Construct Functions

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Answer Key & Explanations

Construct Functions · Grade 8 · Worksheet 2

1. A line passes through points (3, 8) and (7, 20). Find the slope of the line. Answer: 3 Solution: Identify the coordinates: (x1, y1) = (3, 8) and (x2, y2) = (7, 20) Use the slope formula: slope = (y2 - y1) / (x2 - x1) Substitute the values: slope = (20 - 8) / (7 - 3) Calculate the numerator: 20 - 8 = 12 Calculate the denominator: 7 - 3 = 4 Divide: 12 ÷ 4 = 3 The slope of the line is 3.
   Full step-by-step solution

   Step 1: Identify the coordinates: (x1, y1) = (3, 8) and (x2, y2) = (7, 20) Step 2: Use the slope formula: slope = (y2 - y1) / (x2 - x1) Step 3: Substitute the values: slope = (20 - 8) / (7 - 3) Step 4: Calculate the numerator: 20 - 8 = 12 Step 5: Calculate the denominator: 7 - 3 = 4 Step 6: Divide: 12 ÷ 4 = 3 The slope of the line is 3.

2. A local library is tracking book borrowing patterns. They found that the number of fiction books borrowed each week (y) has a linear relationship with the number of new members (x). When there are 20 new members, 180 fiction books are borrowed. When there are 35 new members, 270 fiction books are borrowed. Write a linear equation in slope-intercept form that represents this relationship. How many fiction books would be borrowed if there were 50 new members? Answer: 360 Solution: Identify the two points: (20, 180) and (35, 270) Calculate the slope: m = (270 - 180) / (35 - 20) = 90 / 15 = 6 Use point-slope form with (20, 180): y - 180 = 6(x - 20) Convert to slope-intercept form: y - 180 = 6x - 120 → y = 6x + 60 Substitute x = 50 into the equation: y = 6(50) + 60 = 300 +…
   Full step-by-step solution

   Step 1: Identify the two points: (20, 180) and (35, 270) Step 2: Calculate the slope: m = (270 - 180) / (35 - 20) = 90 / 15 = 6 Step 3: Use point-slope form with (20, 180): y - 180 = 6(x - 20) Step 4: Convert to slope-intercept form: y - 180 = 6x - 120 → y = 6x + 60 Step 5: Substitute x = 50 into the equation: y = 6(50) + 60 = 300 + 60 = 360 Step 6: The number of fiction books borrowed would be 360.

3. Mason earns $27 per hour as a tutor. Write a linear function f(x) that represents his total earnings after working x hours. Answer: f(x) = 27x Solution: Identify the rate of change. Mason earns $27 per hour, so the slope (m) is 27. Identify the starting value.
   Full step-by-step solution

   Step 1: Identify the rate of change. Mason earns $27 per hour, so the slope (m) is 27. Step 2: Identify the starting value. Since he earns nothing when he works 0 hours, the y-intercept (b) is 0. Step 3: Write the function in slope-intercept form f(x) = mx + b. Step 4: Substitute m = 27 and b = 0: f(x) = 27x + 0. Step 5: Simplify: f(x) = 27x. The answer is f(x) = 27x.

4. Aisha is comparing two internet plans. Plan A has a $30 monthly fee plus $0.50 per gigabyte of data used. Plan B has a $20 monthly fee plus $0.75 per gigabyte of data used. After how many gigabytes of data usage will both plans cost the same amount? Answer: 40 Solution: Write equations for both plans. Let x = gigabytes of data used.
   Full step-by-step solution

   Step 1: Write equations for both plans. Let x = gigabytes of data used. Plan A: Cost = 30 + 0.50x Plan B: Cost = 20 + 0.75x Step 2: Set the equations equal to find when costs are the same: 30 + 0.50x = 20 + 0.75x Step 3: Subtract 20 from both sides: 10 + 0.50x = 0.75x Step 4: Subtract 0.50x from both sides: 10 = 0.25x Step 5: Divide both sides by 0.25: x = 10 ÷ 0.25 x = 40 The answer is 40 gigabytes.

5. A line passes through points (8, 17) and (14, 35). Write the equation of the line in the form f(x) = mx + b. Answer: f(x) = 3x - 7 Solution: Find the slope m = (35 - 17) / (14 - 8) = 18 / 6 = 3. Use point (8, 17) in y = mx + b: 17 = 3(8) + b → 17 = 24 + b → b = 17 - 24 = -7. Write the equation: f(x) = 3x - 7.
   Full step-by-step solution

   Step 1: Find the slope m = (35 - 17) / (14 - 8) = 18 / 6 = 3. Step 2: Use point (8, 17) in y = mx + b: 17 = 3(8) + b → 17 = 24 + b → b = 17 - 24 = -7. Step 3: Write the equation: f(x) = 3x - 7. The answer is f(x) = 3x - 7.

6. Mere is saving money for a new bike. She starts with $24 and saves $8 each week. Write a linear function f(x) to represent the total amount of money Mere has after x weeks. Answer: f(x) = 8x + 24 Solution: Identify the initial value (y-intercept). Mere starts with $24, so b = 24. Identify the rate of change (slope).
   Full step-by-step solution

   Step 1: Identify the initial value (y-intercept). Mere starts with $24, so b = 24. Step 2: Identify the rate of change (slope). She saves $8 each week, so m = 8. Step 3: Write the linear function in the form f(x) = mx + b. Step 4: Substitute m = 8 and b = 24: f(x) = 8x + 24. The answer is f(x) = 8x + 24.

7. Charlotte earns $12 per hour babysitting and a $7 travel fee per job. Write a linear function f(x) to represent her total earnings for x hours of babysitting. Answer: f(x) = 12x + 7 Solution: Identify the rate of change (slope). Charlotte earns $12 per hour, so the slope m = 12. Identify the fixed amount (y-intercept).
   Full step-by-step solution

   Step 1: Identify the rate of change (slope). Charlotte earns $12 per hour, so the slope m = 12. Step 2: Identify the fixed amount (y-intercept). She gets a $7 travel fee per job, which does not depend on hours, so b = 7. Step 3: Write the function in the form f(x) = mx + b. Step 4: Substitute m = 12 and b = 7 to get f(x) = 12x + 7. The answer is f(x) = 12x + 7.

8. Mere is saving money for a new bike. She starts with $14 and saves $6 each week. Write a linear function f(x) that represents the total amount of money Mere has saved after x weeks. Answer: f(x) = 6x + 14 Solution: Identify the starting amount (y-intercept). Mere starts with $14, so b = 14. Identify the amount saved each week (slope).
   Full step-by-step solution

   Step 1: Identify the starting amount (y-intercept). Mere starts with $14, so b = 14. Step 2: Identify the amount saved each week (slope). She saves $6 per week, so m = 6. Step 3: Write the linear function in the form f(x) = mx + b. Step 4: Substitute m = 6 and b = 14: f(x) = 6x + 14. The answer is f(x) = 6x + 14.