Function Notation

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Answer Key & Explanations

Function Notation · Grade 9 · Worksheet 3

1. Mason is tracking the fuel efficiency of his motorcycle. The distance traveled in kilometers after t hours is modeled by the function D(t) = 72t - 2t². How far does Mason travel after exactly 7 hours? Answer: 406 Solution: Write the function: D(t) = 72t - 2t² Substitute t = 7 into the function: D(7) = 72(7) - 2(7)² Calculate the exponent first: (7)² = 49 Multiply: 72 × 7 = 504 and 2 × 49 = 98 Subtract: 504 - 98 = 406 The answer is 406 kilometers.
   Full step-by-step solution

   Step 1: Write the function: D(t) = 72t - 2t² Step 2: Substitute t = 7 into the function: D(7) = 72(7) - 2(7)² Step 3: Calculate the exponent first: (7)² = 49 Step 4: Multiply: 72 × 7 = 504 and 2 × 49 = 98 Step 5: Subtract: 504 - 98 = 406 The answer is 406 kilometers.

2. A graph on a coordinate plane shows the function f(x) = -3x^2 + 15x - 11. Aroha draws a vertical line at x = 7 and marks the point where it meets the curve. Evaluate f(7) to find the y-coordinate of that point. Answer: -53 Solution: Identify the function: f(x) = -3x^2 + 15x - 11. Substitute x = 7 into the function: f(7) = -3(7)^2 + 15(7) - 11. Calculate the exponent: (7)^2 = 49.
   Full step-by-step solution

   Step 1: Identify the function: f(x) = -3x^2 + 15x - 11. Step 2: Substitute x = 7 into the function: f(7) = -3(7)^2 + 15(7) - 11. Step 3: Calculate the exponent: (7)^2 = 49. Step 4: Perform the multiplications: -3 * 49 = -147, and 15 * 7 = 105. Step 5: Add and subtract in order: -147 + 105 = -42, then -42 - 11 = -53. Step 6: Therefore, f(7) = -53.

3. Mason is analyzing the speed of a particle moving along a straight line. The velocity of the particle, in meters per second, at time t seconds is given by the function v(t) = 2t² - 17t + 27. Mason needs to determine the velocity of the particle exactly 7 seconds after it starts moving. What is v(7)? Answer: 6 Solution: Start with the function v(t) = 2t² - 17t + 27. Substitute t = 7 into the function: v(7) = 2(7)² - 17(7) + 27. Calculate the exponent first: 7² = 49.
   Full step-by-step solution

   Step 1: Start with the function v(t) = 2t² - 17t + 27. Step 2: Substitute t = 7 into the function: v(7) = 2(7)² - 17(7) + 27. Step 3: Calculate the exponent first: 7² = 49. Step 4: Perform the multiplications: 2 × 49 = 98 and -17 × 7 = -119. Step 5: Combine the terms: 98 - 119 + 27 = -21 + 27 = 6. Step 6: The velocity of the particle after 7 seconds is 6 meters per second. The answer is 6.

4. f(x) = 6x² - 8x + 11, f(4) = ? Answer: 75 Solution: Start with the function f(x) = 6x² - 8x + 11 Substitute x = 4 into the function: f(4) = 6(4)² - 8(4) + 11 Calculate the exponent first: (4)² = 16 Multiply: 6 × 16 = 96 and -8 × 4 = -32 Combine all terms: 96 - 32 + 11 Perform the operations from left to right: 96 - 32 = 64, then 64 + 11 = 75 The…
   Full step-by-step solution

   Step 1: Start with the function f(x) = 6x² - 8x + 11 Step 2: Substitute x = 4 into the function: f(4) = 6(4)² - 8(4) + 11 Step 3: Calculate the exponent first: (4)² = 16 Step 4: Multiply: 6 × 16 = 96 and -8 × 4 = -32 Step 5: Combine all terms: 96 - 32 + 11 Step 6: Perform the operations from left to right: 96 - 32 = 64, then 64 + 11 = 75 The answer is 75.

5. Ava is studying the trajectory of a model rocket launched from a platform. The height of the rocket in meters above the ground after t seconds is given by the function h(t) = -5t² + 41t + 6. Ava needs to record the height of the rocket exactly 6 seconds after launch. What is h(6)? Answer: 72 Solution: Start with the function h(t) = -5t² + 41t + 6. Substitute t = 6 into the function: h(6) = -5(6)² + 41(6) + 6. Calculate the exponent first: (6)² = 36.
   Full step-by-step solution

   Step 1: Start with the function h(t) = -5t² + 41t + 6. Step 2: Substitute t = 6 into the function: h(6) = -5(6)² + 41(6) + 6. Step 3: Calculate the exponent first: (6)² = 36. Step 4: Multiply: -5 × 36 = -180 and 41 × 6 = 246. Step 5: Combine all terms: -180 + 246 + 6 = 66 + 6 = 72. Step 6: The height of the rocket after 6 seconds is 72 meters. The answer is 72.

