B1-B3

Linear Functions and Their Representations

Definition of a Function

A function is a relation in which each input (x-value) has exactly one output (y-value). Functions can be represented in various forms, including tables, graphs, equations, and verbal descriptions.

• Invertible (One-to-One) Functions: A function is invertible if each output is paired with exactly one input. If invertible, the inverse function can be found by switching x and y and solving for y.

• Example: Given a table of values, determine if the relation is a function and if it is invertible.

Representations of Functions

• Table: Lists input-output pairs.

• Graph: Plots points (x, y) on the coordinate plane.

• Equation: Expresses the relationship algebraically, e.g., $f(x) = 2x + 3$.

• Verbal Description: Describes the relationship in words.

Linear Functions

Definition and Standard Form

A linear function is a function of the form $f(x) = mx + b$ or $y = mx + b$, where m is the slope and b is the y-intercept. The graph of a linear function is a straight line.

• Slope (m): Measures the steepness of the line.

• Y-intercept (b): The point where the line crosses the y-axis.

Finding the Slope

The slope of a non-vertical line through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

• Positive Slope: Line rises from left to right.

• Negative Slope: Line falls from left to right.

• Zero Slope: Horizontal line.

• Undefined Slope: Vertical line.

Type of Slope	Graphical Description	Algebraic Property
Positive	Line rises left to right	$m > 0$
Negative	Line falls left to right	$m < 0$
Zero	Horizontal line	$m = 0$
Undefined	Vertical line	Division by zero

Forms of Linear Equations

• Slope-Intercept Form: $y = mx + b$

• Point-Slope Form: $y - y_1 = m(x - x_1)$

Form	Equation	When to Use
Slope-Intercept	$y = mx + b$	When slope and y-intercept are known
Point-Slope	$y - y_1 = m(x - x_1)$	When slope and a point are known

Finding the Equation of a Line

1. Find the slope $m$ using two points.

2. Use point-slope or slope-intercept form to write the equation.

3. Solve for $b$ if necessary.

Example: Find the equation of the line passing through $(3, 4)$ and $(2, -2)$.

$m = \frac{-2 - 4}{2 - 3} = \frac{-6}{-1} = 6$ Using point-slope form: $y - 4 = 6(x - 3)$

Identifying Linear Functions from Tables

A table represents a linear function if the change in $y$ divided by the change in $x$ is constant for all pairs of points.

Example: If $x$ increases by 1 and $y$ increases by 2 each time, the function is linear with $m = 2$.

Parallel and Perpendicular Lines

Slopes of Parallel and Perpendicular Lines

• Parallel Lines: Have the same slope ($m_1 = m_2$).

• Perpendicular Lines: Slopes are negative reciprocals ($m_1 = -1/m_2$).

Example: A line parallel to $y = 2x + 3$ has slope $m = 2$. A line perpendicular to it has slope $m = -1/2$.

Applications of Linear Functions

Modeling with Linear Functions

Linear functions can model real-world situations where there is a constant rate of change. Examples include cost, distance, and population growth (in simple cases).

• Example: The cost $C$ to plant seeds is $C(a) = 3000 + 200a$, where $a$ is the number of acres planted.

• Interpreting the Slope: The slope represents the rate of change (e.g., cost per acre).

• Interpreting the Y-intercept: The y-intercept represents the initial value (e.g., fixed cost).

Solving Application Problems

1. Identify the variables and their relationship.

2. Write the linear equation based on the context.

3. Interpret the slope and intercept in the context of the problem.

Example: If a runner starts 4 miles from home and runs at 1 mile per minute, the distance from home after $t$ minutes is $d(t) = 4 - t$.

Practice Problems

• Find the slope of the line passing through two points.

• Determine if a table represents a linear function.

• Write the equation of a line given a point and a slope.

• Model real-world scenarios with linear equations and interpret the results.

Summary Table: Key Properties of Linear Functions

Property	Description	Example
Slope	Rate of change of y with respect to x	$m = 2$ in $y = 2x + 3$
Y-intercept	Value of y when $x = 0$	$b = 3$ in $y = 2x + 3$
Parallel Lines	Same slope	$y = 2x + 1$ and $y = 2x - 4$
Perpendicular Lines	Slopes are negative reciprocals	$y = 2x + 3$ and $y = -\frac{1}{2}x + 1$

Additional info: These notes cover the identification and construction of linear functions from tables, graphs, equations, and verbal descriptions, as well as applications and interpretation of slope and intercepts in context. Practice problems reinforce these concepts.