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Function Notation

Grade 9 · Algebra · Worksheet 1

  1. Isabella is a materials engineer testing the thermal expansion of a metal rod. The length L of the rod in millimeters after being heated to a temperature T degrees Celsius is modeled by the function L(T) = 250 + 0.027T + 0.00012T². The testing protocol requires her to measure the rod's length when the temperature reaches exactly 70°C. What is L(70)? Answer: ______________
  2. A right triangle is drawn on a coordinate plane with vertices at (0,0), (4,0), and (4,3). The function f(x) = √(x² + 9) represents the distance from the point (0,0) to any point (x,3) along the vertical line x = 4. Evaluate f(4) to find the length of the hypotenuse of this triangle. Answer: ______________
  3. f(x) = 3x² - 2x + 7, f(-2) = ? Answer: ______________
  4. A rectangular garden is drawn on a coordinate plane with vertices at (0,0), (15,0), (15,10), and (0,10). A function f(x) = 2x^2 - 25x + 75 represents the area of a triangle formed by connecting the point (x,0) to the top-left vertex (0,10) and the top-right vertex (15,10). Evaluate f(5). Answer: ______________
  5. Tane is a botanist studying the growth of a rare fern species. The height H of the fern in centimeters after t days is modeled by the function H(t) = 35 + 2t - 0.01t². Tane wants to record the height of the fern exactly 15 days after the study began. What is H(15)? Answer: ______________
  6. Charlotte is a marine biologist studying the population of a certain fish species in a lake. The population size P(t) after t years is modeled by the function P(t) = 2t² - 7t + 122. Charlotte needs to record the population size exactly 7 years after the study begins. What is P(7)? Answer: ______________
  7. Charlotte is analyzing the revenue from her online tutoring service. The monthly revenue in dollars, R(x), is modeled by the function R(x) = -0.4x² + 28x - 40, where x represents the number of hours of tutoring provided. If Charlotte provides 15 hours of tutoring this month, what is the monthly revenue R(15)? Answer: ______________
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Answer Key & Explanations

Function Notation · Grade 9 · Worksheet 1

  1. Isabella is a materials engineer testing the thermal expansion of a metal rod. The length L of the rod in millimeters after being heated to a temperature T degrees Celsius is modeled by the function L(T) = 250 + 0.027T + 0.00012T². The testing protocol requires her to measure the rod's length when the temperature reaches exactly 70°C. What is L(70)? Answer: 252.37 Solution: Write the function: L(T) = 250 + 0.027T + 0.00012T² Substitute T = 70: L(70) = 250 + 0.027(70) + 0.00012(70)² Calculate the exponent first: 70² = 4900 0.027 × 70 = 1.89 0.00012 × 4900 = 0.588 Add all terms: 250 + 1.89 + 0.588 = 252.378 Round to two decimal places (as is typical for measurement):…
    Full step-by-step solution

    Step 1: Write the function: L(T) = 250 + 0.027T + 0.00012T² Step 2: Substitute T = 70: L(70) = 250 + 0.027(70) + 0.00012(70)² Step 3: Calculate the exponent first: 70² = 4900 Step 4: Perform the multiplications: 0.027 × 70 = 1.89 0.00012 × 4900 = 0.588 Step 5: Add all terms: 250 + 1.89 + 0.588 = 252.378 Step 6: Round to two decimal places (as is typical for measurement): 252.38 mm The answer is 252.37 (if exact) or 252.38 (rounded). Since the problem uses up to 5 decimal places in the coefficients, we give the exact result: 252.37.

  2. A right triangle is drawn on a coordinate plane with vertices at (0,0), (4,0), and (4,3). The function f(x) = √(x² + 9) represents the distance from the point (0,0) to any point (x,3) along the vertical line x = 4. Evaluate f(4) to find the length of the hypotenuse of this triangle. Answer: 5 Solution: f(x) = sqrt(x^2 + 9) This represents the distance from (0,0) to (x,3) along the vertical line x = 4. But in the problem, the triangle has vertices at (0,0), (4,0), and (4,3). Identify the hypotenuse.
    Full step-by-step solution

    Let's go step-by-step. We are given the function: f(x) = sqrt(x^2 + 9) This represents the distance from (0,0) to (x,3) along the vertical line x = 4. But in the problem, the triangle has vertices at (0,0), (4,0), and (4,3). Step 1: Identify the hypotenuse. The hypotenuse is the side opposite the right angle. The right angle is at (4,0) because the sides along x=4 (vertical) and along y=0 (horizontal) meet there. So the hypotenuse is from (0,0) to (4,3). Step 2: Use the function to find the length of the hypotenuse. The function f(x) = sqrt(x^2 + 9) gives the distance from (0,0) to (x,3). For the point (4,3), we set x = 4. Step 3: Substitute x = 4 into f(x). f(4) = sqrt(4^2 + 9) = sqrt(16 + 9) = sqrt(25) Step 4: Simplify. sqrt(25) = 5 Step 5: Conclusion. The length of the hypotenuse is 5. Answer: 5

  3. f(x) = 3x² - 2x + 7, f(-2) = ? Answer: 23 Solution: Start with the function f(x) = 3x² - 2x + 7 Substitute x = -2 into the function: f(-2) = 3(-2)² - 2(-2) + 7 Calculate the exponent first: (-2)² = 4 Multiply: 3 × 4 = 12 and -2 × (-2) = 4 Rewrite the expression: 12 + 4 + 7 Add from left to right: 12 + 4 = 16, then 16 + 7 = 23 The answer is 23.
    Full step-by-step solution

    Step 1: Start with the function f(x) = 3x² - 2x + 7 Step 2: Substitute x = -2 into the function: f(-2) = 3(-2)² - 2(-2) + 7 Step 3: Calculate the exponent first: (-2)² = 4 Step 4: Multiply: 3 × 4 = 12 and -2 × (-2) = 4 Step 5: Rewrite the expression: 12 + 4 + 7 Step 6: Add from left to right: 12 + 4 = 16, then 16 + 7 = 23 The answer is 23.

