function f(x) = 2x² - 12x + 20, we can use the vertex formula. For a quadratic in the form f(x) = ax² + bx + c, the x-coordinate of the vertex is given by x = -b / (2a). Step 1: Identify the coefficients a, b, and c from the function. a = 2 b = -12 c = 20 Step 2: Calculate the x-coordinate of the vertex using the formula x = -b / (2a). x = -(-12) / (2 * 2) x = 12 / 4 x = 3 So, the minimum number of users occurs 3 months after launch. Step 3: Calculate the y-coordinate of the vertex by substituting x = 3 back into the original function f(x). f(3) = 2*(3)² - 12*(3) + 20 f(3) = 2*(9) - 12*(3) + 20 f(3) = 18 - 36 + 20 f(3) = (18 + 20) - 36 f(3) = 38 - 36 f(3) = 2 Since f(x) represents thousands of users, the minimum number of users is 2,000. Step 4: State the vertex as an ordered pair (x, y). The vertex is (3, 2). This means the minimum number of users is 2,000, and it occurs 3 months after the app's launch.