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Construct Linear Functions

Grade 9 · Algebra · Worksheet 3

  1. Points (11, 47) and (17, 83). Find f(x) = mx + b. Answer: ______________
  2. A right triangle is drawn on a coordinate plane with vertices at (0,0), (4,0), and (4,3). A line is drawn from the vertex at (4,3) perpendicular to the hypotenuse, intersecting it at point P. What are the coordinates of point P? Answer: ______________
  3. Points (12, 47) and (20, 71). Find f(x) = mx + b. Answer: ______________
  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: ______________
  5. A coordinate plane shows a line passing through two points: (1, 11) and (6, 36). The line represents a linear function f(x). What is the equation of this linear function in slope-intercept form? Answer: ______________
  6. Noah is tracking the distance a delivery drone travels over time during a test flight. At 2 seconds, the drone has traveled 14 meters from its starting point. At 7 seconds, it has traveled 44 meters. Assuming the drone moves at a constant speed, construct a linear function d(t) that gives the distance in meters from the starting point after t seconds. Then use this function to predict the distance traveled after 10 seconds. Answer: ______________
  7. 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: ______________
  8. f(x) = 2x² - 3x + 7, f(-2) = ? Answer: ______________
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Answer Key & Explanations

Construct Linear Functions · Grade 9 · Worksheet 3

  1. Points (11, 47) and (17, 83). Find f(x) = mx + b. Answer: f(x) = 6x - 19 Solution: Find the slope m using m = (y2 - y1)/(x2 - x1). m = (83 - 47)/(17 - 11) = 36/6 = 6. Use point-slope form with point (11, 47): y - 47 = 6(x - 11).
    Full step-by-step solution

    Step 1: Find the slope m using m = (y2 - y1)/(x2 - x1). m = (83 - 47)/(17 - 11) = 36/6 = 6. Step 2: Use point-slope form with point (11, 47): y - 47 = 6(x - 11). Step 3: Simplify: y - 47 = 6x - 66. Step 4: Solve for y: y = 6x - 66 + 47 = 6x - 19. Step 5: Therefore, f(x) = 6x - 19. The answer is f(x) = 6x - 19.

  2. A right triangle is drawn on a coordinate plane with vertices at (0,0), (4,0), and (4,3). A line is drawn from the vertex at (4,3) perpendicular to the hypotenuse, intersecting it at point P. What are the coordinates of point P? Answer: (2.56, 1.92) Solution: In a right triangle, when a perpendicular is drawn from the right angle to the hypotenuse, it creates two smaller triangles that are similar to the original triangle and to each other.
    Full step-by-step solution

    In a right triangle, when a perpendicular is drawn from the right angle to the hypotenuse, it creates two smaller triangles that are similar to the original triangle and to each other. This geometric relationship allows us to find the coordinates of the foot of the perpendicular using ratios of the triangle's side lengths and the concept of section formula.

  3. Points (12, 47) and (20, 71). Find f(x) = mx + b. Answer: f(x) = 3x + 11 Solution: Find the slope m using the formula m = (y2 - y1) / (x2 - x1). Using (12, 47) as (x1, y1) and (20, 71) as (x2, y2): m = (71 - 47) / (20 - 12) = 24 / 8 = 3.
    Full step-by-step solution

    Step 1: Find the slope m using the formula m = (y2 - y1) / (x2 - x1). Using (12, 47) as (x1, y1) and (20, 71) as (x2, y2): m = (71 - 47) / (20 - 12) = 24 / 8 = 3. Step 2: Use the slope m = 3 and one point, say (12, 47), in the equation y = mx + b to find b: 47 = 3(12) + b → 47 = 36 + b → b = 47 - 36 = 11. Step 3: Write the linear function: f(x) = 3x + 11. The answer is f(x) = 3x + 11.

  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. A coordinate plane shows a line passing through two points: (1, 11) and (6, 36). The line represents a linear function f(x). What is the equation of this linear function in slope-intercept form? Answer: f(x) = 5x + 6 Solution: Identify the two points: (x1, y1) = (1, 11) and (x2, y2) = (6, 36). Calculate the slope m using the formula m = (y2 - y1) / (x2 - x1). m = (36 - 11) / (6 - 1) = 25 / 5 = 5.
    Full step-by-step solution

    Step 1: Identify the two points: (x1, y1) = (1, 11) and (x2, y2) = (6, 36). Step 2: Calculate the slope m using the formula m = (y2 - y1) / (x2 - x1). Step 3: m = (36 - 11) / (6 - 1) = 25 / 5 = 5. Step 4: Use the point-slope form with point (1, 11): y - 11 = 5(x - 1). Step 5: Simplify to slope-intercept form: y - 11 = 5x - 5, so y = 5x + 6. Step 6: Write using function notation: f(x) = 5x + 6. The answer is f(x) = 5x + 6.

  6. Noah is tracking the distance a delivery drone travels over time during a test flight. At 2 seconds, the drone has traveled 14 meters from its starting point. At 7 seconds, it has traveled 44 meters. Assuming the drone moves at a constant speed, construct a linear function d(t) that gives the distance in meters from the starting point after t seconds. Then use this function to predict the distance traveled after 10 seconds. Answer: 62 meters Solution: Identify the points. The drone passes through (2, 14) and (7, 44), where t is time and d is distance. Find the slope m = (44 - 14) / (7 - 2) = 30 / 5 = 6.
    Full step-by-step solution

    Step 1: Identify the points. The drone passes through (2, 14) and (7, 44), where t is time and d is distance. Step 2: Find the slope m = (44 - 14) / (7 - 2) = 30 / 5 = 6. The drone travels 6 meters per second. Step 3: Use slope-intercept form d(t) = mt + b. Plug in (2, 14): 14 = 6(2) + b, so 14 = 12 + b, thus b = 2. Step 4: The linear function is d(t) = 6t + 2. Step 5: Predict distance at t = 10: d(10) = 6(10) + 2 = 60 + 2 = 62 meters. The answer is 62 meters.

  7. 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).

  8. f(x) = 2x² - 3x + 7, f(-2) = ? Answer: 21 Solution: Write the function: f(x) = 2x² - 3x + 7 Substitute x = -2 into the function: f(-2) = 2(-2)² - 3(-2) + 7 Calculate (-2)² = 4: f(-2) = 2(4) - 3(-2) + 7 Multiply 2 × 4 = 8: f(-2) = 8 - 3(-2) + 7 Multiply -3 × (-2) = 6: f(-2) = 8 + 6 + 7 Add the numbers: 8 + 6 = 14, then 14 + 7 = 21 The final answer…
    Full step-by-step solution

    Step 1: Write the function: f(x) = 2x² - 3x + 7 Step 2: Substitute x = -2 into the function: f(-2) = 2(-2)² - 3(-2) + 7 Step 3: Calculate (-2)² = 4: f(-2) = 2(4) - 3(-2) + 7 Step 4: Multiply 2 × 4 = 8: f(-2) = 8 - 3(-2) + 7 Step 5: Multiply -3 × (-2) = 6: f(-2) = 8 + 6 + 7 Step 6: Add the numbers: 8 + 6 = 14, then 14 + 7 = 21 Step 7: The final answer is f(-2) = 21