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Construct Functions

Grade 8 · Algebra · Worksheet 3

  1. A local coffee shop is analyzing their daily revenue. They notice that when they charge $3.50 per coffee, they sell 120 cups per day, and when they lower the price to $2.50 per coffee, they sell 180 cups per day. Assuming a linear relationship between price and number of cups sold, what price should they charge to sell exactly 150 cups per day? Answer: ______________
  2. A right triangle is drawn on a coordinate plane with vertices at (1, 2), (1, 6), and (4, 2). A line is drawn from the vertex at (1, 6) to the midpoint of the hypotenuse. What is the slope of this line? Answer: ______________
  3. A rectangular garden is being planned with a length that is 3 meters more than twice its width. The perimeter of the garden is 36 meters. If you visualize this rectangle on a coordinate plane, with the bottom-left corner at the origin (0,0) and the width along the x-axis, what are the coordinates of all four corners of the rectangle? Answer: ______________
  4. Mere is helping her family's catering business track costs. For a recent event, they prepared 24 platters and the total cost was $180. For another event, they prepared 40 platters and the total cost was $260. Assuming the relationship between the number of platters (x) and the total cost in dollars (y) is linear, write a linear equation in slope-intercept form that models this relationship. Answer: ______________
  5. Emma is tracking the depth of water in a tank as it drains at a constant rate. She draws the situation on a coordinate plane, where the x-axis represents time in hours and the y-axis represents the water depth in inches. The line passes through the points (0, 50) and (10, 0). Write a linear function f(x) that models the water depth after x hours. Answer: ______________
  6. Emma earns $15 per hour babysitting and a flat $5 travel fee per job. Write a linear function f(x) to represent her total earnings for x hours of babysitting. Answer: ______________
  7. A line passes through points (3, 7) and (6, 13). Find the slope of the line. Answer: ______________
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Answer Key & Explanations

Construct Functions · Grade 8 · Worksheet 3

  1. A local coffee shop is analyzing their daily revenue. They notice that when they charge $3.50 per coffee, they sell 120 cups per day, and when they lower the price to $2.50 per coffee, they sell 180 cups per day. Assuming a linear relationship between price and number of cups sold, what price should they charge to sell exactly 150 cups per day? Answer: 3.00 Solution: Identify the two known points: (3.50, 120) and (2.50, 180) Calculate the slope: m = (180 - 120) / (2.50 - 3.50) = 60 / (-1.00) = -60 Use point-slope form with the first point: y - 120 = -60(x - 3.50) Convert to slope-intercept form: y = -60x + 210 + 120 = -60x + 330 Set y = 150 and solve for x:…
    Full step-by-step solution

    Step 1: Identify the two known points: (3.50, 120) and (2.50, 180) Step 2: Calculate the slope: m = (180 - 120) / (2.50 - 3.50) = 60 / (-1.00) = -60 Step 3: Use point-slope form with the first point: y - 120 = -60(x - 3.50) Step 4: Convert to slope-intercept form: y = -60x + 210 + 120 = -60x + 330 Step 5: Set y = 150 and solve for x: 150 = -60x + 330 Step 6: Subtract 330 from both sides: -180 = -60x Step 7: Divide both sides by -60: x = 3.00 The answer is $3.00.

  2. A right triangle is drawn on a coordinate plane with vertices at (1, 2), (1, 6), and (4, 2). A line is drawn from the vertex at (1, 6) to the midpoint of the hypotenuse. What is the slope of this line? Answer: -4/3 Solution: Identify the hypotenuse. The vertices are (1, 2), (1, 6), and (4, 2). The hypotenuse is between (1, 6) and (4, 2).
    Full step-by-step solution

    Step 1: Identify the hypotenuse. The vertices are (1, 2), (1, 6), and (4, 2). The hypotenuse is between (1, 6) and (4, 2). Step 2: Find the midpoint of the hypotenuse using the midpoint formula: ((x₁ + x₂)/2, (y₁ + y₂)/2). Midpoint = ((1 + 4)/2, (6 + 2)/2) = (5/2, 8/2) = (2.5, 4) Step 3: Calculate the slope between (1, 6) and (2.5, 4) using the slope formula: (y₂ - y₁)/(x₂ - x₁). Slope = (4 - 6)/(2.5 - 1) = (-2)/(1.5) = -2/(3/2) = -4/3 Step 4: The slope of the line is -4/3.

