Linear functions are the building blocks of algebra, describing relationships that change at a constant rate. They're everywhere in daily life, from calculating phone bills to predicting population growth. Understanding their forms and features is key to grasping more complex math concepts.

Mastering linear functions opens doors to analyzing real-world scenarios. You'll learn to interpret slopes as rates of change, graph lines, and solve problems using different equation forms. These skills will help you make sense of data, predict trends, and tackle more advanced math topics.

Linear Functions

Forms of linear functions

• Linear functions represented in various forms:
  • Slope-intercept form $y = mx + b$ represents slope $m$ and y-intercept $b$
  • Point-slope form $y - y_1 = m(x - x_1)$ uses point $(x_1, y_1)$ on the line and slope $m$
  • Standard form $Ax + By = C$ has constants $A$, $B$, and $C$, with $A$ and $B$ not both zero
• Interpret parameters in each form:
  • Slope-intercept: $m$ is rate of change, $b$ is initial value or starting point
  • Point-slope: $m$ is rate of change, $(x_1, y_1)$ is known point on line
  • Standard: $\frac{-A}{B}$ is slope, $\frac{C}{B}$ is y-intercept when $B \neq 0$

Slope as rate of change

• Slope represents rate of change in linear function
  • Calculated as change in y-value divided by change in x-value: $m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$
  • Positive slope indicates increasing function, negative slope decreasing function
  • Zero slope is horizontal line (no change in y-value), undefined slope is vertical line (no change in x-value)
• Interpret slope in real-world contexts
  • Distance-time graph: slope represents velocity or speed
• Direct variation is a special case where $y = kx$, with $k$ being the constant of variation (slope)

Equations from given information

• Slope and y-intercept given: Use slope-intercept form $y = mx + b$
• Slope and point given: Use point-slope form $y - y_1 = m(x - x_1)$, then convert to slope-intercept
• Two points given:
  1. Calculate slope using $m = \frac{y_2 - y_1}{x_2 - x_1}$
  2. Use point-slope form
  3. Convert to slope-intercept form
• Table of values given: Identify two points, calculate slope, use point-slope form
• Graph given: Identify slope and y-intercept, use slope-intercept form

Graphing and key features

• Graphing linear function:
  1. Plot y-intercept $(0, b)$
  2. Use slope to find additional points: rise over run, or $\frac{\Delta y}{\Delta x}$
  3. Connect points with straight line
• Interpret key features of graph:
  • x-intercept: point where line crosses x-axis $(y = 0)$
  • y-intercept: point where line crosses y-axis $(x = 0)$
  • Slope: steepness and direction of line

Relationships between lines

• Parallel lines have same slope but different y-intercepts
  • Equation of line parallel to $y = mx + b$ is $y = mx + c$, where $c \neq b$
• Perpendicular lines have slopes that are negative reciprocals
  • If line 1 has slope $m_1$, perpendicular line has slope $m_2 = -\frac{1}{m_1}$
  • Line 1 with slope 2 has perpendicular line with slope $-\frac{1}{2}$

Real-world applications of linear functions

• Identify given information and unknown variable in problem
• Create linear equation that models situation
• Solve equation for unknown variable
• Interpret solution in context of problem
• Check if solution makes sense in real-world context
• Linear interpolation can be used to estimate values between known data points

Representations of linear functions

• Algebraic representations (equations):
  • Slope-intercept, point-slope, and standard forms provide specific information about line (slope, y-intercept, points)
• Graphical representations:
  • Visual depiction of line shows x-intercept, y-intercept, and slope for quick identification of key features
• Tabular representations (tables of values):
  • List specific points on line, can be used to identify patterns and calculate slope
• Verbal representations (word problems):
  • Describe real-world situation involving linear relationship, require translation into algebraic representation to solve

Function notation and domain/range

• Function notation $f(x)$ is used to represent the output of a linear function for a given input $x$
• Domain is the set of all possible input values (x-values) for the function
• Range is the set of all possible output values (y-values) for the function