4.1 Linear Functions

Linear functions are the building blocks of algebra, describing straight-line relationships between variables. They're everywhere in real life, from calculating costs to predicting travel times. Understanding their components and forms helps us model and analyze countless everyday situations.

Mastering linear functions opens doors to more complex math concepts. By grasping slopes, intercepts, and different equation forms, you'll develop problem-solving skills applicable across various fields. It's a fundamental tool for making sense of the world around us.

Linear Functions

Representation of linear functions

• Definition of a linear function: $f(x) = mx + b$ (function notation)
  • $m$ represents the slope, rate of change of $y$ with respect to $x$
  • $b$ represents the y-intercept, value of $y$ when $x = 0$
• Slope-intercept form: $y = mx + b$
  • Directly provides slope ($m$) and y-intercept ($b$)
• Point-slope form: $y - y_1 = m(x - x_1)$
  • Utilizes a known point $(x_1, y_1)$ and slope $m$ to define the line
• Standard form: $Ax + By = C$
  • $A$, $B$, and $C$ are constants, $A$ and $B$ cannot both be zero
• Graphing linear functions using slope-intercept form
  • Plot y-intercept $(0, b)$, then use slope $m$ to find additional points
  • Rise over run: move vertically by $m$ units for every 1 unit horizontally
• Graphing linear functions using point-slope form
  • Plot given point $(x_1, y_1)$, then use slope $m$ to find additional points
• Graphing linear functions using standard form
  • Find x-intercept by setting $y = 0$ and solving for $x$
  • Find y-intercept by setting $x = 0$ and solving for $y$
  • Plot intercepts and connect with a straight line

Slope as rate of change

• Slope measures the steepness and direction of a line
  • Positive slope: $y$ increases as $x$ increases (uphill)
  • Negative slope: $y$ decreases as $x$ increases (downhill)
  • Zero slope: $y$ remains constant as $x$ changes (horizontal)
• Real-world applications of slope
  • Distance-time graphs: slope represents velocity (miles per hour, meters per second)
  • Cost-quantity graphs: slope represents unit price (dollars per item)
  • Temperature-time graphs: slope represents rate of temperature change (℃ per minute)

Equations for linear functions

• Writing equations given slope and y-intercept
  • Use slope-intercept form: $y = mx + b$
• Writing equations given slope and a point
  1. Use point-slope form: $y - y_1 = m(x - x_1)$
  2. Simplify and convert to slope-intercept form
• Writing equations given two points
  1. Calculate slope using $m = \frac{y_2 - y_1}{x_2 - x_1}$
  2. Use point-slope form with calculated slope and one of the given points
  3. Simplify and convert to slope-intercept form
• Analyzing equations in slope-intercept form
  • Identify slope $m$ and y-intercept $b$ directly from the equation
• Determining x-intercept
  • Set $y = 0$ and solve for $x$

Behavior of linear functions

• Increasing function: positive slope ($m > 0$)
  • As $x$ increases, $y$ increases
• Decreasing function: negative slope ($m < 0$)
  • As $x$ increases, $y$ decreases
• Constant function: zero slope ($m = 0$)
  • As $x$ changes, $y$ remains constant (horizontal line)

Parallel and perpendicular lines

• Parallel lines have the same slope but different y-intercepts
  • Equation of a line parallel to $y = mx + b$ passing through $(x_1, y_1)$: $y - y_1 = m(x - x_1)$
• Perpendicular lines have slopes that are negative reciprocals of each other
  • If $m_1$ and $m_2$ are slopes of perpendicular lines, then $m_1 \cdot m_2 = -1$
  • Equation of a line perpendicular to $y = mx + b$ passing through $(x_1, y_1)$: $y - y_1 = -\frac{1}{m}(x - x_1)$

Applications of linear functions

• Modeling real-world situations
  1. Identify variables and constants
  2. Determine appropriate equation form based on given information
  3. Interpret results in the context of the problem
• Examples of practical problems
  • Break-even analysis: finding production level where revenue equals costs
  • Total cost calculation: accounting for base price, tax, and shipping
  • Time-distance problems: determining travel time at constant speed

Technology for linear relationships

• Graphing calculators
  • Enter and graph linear functions
  • Find equation of a line given two points or a point and slope
  • Determine intersection point(s) of two lines
• Spreadsheet software (Microsoft Excel, Google Sheets)
  • Create scatter plots of data points
  • Add trendlines to visualize linear relationships
  • Use built-in functions for slope, y-intercept, and correlation coefficient
• Online graphing tools (Desmos, GeoGebra)
  • Graph linear functions and manipulate parameters
  • Explore effects of changing slope and y-intercept on the graph
  • Create sliders to dynamically adjust equation coefficients

Linear Functions and the Coordinate Plane

• The coordinate plane is used to visualize linear functions graphically
• Domain: the set of all possible input values (x-coordinates) for a linear function
• Range: the set of all possible output values (y-coordinates) for a linear function
• Linear equation: an equation that represents a straight line on the coordinate plane
• Independent variable: typically represented by x, it's the input of the function
• Dependent variable: typically represented by y, it's the output of the function and depends on the value of x