Construct Linear Functions

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Answer Key & Explanations

Construct Linear Functions · Grade 9 · Worksheet 2

1. A research scientist is studying the decay of a radioactive isotope. The remaining mass M(t) in grams after t days is modeled by the function M(t) = 80 × (1/2)^(t/15). The scientist needs to determine after how many days only 10 grams of the isotope will remain. Find the time t when this occurs. Answer: 45 Solution: Step 1: Set up the equation using the given function and target mass: 80 × (1/2)^(t/15) = 10 Step 2: Divide both sides by 80: (1/2)^(t/15) = 10/80 = 1/8 Step 3: Recognize that 1/8 = (1/2)^3 Step 4: Set the exponents equal to each other: t/15 = 3 Step 5: Multiply both sides by 15: t = 3 × 15 = 45…
   Full step-by-step solution

   Step 1: Set up the equation using the given function and target mass: 80 × (1/2)^(t/15) = 10 Step 2: Divide both sides by 80: (1/2)^(t/15) = 10/80 = 1/8 Step 3: Recognize that 1/8 = (1/2)^3 Step 4: Set the exponents equal to each other: t/15 = 3 Step 5: Multiply both sides by 15: t = 3 × 15 = 45 Step 6: Verify by substituting back: M(45) = 80 × (1/2)^(45/15) = 80 × (1/2)^3 = 80 × 1/8 = 10 The answer is 45 days.

2. A right triangle is drawn on a coordinate plane with vertices at (0,0), (4,0), and (4,3). A line representing the function f(x) is drawn along the hypotenuse of this triangle. What is the equation of this linear function in slope-intercept form? Answer: f(x) = -3/4x + 3 Solution: The slope-intercept form of a linear function is y = mx + b, where m represents the slope and b represents the y-intercept.
   Full step-by-step solution

   The slope-intercept form of a linear function is y = mx + b, where m represents the slope and b represents the y-intercept. The slope can be determined from any two points on the line using the formula (y₂ - y₁)/(x₂ - x₁). The y-intercept is the point where the line crosses the y-axis. For example, if you had a line passing through points (1,2) and (3,4), you would first calculate the slope, then use one point to solve for the y-intercept.

3. Isabella is tracking the growth of her savings over time. She starts with a certain amount in her bank account and adds the same fixed amount each week. After 4 weeks, she has $260. After 9 weeks, she has $410. Determine the linear function S(w) that models the total amount of money in her account after w weeks, and use it to find how much money she initially had in the account. Answer: 140 Solution: Identify the two data points: (4, 260) and (9, 410). Calculate the slope (rate of change) using the formula m = (y2 - y1) / (x2 - x1). m = (410 - 260) / (9 - 4) = 150 / 5 = 30.
   Full step-by-step solution

   Step 1: Identify the two data points: (4, 260) and (9, 410). Step 2: Calculate the slope (rate of change) using the formula m = (y2 - y1) / (x2 - x1). m = (410 - 260) / (9 - 4) = 150 / 5 = 30. Step 3: The slope m = 30 means she adds $30 each week. Step 4: Use the slope-intercept form S(w) = mw + b. Substitute one point, say (4, 260), to find b: 260 = 30(4) + b -> 260 = 120 + b -> b = 140. Step 5: The linear function is S(w) = 30w + 140. Step 6: The initial amount is the y-intercept b, which is $140. The answer is 140.

4. Points (11, 47) and (17, 83). Find f(x) = mx + b. Answer: f(x) = 6x - 19 Solution: Find the slope m using m = (y2 - y1)/(x2 - x1). m = (83 - 47)/(17 - 11) = 36/6 = 6. Use point-slope form with point (11, 47): y - 47 = 6(x - 11).
   Full step-by-step solution

   Step 1: Find the slope m using m = (y2 - y1)/(x2 - x1). m = (83 - 47)/(17 - 11) = 36/6 = 6. Step 2: Use point-slope form with point (11, 47): y - 47 = 6(x - 11). Step 3: Simplify: y - 47 = 6x - 66. Step 4: Solve for y: y = 6x - 66 + 47 = 6x - 19. Step 5: Therefore, f(x) = 6x - 19. The answer is f(x) = 6x - 19.

