crams
Honors Pre-Calculus
Table of Contents
Unit 2 Overview: Linear Functions
2.1 Linear Functions
2.2 Graphs of Linear Functions
2.3 Modeling with Linear Functions
2.4 Fitting Linear Models to Data

Linear functions are the building blocks of algebra. They're straight lines on a graph, showing a constant rate of change between two variables. Understanding their key features, like slope and intercepts, is crucial for solving real-world problems.

Graphing linear functions helps visualize relationships. You can determine equations from graphs, and vice versa. This skill is essential for analyzing parallel and perpendicular lines, solving systems of equations, and exploring function properties.

Graphing and Interpreting Linear Functions

Key features of linear functions

  • Linear functions have the general form $y = mx + b$, where $m$ represents the slope and $b$ represents the y-intercept
    • Slope ($m$) indicates the rate of change of the function
      • Calculated using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ or the rise over run method
      • Positive slope signifies an increasing function (upward sloping line), while negative slope signifies a decreasing function (downward sloping line)
    • y-intercept ($b$) represents the point at which the line intersects the y-axis
      • Occurs when the x-coordinate is equal to 0
    • x-intercept represents the point at which the line intersects the x-axis
      • Occurs when the y-coordinate is equal to 0
  • To graph a linear function, plot the y-intercept and use the slope to determine additional points on the line
  • Linear equations can be expressed using function notation (e.g., f(x) = mx + b)

Equations from linear graphs

  • To determine the equation of a line from its graph, identify the slope and y-intercept
    • Slope can be calculated using two distinct points on the line: $m = \frac{y_2 - y_1}{x_2 - x_1}$
    • y-intercept is the point at which the line intersects the y-axis
  • Substitute the calculated slope and y-intercept into the slope-intercept form of the equation: $y = mx + b$

Relationships Between Lines

Parallel and perpendicular lines

  • Parallel lines have identical slopes but different y-intercepts
    • General form: $y = m_1x + b_1$ and $y = m_1x + b_2$, where $m_1$ is the same for both lines
  • Perpendicular lines have slopes that are negative reciprocals of each other
    • If line 1 has slope $m_1$, then line 2 has slope $m_2 = -\frac{1}{m_1}$

Solutions to linear systems

  • For a line parallel to $y = mx + b$, use the same slope $m$ and a different y-intercept
    • Example: If the given line is $y = 2x + 3$, a parallel line could be $y = 2x - 1$
  • For a line perpendicular to $y = mx + b$, use the negative reciprocal of the slope $m$ and any y-intercept
    • Example: If the given line is $y = 2x + 3$, a perpendicular line could be $y = -\frac{1}{2}x + 1$

Solving Systems of Linear Equations

Solutions to linear systems

  • Graphical method: Graph both linear equations on the same coordinate plane
    1. The solution is the point of intersection of the two lines
    2. If the lines are parallel, there is no solution (inconsistent system)
    3. If the lines are the same, there are infinitely many solutions (dependent system)
  • Algebraic methods: Elimination and substitution
    1. Elimination: Add or subtract the equations to eliminate one variable, then solve for the remaining variable
    2. Substitution: Solve one equation for a variable and substitute the result into the other equation
  • Interpret the solution in the context of the problem, if applicable (real-world scenarios)

Function Properties and Representation

Domain and Range

  • The domain of a linear function is the set of all possible input values (x-coordinates)
  • The range of a linear function is the set of all possible output values (y-coordinates)
  • For most linear functions, both the domain and range are all real numbers, unless there are specific restrictions given in the problem context

Coordinate Plane

  • Linear functions are typically graphed on a coordinate plane, which consists of two perpendicular number lines (x-axis and y-axis)
  • Each point on the coordinate plane represents an ordered pair (x, y)
  • The coordinate plane is divided into four quadrants, numbered counterclockwise from the upper right

Key Terms to Review (22)