  3. A rectangular garden is being planned with a length that is 3 meters more than twice its width. The perimeter of the garden is 36 meters. If you visualize this rectangle on a coordinate plane, with the bottom-left corner at the origin (0,0) and the width along the x-axis, what are the coordinates of all four corners of the rectangle? Answer: (0,0), (5,0), (5,13), (0,13) Solution: Let the width be \( w \) meters (along the x-axis). Let the length be \( l \) meters (along the y-axis). Length is 3 meters more than twice the width.
    Full step-by-step solution

    Let's go step-by-step. --- **Step 1: Define variables** Let the width be \( w \) meters (along the x-axis). Let the length be \( l \) meters (along the y-axis). We are told: Length is 3 meters more than twice the width. So: \( l = 2w + 3 \) --- **Step 2: Use the perimeter formula** Perimeter of a rectangle: \( P = 2 \times (l + w) \) Given \( P = 36 \): \( 2(l + w) = 36 \) Divide both sides by 2: \( l + w = 18 \) --- **Step 3: Substitute \( l \) from Step 1 into Step 2** \( (2w + 3) + w = 18 \) \( 3w + 3 = 18 \) \( 3w = 15 \) \( w = 5 \) --- **Step 4: Find \( l \)** \( l = 2w + 3 = 2(5) + 3 = 10 + 3 = 13 \) So width \( w = 5 \) m, length \( l = 13 \) m. --- **Step 5: Place rectangle on coordinate plane** Bottom-left corner at (0,0). Width along x-axis: bottom side from (0,0) to (5,0). Length along y-axis: left side from (0,0) to (0,13). So corners are: A = (0,0) B = (5,0) C = (5,13) D = (0,13) --- **Step 6: Verify** Perimeter: \( 2 \times (5 + 13) = 2 \times 18 = 36 \) ✅ Length 13 is indeed 3 more than twice width 5: \( 2 \times 5 + 3 = 13 \) ✅ --- **Final answer:** (0,0), (5,0), (5,13), (0,13)

  4. Mere is helping her family's catering business track costs. For a recent event, they prepared 24 platters and the total cost was $180. For another event, they prepared 40 platters and the total cost was $260. Assuming the relationship between the number of platters (x) and the total cost in dollars (y) is linear, write a linear equation in slope-intercept form that models this relationship. Answer: y = 5x + 60 Solution: Identify the two points from the problem: (24, 180) and (40, 260). Calculate the slope (m) using the formula m = (y2 - y1) / (x2 - x1). m = (260 - 180) / (40 - 24) = 80 / 16 = 5.
    Full step-by-step solution

    Step 1: Identify the two points from the problem: (24, 180) and (40, 260). Step 2: Calculate the slope (m) using the formula m = (y2 - y1) / (x2 - x1). m = (260 - 180) / (40 - 24) = 80 / 16 = 5. Step 3: Use the point-slope form with one point, say (24, 180): y - 180 = 5(x - 24). Step 4: Distribute the 5: y - 180 = 5x - 120. Step 5: Add 180 to both sides: y = 5x + 60. The equation is y = 5x + 60.

  5. Emma is tracking the depth of water in a tank as it drains at a constant rate. She draws the situation on a coordinate plane, where the x-axis represents time in hours and the y-axis represents the water depth in inches. The line passes through the points (0, 50) and (10, 0). Write a linear function f(x) that models the water depth after x hours. Answer: f(x) = -5x + 50 Solution: Identify the given points. The line passes through (0, 50) and (10, 0). The point (0, 50) is the y-intercept, so b = 50.
    Full step-by-step solution

    Step 1: Identify the given points. The line passes through (0, 50) and (10, 0). The point (0, 50) is the y-intercept, so b = 50. Step 2: Calculate the slope m using the formula m = (y2 - y1) / (x2 - x1). Using (0, 50) and (10, 0): m = (0 - 50) / (10 - 0) = -50 / 10 = -5 The slope is -5, meaning the depth decreases by 5 inches per hour. Step 3: Write the linear function in slope-intercept form f(x) = mx + b. f(x) = -5x + 50 Step 4: Verify by checking that both points satisfy the function: For x = 0: f(0) = -5(0) + 50 = 50 ✓ For x = 10: f(10) = -5(10) + 50 = -50 + 50 = 0 ✓ The answer is f(x) = -5x + 50.

  6. Emma earns $15 per hour babysitting and a flat $5 travel fee per job. Write a linear function f(x) to represent her total earnings for x hours of babysitting. Answer: f(x) = 15x + 5 Solution: Identify the constant part. Emma gets a flat $5 travel fee per job, no matter how many hours she works. This is the y-intercept (b).
    Full step-by-step solution

    Step 1: Identify the constant part. Emma gets a flat $5 travel fee per job, no matter how many hours she works. This is the y-intercept (b). Step 2: Identify the rate of change. She earns $15 per hour, so for each hour (x), she earns 15x dollars. This is the slope (m). Step 3: Write the linear function in the form f(x) = mx + b. Step 4: Substitute m = 15 and b = 5. Step 5: f(x) = 15x + 5. The answer is f(x) = 15x + 5.

  7. A line passes through points (3, 7) and (6, 13). Find the slope of the line. Answer: 2 Solution: Identify the coordinates: (3, 7) and (6, 13) Calculate the change in y-values: 13 - 7 = 6 Calculate the change in x-values: 6 - 3 = 3 Divide the change in y by the change in x: 6 ÷ 3 = 2 The slope of the line is 2.
    Full step-by-step solution

    Step 1: Identify the coordinates: (3, 7) and (6, 13) Step 2: Calculate the change in y-values: 13 - 7 = 6 Step 3: Calculate the change in x-values: 6 - 3 = 3 Step 4: Divide the change in y by the change in x: 6 ÷ 3 = 2 Step 5: The slope of the line is 2.