5. Points (14, 23) and (29, 68). Find f(x) = mx + b. Answer: f(x) = 3x - 19 Solution: Find the slope m = (y2 - y1) / (x2 - x1) = (68 - 23) / (29 - 14) = 45 / 15 = 3. Use point (14, 23) and slope m = 3 in y = mx + b: 23 = 3(14) + b. Simplify: 23 = 42 + b.
   Full step-by-step solution

   Step 1: Find the slope m = (y2 - y1) / (x2 - x1) = (68 - 23) / (29 - 14) = 45 / 15 = 3. Step 2: Use point (14, 23) and slope m = 3 in y = mx + b: 23 = 3(14) + b. Step 3: Simplify: 23 = 42 + b. Step 4: Solve for b: b = 23 - 42 = -19. Step 5: The linear function is f(x) = 3x - 19.

6. f(x) = 2x + 3, g(x) = x² - 1, (f∘g)(2) = ? Answer: 9 Solution: f(x) = 2x + 3 g(x) = x² - 1 We need to find (f∘g)(2). (f∘g)(2) means f(g(2)). So first we find g(2), then plug that result into f.
   Full step-by-step solution

   We are given: f(x) = 2x + 3 g(x) = x² - 1 We need to find (f∘g)(2). Step 1: Understand the notation (f∘g)(2) means f(g(2)). So first we find g(2), then plug that result into f. Step 2: Compute g(2) g(x) = x² - 1 g(2) = (2)² - 1 g(2) = 4 - 1 g(2) = 3 Step 3: Compute f(g(2)) We found g(2) = 3, so now we compute f(3). f(x) = 2x + 3 f(3) = 2(3) + 3 f(3) = 6 + 3 f(3) = 9 Step 4: Conclusion (f∘g)(2) = 9 Final answer: 9

7. Isabella is conducting a physics experiment where she heats a metal rod and measures its length at different temperatures. At a temperature of 12°C, the rod measures 102 cm. At a temperature of 27°C, the rod measures 107 cm. Assuming the relationship between temperature (t in °C) and length (L in cm) is linear, construct a linear function L(t) that models this relationship. Then, use your function to predict the length of the rod at a temperature of 42°C. Answer: L(t) = (1/3)t + 98; L(42) = 112 cm Solution: Identify the two points as (t, L): (12, 102) and (27, 107). Calculate the slope m = (107 - 102) / (27 - 12) = 5 / 15 = 1/3. Use point-slope form: L - 102 = (1/3)(t - 12).
   Full step-by-step solution

   Step 1: Identify the two points as (t, L): (12, 102) and (27, 107). Step 2: Calculate the slope m = (107 - 102) / (27 - 12) = 5 / 15 = 1/3. Step 3: Use point-slope form: L - 102 = (1/3)(t - 12). Step 4: Simplify to slope-intercept form: L - 102 = (1/3)t - 4, so L = (1/3)t + 98. Step 5: Substitute t = 42: L(42) = (1/3)(42) + 98 = 14 + 98 = 112. The linear function is L(t) = (1/3)t + 98, and the predicted length at 42°C is 112 cm.

8. A right triangle is drawn on a coordinate plane with vertices at (0,0), (4,0), and (4,3). A line is drawn from the vertex at (4,3) perpendicular to the hypotenuse, intersecting it at point P. What are the coordinates of point P? Answer: (2.56, 1.92) Solution: In a right triangle, when a perpendicular is drawn from the right angle to the hypotenuse, it creates two smaller triangles that are similar to the original triangle and to each other.
   Full step-by-step solution

   In a right triangle, when a perpendicular is drawn from the right angle to the hypotenuse, it creates two smaller triangles that are similar to the original triangle and to each other. This geometric relationship allows us to find the coordinates of the foot of the perpendicular using ratios of the triangle's side lengths and the concept of section formula.