Domain: The domain of a function refers to the set of all possible input values for the function. It represents the range of values that the independent variable can take on. The domain is a crucial concept in understanding the behavior and properties of various mathematical functions.
Range: The range of a function refers to the set of all possible output values or the set of all values that the function can attain. It represents the vertical extent or the interval of values that the function can produce as the input variable changes. The range is an important concept in the study of functions and their properties, as it provides information about the behavior and characteristics of the function.
Y-intercept: The y-intercept is the point where a line or curve intersects the y-axis, representing the value of the function when the independent variable (x) is equal to zero. It is a critical parameter that describes the behavior of various functions, including linear, quadratic, polynomial, and exponential functions.
X-Intercept: The x-intercept of a graph is the point where the graph intersects the x-axis, indicating the value of x when the function's output or y-value is zero. It represents the horizontal location where a curve or line crosses the x-axis.
Increasing Function: An increasing function is a function whose value increases as the input variable increases. In other words, as the independent variable gets larger, the dependent variable also gets larger. This concept is fundamental to understanding the behavior and rates of change of various types of functions.
Decreasing Function: A decreasing function is a function where the output values decrease as the input values increase. In other words, as the independent variable increases, the dependent variable decreases. This term is particularly relevant in the context of understanding rates of change and the behavior of graphs, as well as analyzing the properties of linear functions.
Slope: Slope is a measure of the steepness or incline of a line. It represents the rate of change in the vertical direction (y-coordinate) compared to the change in the horizontal direction (x-coordinate) along a line. Slope is a fundamental concept in understanding the properties and behaviors of linear functions.
Linear Functions: A linear function is a mathematical function that can be represented by a straight line on a graph. These functions have a constant rate of change, meaning the change in the dependent variable is proportional to the change in the independent variable.
Function Notation: Function notation is a way of representing functions using symbolic expressions, where the function name is followed by an input value enclosed in parentheses. It provides a concise and efficient way to denote the relationship between the input and output of a function, allowing for the evaluation and manipulation of functions.
Slope-Intercept Form: The slope-intercept form is a way to express the equation of a linear function, where the slope and y-intercept of the line are explicitly shown. This form allows for easy interpretation of the line's behavior and is widely used in the study of linear functions, their graphs, and the fitting of linear models to data.
Perpendicular Lines: Perpendicular lines are lines that intersect at a right angle, forming a 90-degree angle between them. This geometric relationship is a fundamental concept in the study of linear functions and their graphical representations.
Parallel Lines: Parallel lines are a pair of lines in a plane that never intersect, maintaining a constant distance between them. This geometric concept is fundamental to understanding various topics in mathematics, including linear functions, graphs of linear functions, and systems of linear equations.
Rise Over Run: Rise over run, also known as the slope, is a way to quantify the steepness or incline of a line on a coordinate plane. It represents the change in the vertical position (rise) of a point on the line compared to the change in its horizontal position (run), providing a measure of the line's slope or rate of change.
Rate of Change: The rate of change is a measure of how a quantity changes over time or with respect to another variable. It represents the slope or steepness of a line or curve, and it is a fundamental concept in understanding the behavior of functions and their graphical representations.
Negative Reciprocals: Negative reciprocals are the inverse of positive numbers, where the sign is changed from positive to negative. They are commonly used in the context of linear functions to describe the relationship between variables and their graphical representations.
Inconsistent System: An inconsistent system is a system of linear equations that has no solution, meaning there are no values for the variables that satisfy all the equations in the system simultaneously. This concept is crucial in understanding the behavior of systems of linear equations and their graphical representations.
Elimination Method: The elimination method, also known as the substitution method, is a technique used to solve systems of linear equations by eliminating one of the variables through algebraic manipulation. This method allows for the determination of the values of the variables that satisfy all the equations in the system simultaneously.
Dependent System: A dependent system is a set of linear equations or functions where the variables are interdependent, meaning the value of one variable depends on the values of the other variables in the system. This concept is crucial in understanding the behavior and solutions of linear functions and systems of linear equations.
Coordinate Plane: The coordinate plane is a two-dimensional grid used to represent and analyze the relationships between variables. It consists of a horizontal x-axis and a vertical y-axis that intersect at the origin, forming four quadrants that allow for the precise location and visualization of points, lines, and other geometric shapes.
Point of Intersection: The point of intersection is the location where two or more lines, curves, or functions meet and share a common point. It represents the solution to a system of equations or the point at which the graphs of linear functions intersect.
Substitution Method: The substitution method is a technique used to solve systems of linear equations by isolating one variable in one equation and then substituting that expression into the other equation(s) to find the values of the remaining variables. This method is applicable in the context of graphing linear functions, as well as solving systems of linear equations in two or three variables, and even systems of nonlinear equations and inequalities in two variables.
Linear Equation: A linear equation is a mathematical equation in which the variables are raised to the first power and the variables are connected by addition, subtraction, or scalar multiplication. These equations represent a straight line when graphed on a coordinate plane.