B1-B3

Linear Functions and Their Representations

Definition and Properties of Linear Functions

A linear function is a function of the form $f(x) = mx + b$, where m is the slope and b is the y-intercept. The graph of a linear function is a straight line. Linear functions can be represented in multiple ways: tables, graphs, equations, and verbal descriptions.

• Function: A relation where each input has exactly one output.

• Invertible Function: A function is invertible if each output is paired with only one input (one-to-one).

• Linear Function: The output changes by a constant amount for each unit increase in input.

Example: The function $f(x) = 2x + 3$ is linear; its graph is a straight line, and its table shows a constant rate of change.

Four Representations of Functions

• Table: Lists input-output pairs.

• Graph: Plots points and shows the shape of the function.

• Equation: Algebraic rule for the function.

• Verbal Description: Describes the relationship in words.

Example: For $f(x) = 2x + 3$, a table, graph, and verbal description can all be constructed.

Slope and Intercept of a Line

Definition of Slope

The slope of a line measures its steepness and is calculated as the ratio of the change in y to the change in x between two points:

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

• Positive Slope: Line rises from left to right.

• Negative Slope: Line falls from left to right.

• Zero Slope: Line is horizontal.

• Undefined Slope: Line is vertical.

Slope Type	Graphical Description	Behavior
Positive	Line rises	Increasing
Negative	Line falls	Decreasing
Zero	Horizontal	Constant
Undefined	Vertical	Not a function

Finding Slope from Two Points

Given two points $(x_1, y_1)$ and $(x_2, y_2)$, the slope is:

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

Example: For points (2, 3) and (4, 7): $m = \frac{7 - 3}{4 - 2} = 2$

Y-Intercept

The y-intercept is the point where the line crosses the y-axis ($x = 0$). In $y = mx + b$, b is the y-intercept.

Example: For $y = 2x + 3$, the y-intercept is 3.

Forms of Linear Equations

Slope-Intercept Form

The most common form for a linear equation is:

$y = mx + b$

• m: Slope

• b: Y-intercept

Point-Slope Form

Given a point $(x_1, y_1)$ and slope $m$:

$y - y_1 = m(x - x_1)$

Example: Through (3, 2) with slope 4: $y - 2 = 4(x - 3)$

Standard Form

Another way to write a linear equation:

$Ax + By = C$

Where $A$, $B$, and $C$ are real numbers, and $A$ and $B$ are not both zero.

Graphing Linear Functions

Graphing from Equation or Table

• Plot the y-intercept $(0, b)$.

• Use the slope to find another point: from $(0, b)$, move right 1 unit and up/down $m$ units.

• Draw a straight line through the points.

Example: For $y = -2x + 4$, plot (0, 4), then (1, 2), and draw the line.

Parallel and Perpendicular Lines

Parallel Lines

• Have the same slope ($m_1 = m_2$).

• Never intersect.

Perpendicular Lines

• Slopes are negative reciprocals: $m_1 \cdot m_2 = -1$.

• Intersect at a right angle.

Example: If a line has slope 2, a perpendicular line has slope $-\frac{1}{2}$.

Identifying Linear Functions from Data

Tables and Linear Functions

• If the change in output is constant for each unit change in input, the function is linear.

• Check differences between consecutive outputs.

Example: Table: x: 1, 2, 3, 4; y: 5, 7, 9, 11. The change in y is always 2, so the function is linear.

Applications of Linear Functions

Modeling Real-World Situations

• Linear functions can model constant rates, such as speed, cost, or growth.

• Given a verbal description, identify the rate (slope) and initial value (y-intercept).

Example: If a taxi charges $3 plus $2 per mile, the cost function is$C(x) = 2x + 3$.

Solving Application Problems

• Translate the problem into a linear equation.

• Identify variables, slope, and intercept.

• Use the equation to answer questions about the situation.

Example: If planting seeds costs $2000 plus $300 per acre,$C(a) = 300a + 2000$.

Practice Problems and Solutions

Sample Problems

• Find the slope of the line passing through (2, -4) and (4, 2): $m = \frac{2 - (-4)}{4 - 2} = 3$

• Determine if the equation $y = 2x^2 + 3$ is linear (it is not, because the exponent is not 1).

• Write the equation of a line through (1, 7) and parallel to $y = 4x - 12$: $y - 7 = 4(x - 1)$

• Find the equation of a line through (3, 2) and perpendicular to $y = -\frac{1}{2}x + 4$: Slope is 2, so $y - 2 = 2(x - 3)$

Summary Table: Forms of Linear Equations

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 a point and slope are known
Standard	$Ax + By = C$	General form, for solving systems

Key Concepts to Remember

• Linear functions have a constant rate of change (slope).

• The graph of a linear function is always a straight line.

• Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals.

• Linear models are useful for describing real-world situations with constant rates.

Additional info: These notes cover the identification, representation, and application of linear functions, including graphical, tabular, algebraic, and verbal forms, as well as the calculation and interpretation of slope and intercepts. Practice problems reinforce these concepts and their use in modeling real-world scenarios.