Construct Functions

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Answer Key & Explanations

Construct Functions · Grade 8 · Worksheet 3

1. Liam is designing a custom skateboard ramp. The ramp's height increases at a constant rate of 0.25 feet for every 1 foot of horizontal distance. If the ramp starts at ground level (0 feet) and needs to reach a height of 4 feet, how many feet horizontally from the starting point will the ramp reach this height? Answer: 16 Solution: We are told the ramp's height increases at a constant rate: 0.25 feet vertically for every 1 foot horizontally. This means the slope (rise over run) is 0.25 / 1. Let y = height in feet, x = horizontal distance in feet.
   Full step-by-step solution

   We are told the ramp's height increases at a constant rate: 0.25 feet vertically for every 1 foot horizontally. This means the slope (rise over run) is 0.25 / 1. Step 1: Understand the relationship between height and horizontal distance. Let y = height in feet, x = horizontal distance in feet. The slope is 0.25, so: y = 0.25 * x Step 2: We know the ramp starts at ground level (0, 0) and must reach a height of 4 feet. Set y = 4 in the equation: 4 = 0.25 * x Step 3: Solve for x. x = 4 / 0.25 Step 4: Calculate 4 divided by 0.25. Dividing by 0.25 is the same as multiplying by 4: 4 / 0.25 = 4 / (1/4) = 4 * 4 = 16 Step 5: Interpret the result. x = 16 feet horizontal distance. So, the ramp reaches a height of 4 feet at 16 feet from the starting point horizontally.

2. Noah earns $9 per hour plus a one-time bonus of $15. Write a function f(x) for his total earnings after working x hours. Answer: f(x) = 9x + 15 Solution: Identify the rate of change (slope). Noah earns $9 per hour, so for each hour x, he earns 9x dollars. This is the variable part.
   Full step-by-step solution

   Step 1: Identify the rate of change (slope). Noah earns $9 per hour, so for each hour x, he earns 9x dollars. This is the variable part. Step 2: Identify the constant (y-intercept). The one-time bonus of $15 does not depend on hours worked, so it is added as a constant. Step 3: Write the function in the form f(x) = mx + b, where m is the rate per hour and b is the constant. Step 4: Substitute m = 9 and b = 15 to get f(x) = 9x + 15. The answer is f(x) = 9x + 15.

3. Aisha is comparing two internet plans. Plan A has a $25 monthly fee plus $0.50 per gigabyte of data used. Plan B has a $40 monthly fee plus $0.25 per gigabyte of data used. After how many gigabytes of data usage will both plans cost the same amount? Answer: 60 Solution: Write equations for both plans. Let x = gigabytes of data used. Plan A: Cost = 25 + 0.50x Plan B: Cost = 40 + 0.25x Set the costs equal to find when they are the same.
   Full step-by-step solution

   Step 1: Write equations for both plans. Let x = gigabytes of data used. Plan A: Cost = 25 + 0.50x Plan B: Cost = 40 + 0.25x Step 2: Set the costs equal to find when they are the same. 25 + 0.50x = 40 + 0.25x Step 3: Subtract 0.25x from both sides. 25 + 0.25x = 40 Step 4: Subtract 25 from both sides. 0.25x = 15 Step 5: Divide both sides by 0.25. x = 15 / 0.25 x = 60 The answer is 60 gigabytes.

4. 2(3x - 5) + 4 = 3(x + 2) - 1 Answer: 3 Solution: When solving equations with variables on both sides, the goal is to get all variable terms on one side and all constant terms on the other.
   Full step-by-step solution

   When solving equations with variables on both sides, the goal is to get all variable terms on one side and all constant terms on the other. Use inverse operations like addition, subtraction, multiplication, or division to maintain balance while simplifying. For example, in an equation like 2(a + 1) = 3(a - 2), you would distribute first, then move terms strategically.

5. Matiu earns $22 per hour at his part-time job. He also receives a weekly bonus of $14 for perfect attendance. Write a linear function f(x) that represents Matiu's total weekly earnings, where x is the number of hours he works in a week. Answer: f(x) = 22x + 14 Solution: Identify the rate of change (slope). Matiu earns $22 per hour, so the slope m = 22. This is the amount added for each additional hour worked.
   Full step-by-step solution

   Step 1: Identify the rate of change (slope). Matiu earns $22 per hour, so the slope m = 22. This is the amount added for each additional hour worked. Step 2: Identify the constant term (y-intercept). The weekly bonus of $14 is paid regardless of hours worked, so b = 14. Step 3: Write the linear function in the form f(x) = mx + b. Step 4: Substitute m = 22 and b = 14: f(x) = 22x + 14. The answer is f(x) = 22x + 14.

6. 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.

7. 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.

8. (3x + 15) ÷ 3 = 10 Answer: 5 Solution: (3x + 15) ÷ 3 = 10 Understand that "÷ 3" means the same as multiplying by 1/3, so we can write: (3x + 15) / 3 = 10 We can simplify the left-hand side by dividing each term in the numerator by 3: (3x)/3 + 15/3 = 10 3x/3 = x 15/3 = 5 x + 5 = 10 Isolate x by subtracting 5 from both sides: x + 5 - 5…
   Full step-by-step solution

   Let's solve the equation step by step. We start with: (3x + 15) ÷ 3 = 10 **Step 1:** Understand that "÷ 3" means the same as multiplying by 1/3, so we can write: (3x + 15) / 3 = 10 **Step 2:** We can simplify the left-hand side by dividing each term in the numerator by 3: (3x)/3 + 15/3 = 10 **Step 3:** Perform the division for each term: 3x/3 = x 15/3 = 5 So we have: x + 5 = 10 **Step 4:** Isolate x by subtracting 5 from both sides: x + 5 - 5 = 10 - 5 x = 5 **Step 5:** Check the solution: Plug x = 5 into the original equation: (3*5 + 15) ÷ 3 = (15 + 15) ÷ 3 = 30 ÷ 3 = 10 This matches the right-hand side, so the solution is correct. **Final answer:** x = 5

9. 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.