6. f(x) = 7x² - 2x + 12, f(7) = ? Answer: 341 Solution: Start with the function f(x) = 7x² - 2x + 12 Substitute x = 7 into the function: f(7) = 7(7)² - 2(7) + 12 Calculate the exponent first: (7)² = 49 Multiply: 7 × 49 = 343 and -2 × 7 = -14 Combine all terms: 343 - 14 + 12 Perform the operations from left to right: 343 - 14 = 329, then 329 + 12 =…
   Full step-by-step solution

   Step 1: Start with the function f(x) = 7x² - 2x + 12 Step 2: Substitute x = 7 into the function: f(7) = 7(7)² - 2(7) + 12 Step 3: Calculate the exponent first: (7)² = 49 Step 4: Multiply: 7 × 49 = 343 and -2 × 7 = -14 Step 5: Combine all terms: 343 - 14 + 12 Step 6: Perform the operations from left to right: 343 - 14 = 329, then 329 + 12 = 341 The answer is 341.

7. A parabola is drawn on a coordinate plane. The function f(x) = -x^2 + 12x - 27 represents the y-coordinate of a point on this parabola. Isabella is analyzing the parabola and needs to find the y-coordinate when x = 7. Evaluate f(7). Answer: 8 Solution: Identify the function: f(x) = -x^2 + 12x - 27. Substitute x = 7 into the function: f(7) = -(7)^2 + 12(7) - 27. Calculate the exponent: (7)^2 = 49.
   Full step-by-step solution

   Step 1: Identify the function: f(x) = -x^2 + 12x - 27. Step 2: Substitute x = 7 into the function: f(7) = -(7)^2 + 12(7) - 27. Step 3: Calculate the exponent: (7)^2 = 49. Step 4: Perform the multiplications: -1 * 49 = -49, and 12 * 7 = 84. Step 5: The expression becomes: -49 + 84 - 27. Step 6: Add and subtract from left to right: -49 + 84 = 35, then 35 - 27 = 8. Step 7: Therefore, f(7) = 8.

8. f(x) = 4x³ - 8x² + 12x - 16, f(2) = ? Answer: 8 Solution: Start with the function f(x) = 4x³ - 8x² + 12x - 16 Substitute x = 2: f(2) = 4(2)³ - 8(2)² + 12(2) - 16 Evaluate exponents: (2)³ = 8, (2)² = 4 Multiply: 4 × 8 = 32, -8 × 4 = -32, 12 × 2 = 24 Combine: 32 - 32 + 24 - 16 Work left to right: 32 - 32 = 0, 0 + 24 = 24, 24 - 16 = 8 The answer is 8.
   Full step-by-step solution

   Step 1: Start with the function f(x) = 4x³ - 8x² + 12x - 16 Step 2: Substitute x = 2: f(2) = 4(2)³ - 8(2)² + 12(2) - 16 Step 3: Evaluate exponents: (2)³ = 8, (2)² = 4 Step 4: Multiply: 4 × 8 = 32, -8 × 4 = -32, 12 × 2 = 24 Step 5: Combine: 32 - 32 + 24 - 16 Step 6: Work left to right: 32 - 32 = 0, 0 + 24 = 24, 24 - 16 = 8 The answer is 8.

9. f(x) = 4x³ - 6x² + 8x - 10, f(2) = ? Answer: 14 Solution: Start with the function f(x) = 4x³ - 6x² + 8x - 10 Substitute x = 2: f(2) = 4(2)³ - 6(2)² + 8(2) - 10 Evaluate exponents: (2)³ = 8, (2)² = 4 Multiply: 4 × 8 = 32, -6 × 4 = -24, 8 × 2 = 16 Combine: 32 - 24 + 16 - 10 Work left to right: 32 - 24 = 8, 8 + 16 = 24, 24 - 10 = 14 The answer is 14.
   Full step-by-step solution

   Step 1: Start with the function f(x) = 4x³ - 6x² + 8x - 10 Step 2: Substitute x = 2: f(2) = 4(2)³ - 6(2)² + 8(2) - 10 Step 3: Evaluate exponents: (2)³ = 8, (2)² = 4 Step 4: Multiply: 4 × 8 = 32, -6 × 4 = -24, 8 × 2 = 16 Step 5: Combine: 32 - 24 + 16 - 10 Step 6: Work left to right: 32 - 24 = 8, 8 + 16 = 24, 24 - 10 = 14 The answer is 14.