Construct Functions

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Answer Key & Explanations

Construct Functions · Grade 8 · Worksheet 1

1. (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

2. Liam is tracking the growth of his sunflower plant. After 3 weeks, it was 15 cm tall, and after 7 weeks, it was 39 cm tall. Assuming the plant grows at a constant rate, write a linear equation in slope-intercept form that represents the plant's height (h) after w weeks. What is the height of the plant after 10 weeks? Answer: 57 cm Solution: - After 3 weeks, height = 15 cm → point (w₁, h₁) = (3, 15) - After 7 weeks, height = 39 cm → point (w₂, h₂) = (7, 39) Slope m = (change in height) / (change in weeks) m = (39 - 15) / (7 - 3) m = 24 / 4 m = 6 So the plant grows 6 cm per week.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Identify the given points** We are told: - After 3 weeks, height = 15 cm → point (w₁, h₁) = (3, 15) - After 7 weeks, height = 39 cm → point (w₂, h₂) = (7, 39) --- **Step 2: Find the slope (rate of growth)** Slope m = (change in height) / (change in weeks) m = (39 - 15) / (7 - 3) m = 24 / 4 m = 6 So the plant grows 6 cm per week. --- **Step 3: Use point-slope form to find the equation** Point-slope form: h - h₁ = m(w - w₁) Using (3, 15) and m = 6: h - 15 = 6(w - 3) --- **Step 4: Convert to slope-intercept form h = m w + b** h - 15 = 6w - 18 h = 6w - 18 + 15 h = 6w - 3 So the linear equation is: h = 6w - 3 --- **Step 5: Find height after 10 weeks** Substitute w = 10 into h = 6w - 3: h = 6(10) - 3 h = 60 - 3 h = 57 cm --- **Final answer:** 57 cm

3. A rectangular garden is drawn on a coordinate plane with corners at points (2, 1), (8, 1), (8, 5), and (2, 5). A diagonal line is drawn from the bottom-left corner (2, 1) to the top-right corner (8, 5). This line represents a path through the garden. What is the slope of this diagonal path? Answer: 2/3 Solution: Identify the coordinates of the two points for the diagonal path. The problem says the diagonal goes from the bottom-left corner (2, 1) to the top-right corner (8, 5).
   Full step-by-step solution

   Step 1: Identify the coordinates of the two points for the diagonal path. The problem says the diagonal goes from the bottom-left corner (2, 1) to the top-right corner (8, 5). So: Point 1: (x1, y1) = (2, 1) Point 2: (x2, y2) = (8, 5) Step 2: Recall the slope formula. Slope = (change in y) / (change in x) = (y2 - y1) / (x2 - x1) Step 3: Calculate the change in y (vertical change). y2 - y1 = 5 - 1 = 4 Step 4: Calculate the change in x (horizontal change). x2 - x1 = 8 - 2 = 6 Step 5: Write the slope as a fraction. Slope = 4 / 6 Step 6: Simplify the fraction. 4/6 = 2/3 Step 7: State the final answer. The slope of the diagonal path is 2/3.

4. 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.

5. Liam is training for a marathon and tracks his running distance over time. He notices that after 2 weeks of training, he can run 8 miles, and after 5 weeks, he can run 17 miles. If his progress follows a linear relationship, how many miles can he run after 7 weeks? Answer: 23 Solution: We are told Liam's progress follows a linear relationship between weeks (x) and miles (y). Identify the given data points. After 2 weeks, he runs 8 miles → (2, 8) After 5 weeks, he runs 17 miles → (5, 17) Find the slope of the line.
   Full step-by-step solution

   We are told Liam's progress follows a linear relationship between weeks (x) and miles (y). Step 1: Identify the given data points. After 2 weeks, he runs 8 miles → (2, 8) After 5 weeks, he runs 17 miles → (5, 17) Step 2: Find the slope of the line. Slope m = (change in y) / (change in x) m = (17 - 8) / (5 - 2) m = 9 / 3 m = 3 This means he increases his running distance by 3 miles per week. Step 3: Use point-slope form to find the equation. Using point (2, 8) and slope m = 3: y - 8 = 3(x - 2) y - 8 = 3x - 6 y = 3x + 2 Step 4: Find the distance after 7 weeks. Plug x = 7 into y = 3x + 2: y = 3(7) + 2 y = 21 + 2 y = 23 Step 5: Conclusion. After 7 weeks, Liam can run 23 miles.

6. Emma is saving money to buy a new bicycle. She already has $75 saved and plans to save $13 each week from her allowance. Write a linear equation in slope-intercept form that represents the total amount of money, y, Emma has saved after x weeks. Then use your equation to determine how many weeks it will take her to save at least $245. Answer: y = 13x + 75; 14 weeks Solution: The starting amount is $75, which is the y-intercept (b = 75). She saves $13 each week, so the slope is 13 (m = 13). The equation in slope-intercept form is y = 13x + 75.
   Full step-by-step solution

   Step 1: The starting amount is $75, which is the y-intercept (b = 75). She saves $13 each week, so the slope is 13 (m = 13). Step 2: The equation in slope-intercept form is y = 13x + 75. Step 3: To find when she has at least $245, set y = 245: 245 = 13x + 75. Step 4: Subtract 75 from both sides: 245 - 75 = 13x, so 170 = 13x. Step 5: Divide both sides by 13: x = 170 / 13 = 13.0769... Since she saves weekly, she needs 14 weeks to reach or exceed $245. Step 6: Check: After 13 weeks, she has 13(13) + 75 = 169 + 75 = $244 (not enough). After 14 weeks, she has 13(14) + 75 = 182 + 75 = $257 (enough). The answer is y = 13x + 75 and 14 weeks.

7. 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 a ride of x miles. Answer: f(x) = x + 6 Solution: Identify the flat fee (y-intercept, b) which is $6. Identify the cost per mile (slope, m) which is $1 per mile. Write the linear function in slope-intercept form: f(x) = mx + b.
   Full step-by-step solution

   Step 1: Identify the flat fee (y-intercept, b) which is $6. Step 2: Identify the cost per mile (slope, m) which is $1 per mile. Step 3: Write the linear function in slope-intercept form: f(x) = mx + b. Step 4: Substitute m = 1 and b = 6: f(x) = 1x + 6. Step 5: Simplify: f(x) = x + 6. The answer is f(x) = x + 6.

8. Ava is saving money for a new laptop. She starts with $50 and saves $15 each week. Write a linear function f(x) that represents the total amount Ava has saved after x weeks. Answer: f(x) = 15x + 50 Solution: Identify the starting amount (y-intercept): Ava starts with $50, so b = 50. Identify the rate of change (slope): She saves $15 each week, so m = 15. Write the linear function in the form f(x) = mx + b: f(x) = 15x + 50.
   Full step-by-step solution

   Step 1: Identify the starting amount (y-intercept): Ava starts with $50, so b = 50. Step 2: Identify the rate of change (slope): She saves $15 each week, so m = 15. Step 3: Write the linear function in the form f(x) = mx + b: f(x) = 15x + 50. The answer is f(x) = 15x + 50.