Construct Functions

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Answer Key & Explanations

Construct Functions · Grade 8 · Worksheet 2

1. A scientist is studying bacterial growth in a lab. The number of bacteria, N, after t hours is given by the linear function N(t) = 12,500t + 8,000. How many hours will it take for the bacteria population to reach 158,000? Answer: 12 Solution: We are given the linear function for the number of bacteria: N(t) = 12,500 * t + 8,000 We want to find t when N(t) = 158,000. Write the equation with the given N(t) value.
   Full step-by-step solution

   We are given the linear function for the number of bacteria: N(t) = 12,500 * t + 8,000 We want to find t when N(t) = 158,000. Step 1: Write the equation with the given N(t) value. 158,000 = 12,500 * t + 8,000 Step 2: Subtract 8,000 from both sides to isolate the term with t. 158,000 - 8,000 = 12,500 * t 150,000 = 12,500 * t Step 3: Divide both sides by 12,500 to solve for t. t = 150,000 / 12,500 Step 4: Simplify the division. First, cancel one zero from numerator and denominator: t = 15,000 / 1,250 Cancel another zero: t = 1,500 / 125 Now, 125 * 12 = 1,500, so: t = 12 Step 5: Interpret the result. It will take 12 hours for the bacteria population to reach 158,000. Final answer: 12

2. A taxi company charges a flat fee of $6 plus $1 per mile. Write a linear function f(x) to represent the total cost for x miles. Answer: f(x) = x + 6 Solution: Identify the flat fee (y-intercept, b). The flat fee is $6, so b = 6. Identify the cost per mile (slope, m).
   Full step-by-step solution

   Step 1: Identify the flat fee (y-intercept, b). The flat fee is $6, so b = 6. Step 2: Identify the cost per mile (slope, m). The cost per mile is $1, so m = 1. Step 3: Write the function in slope-intercept form: f(x) = mx + b. Step 4: Substitute m = 1 and b = 6: f(x) = 1x + 6, which simplifies to f(x) = x + 6. The answer is f(x) = x + 6.

3. (2x + 5) - (x - 3) = ? Answer: x + 8 Solution: (2x + 5) - (x - 3) The minus sign in front of (x - 3) means we subtract both terms inside the parentheses: (2x + 5) - x - (-3) (2x + 5) - x + 3 2x - x = 1x (or just x) 5 + 3 = 8 x + 8 Final Answer: x + 8
   Full step-by-step solution

   Let's solve step by step. We start with: (2x + 5) - (x - 3) **Step 1: Distribute the negative sign** The minus sign in front of (x - 3) means we subtract both terms inside the parentheses: (2x + 5) - x - (-3) That becomes: (2x + 5) - x + 3 **Step 2: Combine like terms** First, combine the x terms: 2x - x = 1x (or just x) **Step 3: Combine the constant terms** 5 + 3 = 8 **Step 4: Write the final expression** x + 8 **Final Answer:** x + 8

4. 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.

5. Liam is tracking his savings for a new video game system. He currently has $120 saved and plans to save $15 each week from his allowance. The system costs $300. Write an equation in slope-intercept form that represents the amount of money Liam has saved after w weeks. After how many weeks will he have enough money to buy the system? Answer: 12 weeks Solution: Liam starts with $120 and saves $15 each week. We want an equation for his total savings after w weeks. The system costs $300.
   Full step-by-step solution

   Step 1: Understand the problem. Liam starts with $120 and saves $15 each week. We want an equation for his total savings after w weeks. The system costs $300. Step 2: Write the equation in slope-intercept form. Slope-intercept form is: y = mx + b Here, y = total savings after w weeks. m = amount saved per week = 15 b = initial savings = 120 x = number of weeks = w So the equation is: y = 15w + 120 Step 3: Find when he has enough money. He needs at least $300, so set y = 300: 300 = 15w + 120 Step 4: Solve for w. Subtract 120 from both sides: 300 - 120 = 15w 180 = 15w Divide both sides by 15: w = 180 / 15 w = 12 Step 5: Interpret the result. After 12 weeks, Liam will have exactly $300 saved. Final answer: 12 weeks

6. Liam is tracking his savings for a new video game system. He starts with $50 in his savings account and adds $15 each week from his allowance. Write a linear equation in slope-intercept form that represents the total amount of money, y, Liam has after x weeks. Answer: y = 15x + 50 Solution: Liam starts with $50 in his savings account. He adds $15 each week from his allowance.
   Full step-by-step solution

   Let's break this down step by step. --- **Step 1: Understand the problem** Liam starts with $50 in his savings account. He adds $15 each week from his allowance. We want a linear equation in slope-intercept form: y = mx + b where: y = total money after x weeks x = number of weeks m = slope (rate of change per week) b = y-intercept (starting amount at week 0) --- **Step 2: Identify the starting amount (y-intercept)** At week 0 (x = 0), Liam has $50. So, when x = 0, y = 50. That means b = 50. --- **Step 3: Identify the slope (amount added per week)** Each week, he adds $15. So the rate of change (slope) m = 15. --- **Step 4: Write the equation in slope-intercept form** Plug m = 15 and b = 50 into y = mx + b: y = 15x + 50 --- **Step 5: Check the equation** After 0 weeks: y = 15(0) + 50 = 50 ✅ After 1 week: y = 15(1) + 50 = 65 ✅ (50 + 15) After 2 weeks: y = 15(2) + 50 = 80 ✅ (50 + 30) The equation matches the situation. --- **Final Answer:** y = 15x + 50

7. Emma earns $17 per hour at her job. She also receives a weekly bonus of $35 for perfect attendance. Write a linear function f(x) that represents her total weekly earnings, where x is the number of hours she works in a week. Answer: f(x) = 17x + 35 Solution: Identify the variable. Let x represent the number of hours Emma works in a week. She earns $17 per hour, so the slope m = 17.
   Full step-by-step solution

   Step 1: Identify the variable. Let x represent the number of hours Emma works in a week. Step 2: Determine the rate of change (slope). She earns $17 per hour, so the slope m = 17. Step 3: Determine the fixed amount (y-intercept). She gets a weekly bonus of $35, which does not depend on hours worked, so the y-intercept b = 35. Step 4: Write the linear function in the form f(x) = mx + b. Step 5: Substitute m = 17 and b = 35 to get f(x) = 17x + 35. The answer is f(x) = 17x + 35.

8. Noah is saving money for a new bicycle. He starts with $26 and saves $11 each week. Write a linear function f(x) that represents the total amount Noah has saved after x weeks. Answer: f(x) = 11x + 26 Solution: Identify the starting amount (y-intercept). Noah starts with $26, so b = 26. Identify the amount saved each week (slope).
   Full step-by-step solution

   Step 1: Identify the starting amount (y-intercept). Noah starts with $26, so b = 26. Step 2: Identify the amount saved each week (slope). He saves $11 each week, so m = 11. Step 3: Write the function in the form f(x) = mx + b. Step 4: Substitute m = 11 and b = 26: f(x) = 11x + 26. The answer is f(x) = 11x + 26.