  4. A rectangular garden is drawn on a coordinate plane with vertices at (0,0), (15,0), (15,10), and (0,10). A function f(x) = 2x^2 - 25x + 75 represents the area of a triangle formed by connecting the point (x,0) to the top-left vertex (0,10) and the top-right vertex (15,10). Evaluate f(5). Answer: 0 Solution: The triangle has vertices at (0,10), (15,10), and (5,0). The base is the horizontal segment from (0,10) to (15,10), so the base length is 15 - 0 = 15.
    Full step-by-step solution

    Step 1: The triangle has vertices at (0,10), (15,10), and (5,0). The base is the horizontal segment from (0,10) to (15,10), so the base length is 15 - 0 = 15. Step 2: The height is the vertical distance from the base (y=10) to the third vertex (y=0), so height = 10 - 0 = 10. Step 3: Area of triangle = (1/2) * base * height = (1/2) * 15 * 10 = 75. Step 4: Now evaluate f(5) = 2(5)^2 - 25(5) + 75 = 2(25) - 125 + 75 = 50 - 125 + 75 = 0. The area formula gives 75, but f(5) = 0 because the function is defined as a different expression. The correct answer is 0.

  5. Tane is a botanist studying the growth of a rare fern species. The height H of the fern in centimeters after t days is modeled by the function H(t) = 35 + 2t - 0.01t². Tane wants to record the height of the fern exactly 15 days after the study began. What is H(15)? Answer: 62.75 Solution: Write the original function: H(t) = 35 + 2t - 0.01t² Substitute t = 15 into the function: H(15) = 35 + 2(15) - 0.01(15)² Calculate the exponent first: (15)² = 225 Multiply: 2 × 15 = 30 and 0.01 × 225 = 2.25 Combine all terms: 35 + 30 - 2.25 = 65 - 2.25 = 62.75 The height of the fern after 15…
    Full step-by-step solution

    Step 1: Write the original function: H(t) = 35 + 2t - 0.01t² Step 2: Substitute t = 15 into the function: H(15) = 35 + 2(15) - 0.01(15)² Step 3: Calculate the exponent first: (15)² = 225 Step 4: Multiply: 2 × 15 = 30 and 0.01 × 225 = 2.25 Step 5: Combine all terms: 35 + 30 - 2.25 = 65 - 2.25 = 62.75 Step 6: The height of the fern after 15 days is 62.75 centimeters. The answer is 62.75.

  6. Charlotte is a marine biologist studying the population of a certain fish species in a lake. The population size P(t) after t years is modeled by the function P(t) = 2t² - 7t + 122. Charlotte needs to record the population size exactly 7 years after the study begins. What is P(7)? Answer: 171 Solution: Start with the function P(t) = 2t² - 7t + 122. Substitute t = 7 into the function: P(7) = 2(7)² - 7(7) + 122. Calculate the exponent first: (7)² = 49.
    Full step-by-step solution

    Step 1: Start with the function P(t) = 2t² - 7t + 122. Step 2: Substitute t = 7 into the function: P(7) = 2(7)² - 7(7) + 122. Step 3: Calculate the exponent first: (7)² = 49. Step 4: Multiply: 2 × 49 = 98 and -7 × 7 = -49. Step 5: Combine the terms: 98 - 49 + 122 = 49 + 122 = 171. The answer is 171.

  7. Charlotte is analyzing the revenue from her online tutoring service. The monthly revenue in dollars, R(x), is modeled by the function R(x) = -0.4x² + 28x - 40, where x represents the number of hours of tutoring provided. If Charlotte provides 15 hours of tutoring this month, what is the monthly revenue R(15)? Answer: 290 Solution: Write the function: R(x) = -0.4x² + 28x - 40 Substitute x = 15 into the function: R(15) = -0.4(15)² + 28(15) - 40 Calculate the exponent first: (15)² = 225 Multiply: -0.4 × 225 = -90 and 28 × 15 = 420 Combine the terms: -90 + 420 - 40 = 330 - 40 = 290 The monthly revenue for 15 hours of tutoring…
    Full step-by-step solution

    Step 1: Write the function: R(x) = -0.4x² + 28x - 40 Step 2: Substitute x = 15 into the function: R(15) = -0.4(15)² + 28(15) - 40 Step 3: Calculate the exponent first: (15)² = 225 Step 4: Multiply: -0.4 × 225 = -90 and 28 × 15 = 420 Step 5: Combine the terms: -90 + 420 - 40 = 330 - 40 = 290 Step 6: The monthly revenue for 15 hours of tutoring is $290. The answer is 290.