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Function Notation

Grade 9 · Algebra · Worksheet 2

  1. Noah is analyzing the profit from his online tutoring business. The daily profit in dollars, P(h), is modeled by the function P(h) = -3h² + 51h - 36, where h represents the number of hours he tutors each day. Due to a scheduling change, Noah tutors for 6 hours today. What is P(6)? Answer: ______________
  2. A local tech company is modeling the profit from their new smartphone app using the function P(x) = -0.02x² + 50x - 800, where x represents the number of app downloads and P(x) represents the profit in dollars. The company wants to determine their profit when they reach 1,200 downloads. Calculate P(1200) to find the profit at this download level. Answer: ______________
  3. f(x) = 2x² - 5x + 3, f(3) = ? Answer: ______________
  4. Noah is a botanist studying the growth of a rare orchid species. The height of the orchid, in centimeters, after t weeks is modeled by the function h(t) = -0.4t² + 12t + 8. Noah wants to record the height of the orchid exactly 9 weeks after planting. What is h(9)? Answer: ______________
  5. f(x) = 7x² - 9x + 11, f(8) = ? Answer: ______________
  6. A drone's altitude above ground is modeled by the function h(t) = -2t² + 12t + 5, where h is height in meters and t is time in seconds after launch. The drone operator needs to know the drone's altitude exactly 3 seconds after launch. What is h(3)? Answer: ______________
  7. f(x) = 2x³ - 5x² + 3x - 8, f(3) = ? Answer: ______________
  8. f(x) = 7x³ - 5x² + 3x - 9, f(2) = ? Answer: ______________
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Answer Key & Explanations

Function Notation · Grade 9 · Worksheet 2

  1. Noah is analyzing the profit from his online tutoring business. The daily profit in dollars, P(h), is modeled by the function P(h) = -3h² + 51h - 36, where h represents the number of hours he tutors each day. Due to a scheduling change, Noah tutors for 6 hours today. What is P(6)? Answer: 162 Solution: Write the function: P(h) = -3h² + 51h - 36 Substitute h = 6: P(6) = -3(6)² + 51(6) - 36 Calculate the exponent first: 6² = 36 Perform multiplication: -3 × 36 = -108 and 51 × 6 = 306 Combine terms: P(6) = -108 + 306 - 36 Add -108 and 306: -108 + 306 = 198 Subtract 36: 198 - 36 = 162 The answer is…
    Full step-by-step solution

    Step 1: Write the function: P(h) = -3h² + 51h - 36 Step 2: Substitute h = 6: P(6) = -3(6)² + 51(6) - 36 Step 3: Calculate the exponent first: 6² = 36 Step 4: Perform multiplication: -3 × 36 = -108 and 51 × 6 = 306 Step 5: Combine terms: P(6) = -108 + 306 - 36 Step 6: Add -108 and 306: -108 + 306 = 198 Step 7: Subtract 36: 198 - 36 = 162 The answer is 162.

  2. A local tech company is modeling the profit from their new smartphone app using the function P(x) = -0.02x² + 50x - 800, where x represents the number of app downloads and P(x) represents the profit in dollars. The company wants to determine their profit when they reach 1,200 downloads. Calculate P(1200) to find the profit at this download level. Answer: 30400 Solution: P(x) = -0.02x² + 50x - 800 Substitute x = 1200 into the function. P(1200) = -0.02 * (1200)² + 50 * 1200 - 800 Calculate (1200)². 1200 * 1200 = 1,440,000 Calculate -0.02 * (1200)².
    Full step-by-step solution

    Let's calculate P(1200) step by step. We are given: P(x) = -0.02x² + 50x - 800 Step 1: Substitute x = 1200 into the function. P(1200) = -0.02 * (1200)² + 50 * 1200 - 800 Step 2: Calculate (1200)². 1200 * 1200 = 1,440,000 Step 3: Calculate -0.02 * (1200)². -0.02 * 1,440,000 = -28,800 Step 4: Calculate 50 * 1200. 50 * 1200 = 60,000 Step 5: Now substitute these results back into the expression. P(1200) = -28,800 + 60,000 - 800 Step 6: Perform the addition and subtraction from left to right. First: -28,800 + 60,000 = 31,200 Then: 31,200 - 800 = 30,400 Final Answer: The profit at 1,200 downloads is $30,400.

  3. f(x) = 2x² - 5x + 3, f(3) = ? Answer: 6 Solution: Start with the function f(x) = 2x² - 5x + 3 Substitute x = 3 into the function: f(3) = 2(3)² - 5(3) + 3 Calculate the exponent first: (3)² = 9 Multiply: 2 × 9 = 18 and -5 × 3 = -15 Combine all terms: 18 - 15 + 3 Calculate from left to right: 18 - 15 = 3, then 3 + 3 = 6 The answer is 6.
    Full step-by-step solution

    Step 1: Start with the function f(x) = 2x² - 5x + 3 Step 2: Substitute x = 3 into the function: f(3) = 2(3)² - 5(3) + 3 Step 3: Calculate the exponent first: (3)² = 9 Step 4: Multiply: 2 × 9 = 18 and -5 × 3 = -15 Step 5: Combine all terms: 18 - 15 + 3 Step 6: Calculate from left to right: 18 - 15 = 3, then 3 + 3 = 6 The answer is 6.

