Functions from Equations: Unlocking the Formula

Algebra 1 Grades High School 3:03 Video
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Lesson Description

Learn how to rewrite an equation to define a function, where one variable's value (the output) depends on another (the input). We'll focus on isolating variables and expressing the relationship using function notation.

Video Resource

How to create a function from an equation (example) | Functions | Algebra I | Khan Academy

Khan Academy

Duration: 3:03
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Key Concepts

  • Solving equations for a specific variable.
  • Understanding the relationship between input and output in a function.
  • Expressing equations in function notation.

Learning Objectives

  • Students will be able to solve a linear equation for one variable in terms of another.
  • Students will be able to represent a given equation as a function using function notation.
  • Students will be able to identify the input and output variables in a function derived from an equation.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the concept of a function as a relationship between two variables: an input and an output. Briefly discuss function notation (e.g., f(x)) and its meaning. Explain that this lesson will show how to create a function from an equation.
  • Video Viewing and Guided Notes (10 mins)
    Play the Khan Academy video "How to create a function from an equation (example) | Functions | Algebra I | Khan Academy". Students should take notes on the steps Sal takes to solve for 'a' and express it as a function of 'b'.
  • Example Problem and Explanation (10 mins)
    Work through a similar example problem on the board. For example: "Given the equation 2x + 5y = 15, express y as a function of x, i.e., find f(x)." Walk through each step, emphasizing the algebraic manipulations and the final function notation.
  • Practice Problems (10 mins)
    Provide students with practice problems where they need to convert equations into functions. Examples: 1. 3p - 4q = 8; express p as a function of q. 2. x/2 + 3y = 6; express y as a function of x. Circulate to assist students and answer questions.
  • Wrap-up and Assessment (5 mins)
    Review the key steps in converting equations into functions. Answer any remaining questions. Assign a short homework assignment with similar problems to reinforce the concepts.

Interactive Exercises

  • Equation-Function Matching
    Create a worksheet with a list of equations and a separate list of corresponding functions. Students must match each equation to its function representation.
  • Online Practice
    Use websites like Khan Academy or IXL for additional practice problems on solving equations for a variable and understanding function notation.

Discussion Questions

  • What does it mean to solve for a variable in an equation?
  • How is an equation different from a function?
  • Why is function notation useful?
  • How can you check if you have correctly created a function from an equation?

Skills Developed

  • Algebraic manipulation
  • Problem-solving
  • Abstract reasoning
  • Understanding of functions

Multiple Choice Questions

Question 1:

Which of the following is the correct first step to solve the equation 2x + 3y = 6 for y?

Correct Answer: Subtract 2x from both sides

Question 2:

If f(x) = -3x + 5, what is the input variable?

Correct Answer: x

Question 3:

Which of the following represents 'y as a function of x'?

Correct Answer: f(x) = y

Question 4:

If you solve the equation 5a - 2b = 10 for a, what is 'a' equal to?

Correct Answer: a = 2 + (2/5)b

Question 5:

What does f(b) represent in the context of this lesson?

Correct Answer: The output value of the function when the input is 'b'

Question 6:

Which operation is used to isolate a variable?

Correct Answer: Inverse operations

Question 7:

In the equation y = f(x), what does 'x' represent?

Correct Answer: The independent variable

Question 8:

If the equation is 4m + n = 12, and you want to express 'm' as a function of 'n', which variable should be isolated?

Correct Answer: m

Question 9:

After solving for a variable to define f(x), you can check the answer by:

Correct Answer: Substituting the function back into the original equation

Question 10:

Given the equation 6w - 2z = 18, which expression correctly represents 'w' as a function of 'z'?

Correct Answer: w = 3 + (1/3)z

Fill in the Blank Questions

Question 1:

The process of rewriting an equation to express one variable in terms of another is called solving for a ___________.

Correct Answer: variable

Question 2:

In function notation, f(x) represents the __________ of the function.

Correct Answer: output

Question 3:

To isolate a variable in an equation, you use __________ operations.

Correct Answer: inverse

Question 4:

When defining a function from an equation, you are expressing one variable as a function of the __________ variable.

Correct Answer: independent

Question 5:

The equation 3x + y = 9 can be written as a function f(x) = _________.

Correct Answer: 9-3x

Question 6:

A function describes how a ___________ variable changes with respect to an independent variable.

Correct Answer: dependent

Question 7:

If an equation is given as 5p = 10 + q, expressing p as a function of q gives p = _________.

Correct Answer: 2 + q/5

Question 8:

The notation f(x) = 2x + 1 represents a __________ function.

Correct Answer: linear

Question 9:

When solving for 'y' in the equation x - 2y = 4, you must __________ 'x' from both sides first.

Correct Answer: subtract

Question 10:

Rewriting an equation as a function is useful for determining the ________ for a given input.

Correct Answer: output

Educational Standards

A1.A.3.1:Solve equations involving several variables for one variable in terms of the others. A1.F.1.3:Write linear functions, using function notation, to represent mathematical models. A1.F.3.1:Identify and generate equivalent representations of linear functions, graphs, tables, and real-world situations.

Teaching Materials

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