Function Notation in the Real World: Balloon Volume

Algebra 1 Grades High School 2:13 Video
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Lesson Description

Learn how to use function notation to solve real-world problems involving volume. This lesson focuses on applying function notation to calculate the volume of a sphere given its radius.

Video Resource

Function notation in context example | Functions and their graphs | Algebra II | Khan Academy

Khan Academy

Duration: 2:13
Watch on YouTube

Key Concepts

  • Function notation
  • Evaluating functions
  • Volume of a sphere
  • Real-world applications of functions

Learning Objectives

  • Students will be able to evaluate a function using function notation.
  • Students will be able to apply function notation to solve real-world problems.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the concept of functions and function notation. Briefly explain that a function is a relationship between inputs and outputs, and function notation provides a concise way to represent this relationship. For example, f(x) represents the output of the function f for the input x.
  • Video Explanation (10 mins)
    Play the Khan Academy video "Function notation in context example | Functions and their graphs | Algebra II | Khan Academy". Pause at key points to emphasize the following: The meaning of V(r) (volume as a function of radius). The process of substituting a value (3 inches) into the function. The calculation of the volume (36π cubic inches). Ensure students understand how the input (radius) relates to the output (volume).
  • Guided Practice (15 mins)
    Work through similar examples with the class. Present a scenario: 'A company's profit, P, as a function of the number of items sold, n, is given by P(n) = 5n - 100. What is the profit if the company sells 50 items?' Guide students through the steps: Identify the input (n = 50). Substitute the value into the function: P(50) = 5(50) - 100. Simplify the expression: P(50) = 250 - 100 = 150. Interpret the result: The profit is $150.
  • Independent Practice (15 mins)
    Provide students with practice problems to solve independently. Examples: 1. The area of a circle as a function of its radius is A(r) = πr². What is the area of a circle with a radius of 4? 2. The distance traveled by a car as a function of time is d(t) = 60t. How far will the car travel in 2.5 hours? Encourage students to show their work and check their answers with a partner.

Interactive Exercises

  • Function Matching
    Create cards with function notations (e.g., f(x) = 2x + 3, g(x) = x²) and cards with corresponding inputs (e.g., x = 2, x = -1). Students match the function notation with the correct input and calculate the output. For example, match f(x) = 2x + 3 with x = 2 and calculate f(2) = 7.

Discussion Questions

  • What does function notation tell us about the relationship between two variables?
  • How can function notation be used to solve real-world problems?
  • Why is it important to understand the units when working with functions?

Skills Developed

  • Applying function notation
  • Problem-solving
  • Algebraic manipulation
  • Interpreting results in context

Multiple Choice Questions

Question 1:

If f(x) = 3x - 2, what is f(4)?

Correct Answer: 10

Question 2:

The area of a square is given by A(s) = s², where s is the side length. If the side length is 5, what is the area?

Correct Answer: 25

Question 3:

If g(x) = x² + 1, what is g(-2)?

Correct Answer: 5

Question 4:

The cost of renting a bike is C(h) = 5h + 2, where h is the number of hours. What is the cost for 3 hours?

Correct Answer: $17

Question 5:

If h(x) = -2x + 5, what is h(0)?

Correct Answer: 5

Question 6:

A function is defined as f(x) = x/2 + 3. What is the value of f(6)?

Correct Answer: 6

Question 7:

Given the function g(x) = x^2 - 4, find g(3).

Correct Answer: 5

Question 8:

The height of a plant in inches after *w* weeks is given by h(w) = 2w + 1. What is the height after 4 weeks?

Correct Answer: 9 inches

Question 9:

If f(x) = -x + 7, what is the value of f(-1)?

Correct Answer: 8

Question 10:

The amount of money earned for working h hours at a job that pays $15 an hour can be represented by m(h)=15h. How much is earned by working 6 hours?

Correct Answer: $90

Fill in the Blank Questions

Question 1:

If f(x) = x + 5, then f(2) = ____.

Correct Answer: 7

Question 2:

The area of a rectangle with a fixed width of 3 is given by A(l) = 3l, where l is the length. If the length is 8, then the area is ____.

Correct Answer: 24

Question 3:

If g(x) = 4x - 1, then g(-1) = ____.

Correct Answer: -5

Question 4:

The cost of a taxi ride is given by C(m) = 2m + 3, where m is the number of miles. If the ride is 5 miles, the cost is ____.

Correct Answer: 13

Question 5:

If h(x) = x/3 + 2, then h(9) = ____.

Correct Answer: 5

Question 6:

Given f(x) = 5 - x, f(3) = ____.

Correct Answer: 2

Question 7:

For the function g(x) = x² - 2x, g(4) = ____.

Correct Answer: 8

Question 8:

The distance, d(t), a car travels in t hours at 50 mph is represented by d(t) = 50t. After 3 hours, the distance is ____ miles.

Correct Answer: 150

Question 9:

If f(x) = -2x + 10, then f(-5) = ____.

Correct Answer: 20

Question 10:

If the function is defined as p(x) = 2x + 7, p(2) is ____.

Correct Answer: 11

Educational Standards

A1.F.3.2:Use function notation; evaluate a function, including nonlinear, at a given point in its domain algebraically and graphically. Interpret the results in terms of the original context. A1.F.1.3:Write linear functions, using function notation, to represent mathematical models. A1.A.3.4:Evaluate linear, absolute value, rational, and radical expressions. Include applying a nonstandard operation such as x ⨀ y=2x+y.

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