Lesson 6: Applications

How do we use the knowledge gained in this lesson to model real applications with functions?

Formula
Function
Input
Output
Independent
Dependent

F.IF.1
A. Understand the concept of a function and use function notation
1. HSF-IF.A.1
Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of
the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the
equation y = f(x).

2. HSF-IF.A.2
Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

B. Interpret functions that arise in applications in terms of the context
5. HSF-IF.B.5
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.?

Lesson Description: This lesson focuses on the end-to-end process of solving real-world problems using linear functions. Students are presented with "word problems" that don't come with a pre-written equation. Key steps include: Building the Model: Identifying the "rate of change" and the "initial value" from a text-based scenario to write a function in the form f(x) = mx + b. Solving for Specific Outcomes: * Finding the value of the function at a specific point (e.g., "What will the profit be in year 5?").Finding the input required for a specific result (e.g., "How many units must be sold to reach $10,000 in revenue?").Interpreting Slopes and Intercepts: Explaining what the "m" and "b" values mean in the specific context (e.g., "The slope represents the cost per gallon of fuel").Multi-Model Comparison: Analyzing two different functions simultaneously to find the Break-Even Point (where the two lines intersect).
Purpose
The purpose of Section 8.6 is to foster quantitative literacy and decision-making. This is the section that answers the common student question, "When will I ever use this?" By working through these applications, students see how algebra is used to compare insurance plans, project business growth, or calculate the time it takes for a medication to leave the bloodstream. It prepares students for the high-level critical thinking required in Middlesex CC’s core curriculum and professional degree programs.
Depth of Knowledge (DOK) Level
DOK Level 3
Level 3 (Strategic Thinking & Complex Reasoning): This section requires students to go beyond the "plug-and-chug" method. They must analyze a situation, design a mathematical model to represent it, and evaluate the meaning of their results. They are often required to make a recommendation based on their math (e.g., "Which rental car company is the better deal if you plan to drive 200 miles?") and justify that choice using evidence from their function.

Model the following with a function:
A local towing company charges $3.25 per mile driven plus a nonrefundable
base fee of $30.00. They tow a maximum of 25 miles.

Class and online work

Use "you try" pg 191 and practice problems to assess students' understanding of the lesson concepts