  4. Noah is a botanist studying the growth of a rare orchid species. The height of the orchid, in centimeters, after t weeks is modeled by the function h(t) = -0.4t² + 12t + 8. Noah wants to record the height of the orchid exactly 9 weeks after planting. What is h(9)? Answer: 83.6 Solution: Start with the function h(t) = -0.4t² + 12t + 8. Substitute t = 9 into the function: h(9) = -0.4(9)² + 12(9) + 8. Calculate the exponent first: (9)² = 81.
    Full step-by-step solution

    Step 1: Start with the function h(t) = -0.4t² + 12t + 8. Step 2: Substitute t = 9 into the function: h(9) = -0.4(9)² + 12(9) + 8. Step 3: Calculate the exponent first: (9)² = 81. Step 4: Multiply: -0.4 × 81 = -32.4 and 12 × 9 = 108. Step 5: Combine all terms: -32.4 + 108 + 8 = 75.6 + 8 = 83.6. Step 6: The height of the orchid after 9 weeks is 83.6 centimeters. The answer is 83.6.

  5. f(x) = 7x² - 9x + 11, f(8) = ? Answer: 387 Solution: Start with the function f(x) = 7x² - 9x + 11 Substitute x = 8 into the function: f(8) = 7(8)² - 9(8) + 11 Calculate the exponent first: (8)² = 64 Multiply: 7 × 64 = 448 and -9 × 8 = -72 Combine all terms: 448 - 72 + 11 Perform the operations from left to right: 448 - 72 = 376, then 376 + 11 =…
    Full step-by-step solution

    Step 1: Start with the function f(x) = 7x² - 9x + 11 Step 2: Substitute x = 8 into the function: f(8) = 7(8)² - 9(8) + 11 Step 3: Calculate the exponent first: (8)² = 64 Step 4: Multiply: 7 × 64 = 448 and -9 × 8 = -72 Step 5: Combine all terms: 448 - 72 + 11 Step 6: Perform the operations from left to right: 448 - 72 = 376, then 376 + 11 = 387 The answer is 387.

  6. A drone's altitude above ground is modeled by the function h(t) = -2t² + 12t + 5, where h is height in meters and t is time in seconds after launch. The drone operator needs to know the drone's altitude exactly 3 seconds after launch. What is h(3)? Answer: 23 Solution: We are given the function for the drone's altitude: h(t) = -2t² + 12t + 5 We need to find h(3), which means we substitute t = 3 into the function. Write the function with t replaced by 3.
    Full step-by-step solution

    We are given the function for the drone's altitude: h(t) = -2t² + 12t + 5 We need to find h(3), which means we substitute t = 3 into the function. Step 1: Write the function with t replaced by 3. h(3) = -2*(3)² + 12*(3) + 5 Step 2: Evaluate the exponent first. (3)² = 9 So h(3) = -2*9 + 12*3 + 5 Step 3: Perform the multiplications. -2 * 9 = -18 12 * 3 = 36 Now h(3) = -18 + 36 + 5 Step 4: Add the numbers from left to right. -18 + 36 = 18 18 + 5 = 23 So the drone's altitude at t = 3 seconds is 23 meters. ANSWER: 23

  7. f(x) = 2x³ - 5x² + 3x - 8, f(3) = ? Answer: 10 Solution: Substitute x = 3 into the function: f(3) = 2(3)³ - 5(3)² + 3(3) - 8 Calculate the exponents first: (3)³ = 27 and (3)² = 9 Multiply coefficients: 2 × 27 = 54, -5 × 9 = -45, 3 × 3 = 9 Write the expression: f(3) = 54 - 45 + 9 - 8 Perform addition and subtraction from left to right: 54 - 45 = 9, 9 +…
    Full step-by-step solution

    Step 1: Substitute x = 3 into the function: f(3) = 2(3)³ - 5(3)² + 3(3) - 8 Step 2: Calculate the exponents first: (3)³ = 27 and (3)² = 9 Step 3: Multiply coefficients: 2 × 27 = 54, -5 × 9 = -45, 3 × 3 = 9 Step 4: Write the expression: f(3) = 54 - 45 + 9 - 8 Step 5: Perform addition and subtraction from left to right: 54 - 45 = 9, 9 + 9 = 18, 18 - 8 = 10 The answer is 10.

  8. f(x) = 7x³ - 5x² + 3x - 9, f(2) = ? Answer: 33 Solution: Start with the function f(x) = 7x³ - 5x² + 3x - 9 Substitute x = 2 into the function: f(2) = 7(2)³ - 5(2)² + 3(2) - 9 Calculate the exponents first: (2)³ = 8 and (2)² = 4 Perform multiplication: 7 × 8 = 56, -5 × 4 = -20, and 3 × 2 = 6 Substitute the results: f(2) = 56 - 20 + 6 - 9 Perform…
    Full step-by-step solution

    Step 1: Start with the function f(x) = 7x³ - 5x² + 3x - 9 Step 2: Substitute x = 2 into the function: f(2) = 7(2)³ - 5(2)² + 3(2) - 9 Step 3: Calculate the exponents first: (2)³ = 8 and (2)² = 4 Step 4: Perform multiplication: 7 × 8 = 56, -5 × 4 = -20, and 3 × 2 = 6 Step 5: Substitute the results: f(2) = 56 - 20 + 6 - 9 Step 6: Perform addition and subtraction from left to right: 56 - 20 = 36, 36 + 6 = 42, 42 - 9 = 33 The answer is 33.