Table of Contents
- Unit Definitions
- Function Notation
- Interpreting Linear Functions Arising in Applications
- Analyzing Linear Functions
- Building Functions
- Systems of Equations/Inequalities
- Constructing and Comparing Linear Models
- Unit Reflection Summary
Unit Definitions
Vocabulary Words
-
Function: A relation where each input has exactly one output.
Example: \( f(x) = 2x + 1 \) is a function since every x-value gives one unique y-value. -
Domain: The set of all possible input values (x-values) for a function.
Example: For \( f(x) = \sqrt{x} \), the domain is \( x \geq 0 \). -
Range: The set of all possible output values (y-values) for a function.
Example: For \( f(x) = 2x + 1 \), the range is all real numbers. -
Linear Function: A function that can be graphically represented as a straight line.
Example: \( f(x) = 3x - 5 \) is linear. -
Slope (m): The rate of change of a function, representing how much y changes for a change in x.
Example: In \( f(x) = 2x + 3 \), the slope \( m \) is 2. -
Y-intercept (b): The point where the line crosses the y-axis, corresponding to \( x=0 \).
Example: In the function \( f(x) = 2x + 3 \), the y-intercept is 3. -
Intercept: Points where the graph intersects the axes (x-intercept, y-intercept).
Example: For \( f(x) = 0.5x + 2 \), the y-intercept is at (0, 2) and the x-intercept at (-4, 0). -
System of Equations: A set of equations with the same variables.
Example:
\[ \begin{align*} y & = 2x + 1 \ y & = -x + 3 \end{align*} \] -
Inequality: A mathematical statement that compares two expressions using \( <, \leq, >, \geq \).
Example: \( 3x + 5 > 2 \). -
Piecewise Function: A function defined by multiple sub-functions, each applying to a certain interval.
Example:
\[ f(x) = \begin{cases} x^2 & \text{if } x < 0 \ x + 1 & \text{if } x \geq 0 \end{cases} \]
Systems of Equations/Inequalities
Contextual Situation for Systems of Equations
Scenario: A farmer has a plot of land where he grows two types of vegetables: tomatoes and carrots. He has a total of 100 plants with a constraint that he can only plant up to 60 tomato plants due to limited space.
Equations:
Let \( t \) represent the number of tomato plants and \( c \) represent the number of carrot plants.
- \( t + c = 100 \) (Total plants)
- \( t \leq 60 \) (Tomato plant constraint)
Explanation of Constraints: The farmer must follow the limitations on the number of tomato plants while maximizing the total number of plants. This creates a feasible region that can be graphed to visualize potential planting combinations.
Contextual Situation for Linear Inequality
Scenario: A student needs to study for a test and wants to ensure she studies at least 3 hours each day, but she can only study for a maximum of 5 hours per day due to other commitments.
Inequality:
Let \( h \) represent hours studied.
The inequality can be expressed as:
\[
3 \leq h \leq 5
\]
Graphing the Solution Set: The graph will show the range between 3 and 5 on the h-axis, indicating the hours a student can study.
Function Notation
Example Illustrating a Function
-
Function: \( f(x) = 3x + 2 \)
- Domain: All real numbers.
- Range: All real numbers.
This is a function because for each value of x, there is a unique value of f(x).
Example Not Illustrating a Function
- Relation: {(1, 2), (1, 3), (2, 4)}
This is not a function because the input \( 1 \) has two different outputs.
Real-World Scenario with Function Notation
Scenario: A taxi company charges a flat fee of $3 plus $2 per mile driven.
Let \( x \) be the number of miles driven.
Function notation:
\[
C(x) = 3 + 2x
\]
Evaluating the Function:
If a taxi drives 5 miles, \( C(5) = 3 + 2(5) = 13 \).
Recursive Formula:
The cost after \( n \) miles:
- \( C(n) = C(n - 1) + 2 \), starting at \( C(0) = 3 \).
Interpreting Linear Functions Arising in Applications
Story: A car rental agency charges $50 for a day’s rental, plus $0.20 per mile driven. The cost function can be modeled as: \[ C(d, m) = 50 + 0.20m \] where \( d \) is the number of days and \( m \) is the miles driven.
Key Features:
- Y-intercept: $50 (base charge)
- Rates of Change: The cost increases by $0.20 for each additional mile.
Graph: (Copy and paste or draw the graph)
The domain reflects the number of days and miles driven. The function is increasing for all valid inputs.
Analyzing Linear Functions
Linear Function:
\[
f(x) = 2x + 3
\]
Graph: (Use technology to graph and print)
The graph will be a straight line: slope = 2, y-intercept = 3.
Two Functions:
- Algebraic Function: \( f(x) = 3x - 1 \)
- Verbal Description: A box with a linear price increase, starting at $10 with an added cost of $5 each time another unit is purchased.
Comparison:
- First function has a steeper slope (3 vs. 2), indicating a faster increase for every x-unit.
Building Functions
Sequence: The sequence of numbers is defined as:
- Start with 2 and add 3 for each new term:
\( a_1 = 2, a_n = a_{n-1} + 3 \).
Recursive Formula:
- \( a_1 = 2 \)
- \( a_n = a_{n-1} + 3 \)
Explicit Formula:
\( a_n = 3n - 1 \).
Constructing and Comparing Linear Models
Word Problem: A company sells two types of daily subscriptions: Basic at $10 and Premium at $20. Let \( b \) and \( p \) be the numbers of subscriptions sold.
- Model in Equations:
\( 10b + 20p \) (Total revenue from subscriptions) - Table/Sequence Illustration:
| Basics Sold | Premiums Sold | Total Revenue | |-------------|----------------|----------------| | 0 | 5 | $100 | | 5 | 0 | $50 | | 3 | 2 | $70 |
Constant Rate of Change:
For each additional Premium subscription sold, revenue increases by $20, while Basic increases by $10.
Graphs: (Construct graphs for each model here)
The slope indicates the revenue increase as subscriptions increase.
Comparison:
- Basic model has a lower slope (10) than Premium (20), suggesting Premium yields more revenue per unit sold.
Reflection / Summary
Learning Journey:
Throughout the unit, understanding linear functions became easier as I practiced their interpretation and relationships. The hardest topic for me was constructing systems of equations, particularly figuring out how to apply constraints accurately.
Standards Understanding:
I feel confident about modeling and analyzing linear functions. However, I would like to delve deeper into systems of inequalities.
Advice for Future Students:
Focus on understanding the relationships between variables. Use real-world contexts to make abstract concepts more tangible. Activities that involved graphing and comparing practical scenarios were especially beneficial in my learning.
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