Construct Linear Functions

lessonbunny.com

Answer Key & Explanations

Construct Linear Functions · Grade 9 · Worksheet 1

1. Mason is saving money to buy a new laptop. He currently has $120 in his savings account and plans to add a fixed amount of money each week from his part-time job. After 4 weeks, he has $220. After 10 weeks, he has $370. Write a linear function f(w) that represents the total amount of money Mason has saved after w weeks, and then use it to determine how many weeks it will take for him to have $520. Answer: 16 weeks Solution: Identify the two data points as ordered pairs (w, f(w)): (4, 220) and (10, 370). Calculate the slope m = (370 - 220) / (10 - 4) = 150 / 6 = 25. This means Mason saves $25 per week.
   Full step-by-step solution

   Step 1: Identify the two data points as ordered pairs (w, f(w)): (4, 220) and (10, 370). Step 2: Calculate the slope m = (370 - 220) / (10 - 4) = 150 / 6 = 25. This means Mason saves $25 per week. Step 3: Use the point-slope form with point (4, 220): f(w) - 220 = 25(w - 4). Step 4: Simplify to slope-intercept form: f(w) = 25w - 100 + 220 = 25w + 120. Step 5: Set f(w) = 520 and solve: 25w + 120 = 520 → 25w = 400 → w = 16. The answer is 16 weeks.

2. A tech startup is modeling their user growth with the function f(x) = 2x² - 12x + 20, where x represents months since launch and f(x) represents thousands of users. The company wants to determine when their user base will reach its minimum point and how many users they'll have at that time. Find the vertex of this quadratic function. Answer: (3, 2) Solution: To find the vertex of the quadratic function f(x) = 2x² - 12x + 20, we can use the vertex formula. For a quadratic in the form f(x) = ax² + bx + c, the x-coordinate of the vertex is given by x = -b/(2a).
   Full step-by-step solution

   To find the vertex of the quadratic function f(x) = 2x² - 12x + 20, we can use the vertex formula. For a quadratic in the form f(x) = ax² + bx + c, the x-coordinate of the vertex is given by x = -b/(2a). Step 1: Identify the coefficients a and b. From the function f(x) = 2x² - 12x + 20, we have: a = 2 b = -12 Step 2: Calculate the x-coordinate of the vertex. x = -b / (2a) Substitute the values of a and b: x = -(-12) / (2 * 2) x = 12 / 4 x = 3 So, the x-coordinate of the vertex is 3. This means the user base reaches its minimum point 3 months after launch. Step 3: Calculate the y-coordinate of the vertex (the minimum number of users). Substitute x = 3 back into the original function f(x). f(3) = 2*(3)² - 12*(3) + 20 First, calculate the square: (3)² = 9 f(3) = 2*9 - 12*3 + 20 Next, perform the multiplications: f(3) = 18 - 36 + 20 Finally, perform the addition and subtraction from left to right: 18 - 36 = -18 -18 + 20 = 2 So, f(3) = 2. Since f(x) represents thousands of users, this means there will be 2,000 users at the minimum point. Step 4: State the vertex. The vertex of the parabola is (3, 2). Therefore, the user base will reach its minimum of 2,000 users 3 months after launch.

3. f(x) = 3x² - 5x + 2, f(4) = ? Answer: 30 Solution: We are given the function f(x) = 3x² - 5x + 2 and asked to find f(4). Substitute x = 4 into the function. f(4) = 3*(4)² - 5*(4) + 2 Evaluate the exponent first.
   Full step-by-step solution

   We are given the function f(x) = 3x² - 5x + 2 and asked to find f(4). Step 1: Substitute x = 4 into the function. f(4) = 3*(4)² - 5*(4) + 2 Step 2: Evaluate the exponent first. (4)² = 16 So f(4) = 3*16 - 5*4 + 2 Step 3: Perform the multiplications. 3*16 = 48 5*4 = 20 So f(4) = 48 - 20 + 2 Step 4: Perform the addition and subtraction from left to right. 48 - 20 = 28 28 + 2 = 30 Therefore, f(4) = 30.

4. A line representing a linear function f(x) passes through the points (9, 16) and (15, 28). What is the equation of this linear function in slope-intercept form? Answer: f(x) = 2x - 2 Solution: Identify the two points: (9, 16) and (15, 28). Calculate the slope using m = (y2 - y1) / (x2 - x1). m = (28 - 16) / (15 - 9) = 12 / 6 = 2.
   Full step-by-step solution

   Step 1: Identify the two points: (9, 16) and (15, 28). Step 2: Calculate the slope using m = (y2 - y1) / (x2 - x1). m = (28 - 16) / (15 - 9) = 12 / 6 = 2. Step 3: Use point-slope form: y - y1 = m(x - x1). Substitute (9, 16) and m = 2. y - 16 = 2(x - 9). Step 4: Simplify to slope-intercept form: y = mx + b. y - 16 = 2x - 18. y = 2x - 18 + 16. y = 2x - 2. Step 5: Write in function notation: f(x) = 2x - 2. The answer is f(x) = 2x - 2.

5. Liam is designing a rectangular garden with a fixed perimeter of 60 meters. He wants to model the area of the garden as a function of its width. If the width is represented by w meters, write a quadratic function A(w) that represents the area of the garden in terms of w. Then determine the dimensions that will maximize the area. Answer: A(w) = -w^2 + 30w, width = 15 m, length = 15 m, maximum area = 225 m² Solution: We have a rectangular garden with perimeter \( P = 60 \) m. Let width = \( w \) meters, length = \( l \) meters.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understand the problem** We have a rectangular garden with perimeter \( P = 60 \) m. Let width = \( w \) meters, length = \( l \) meters. Perimeter formula: \[ P = 2l + 2w \] Substitute \( P = 60 \): \[ 2l + 2w = 60 \] --- **Step 2: Express length in terms of width** Divide both sides by 2: \[ l + w = 30 \] So: \[ l = 30 - w \] --- **Step 3: Write area as a function of width** Area \( A = l \times w \): \[ A(w) = (30 - w) \times w \] \[ A(w) = 30w - w^2 \] Or in standard quadratic form: \[ A(w) = -w^2 + 30w \] --- **Step 4: Find the width that maximizes area** This is a downward-opening parabola (since coefficient of \( w^2 \) is negative). Maximum occurs at vertex \( w = -\frac{b}{2a} \). Here \( a = -1 \), \( b = 30 \): \[ w = -\frac{30}{2(-1)} = -\frac{30}{-2} = 15 \] So width for maximum area is \( w = 15 \) m. --- **Step 5: Find corresponding length and maximum area** Length: \[ l = 30 - w = 30 - 15 = 15 \ \text{m} \] Area: \[ A = 15 \times 15 = 225 \ \text{m}^2 \] --- **Final Answer:** \[ A(w) = -w^2 + 30w \] Dimensions for maximum area: width = 15 m, length = 15 m, maximum area = 225 m².

6. Mere is saving money to buy a new laptop. She has already saved $60 and plans to save an additional $25 each week from her part-time job. Write a linear function f(w) that represents the total amount of money Mere has saved after w weeks. Then, use your function to determine how many weeks it will take her to save $410. Answer: 14 Solution: Identify the initial amount (y-intercept). Mere starts with $60, so b = 60. Identify the weekly savings (slope).
   Full step-by-step solution

   Step 1: Identify the initial amount (y-intercept). Mere starts with $60, so b = 60. Step 2: Identify the weekly savings (slope). She saves $25 per week, so m = 25. Step 3: Write the linear function: f(w) = 25w + 60. Step 4: Set f(w) = 410 and solve for w: 25w + 60 = 410. Step 5: Subtract 60 from both sides: 25w = 350. Step 6: Divide both sides by 25: w = 14. The answer is 14 weeks.

7. A marine biologist is studying the population of dolphins in a protected bay. The dolphin population P(t) after t years is modeled by the function P(t) = 80 × 2^(t/4). The conservation team wants to know when the dolphin population will reach exactly 640 individuals. Solve for the time t when this occurs. Answer: 12 Solution: Set up the equation using the given function: 80 × 2^(t/4) = 640 Divide both sides by 80: 2^(t/4) = 640 ÷ 80 Simplify: 2^(t/4) = 8 Recognize that 8 = 2^3, so: 2^(t/4) = 2^3 Set the exponents equal to each other: t/4 = 3 Multiply both sides by 4: t = 3 × 4 Calculate: t = 12 The answer is 12 years.
   Full step-by-step solution

   Step 1: Set up the equation using the given function: 80 × 2^(t/4) = 640 Step 2: Divide both sides by 80: 2^(t/4) = 640 ÷ 80 Step 3: Simplify: 2^(t/4) = 8 Step 4: Recognize that 8 = 2^3, so: 2^(t/4) = 2^3 Step 5: Set the exponents equal to each other: t/4 = 3 Step 6: Multiply both sides by 4: t = 3 × 4 Step 7: Calculate: t = 12 The answer is 12 years.