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
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- Linear functions represented in various forms:
- Slope-intercept form represents slope and y-intercept
- Point-slope form uses point on the line and slope
- Standard form has constants , , and , with and not both zero
- Interpret parameters in each form:
- Slope-intercept: is rate of change, is initial value or starting point
- Point-slope: is rate of change, is known point on line
- Standard: is slope, is y-intercept when
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:
- 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 , with being the constant of variation (slope)
Equations from given information
- Slope and y-intercept given: Use slope-intercept form
- Slope and point given: Use point-slope form , then convert to slope-intercept
- Two points given:
- Calculate slope using
- Use point-slope form
- 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:
- Plot y-intercept
- Use slope to find additional points: rise over run, or
- Connect points with straight line
- Interpret key features of graph:
- x-intercept: point where line crosses x-axis
- y-intercept: point where line crosses y-axis
- Slope: steepness and direction of line
Relationships between lines
- Parallel lines have same slope but different y-intercepts
- Equation of line parallel to is , where
- Perpendicular lines have slopes that are negative reciprocals
- If line 1 has slope , perpendicular line has slope
- Line 1 with slope 2 has perpendicular line with slope
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 is used to represent the output of a linear function for a given input
- 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
Key Terms to Review (43)
Compression: Compression refers to a transformation that reduces the distance between points in a graph. It often results in the graph appearing 'squeezed' either horizontally or vertically.
Average rate of change: The average rate of change of a function between two points is the change in the function's value divided by the change in the input values. It represents the slope of the secant line connecting these points on the graph.
Decreasing function: A decreasing function is one where the value of the function decreases as the input increases. For any two points $x_1$ and $x_2$ where $x_1 < x_2$, $f(x_1) \geq f(x_2)$.
Direct variation: Direct variation describes a linear relationship between two variables where one variable is a constant multiple of the other. Mathematically, it is expressed as $y = kx$, where $k$ is the constant of variation.
Domain: The domain of a function is the complete set of possible input values (x-values) that allow the function to work within its constraints. It specifies the range of x-values for which the function is defined.
Function notation: Function notation is a way to represent functions in the form $f(x)$, where $f$ names the function and $x$ represents the input variable. It provides a concise and clear way to denote the output of a function given an input.
Horizontal line: A horizontal line is a straight line that runs left to right and has a constant y-value for all points. Its slope is zero because there is no vertical change as you move along the line.
Increasing linear function: An increasing linear function is a function of the form $f(x) = mx + b$ where $m > 0$. This means that as $x$ increases, $f(x)$ also increases.
Linear equation: A linear equation in one variable is an algebraic equation that can be written in the form $ax + b = 0$, where $a$ and $b$ are constants, and $x$ is the variable. The solution to the equation is the value of $x$ that makes the equation true.
Linear function: A linear function is a mathematical function that creates a straight line when graphed. It can be expressed in the form $f(x) = mx + b$, where $m$ represents the slope and $b$ represents the y-intercept.
Perpendicular lines: Perpendicular lines are lines that intersect at a right angle, which is 90 degrees. Their slopes are negative reciprocals of each other in the coordinate plane.
Point-slope formula: The point-slope formula is a method used to find the equation of a line given a point on the line and its slope. It is expressed as $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope.
Slope: Slope measures the steepness and direction of a line, typically defined as the ratio of the vertical change to the horizontal change between two points on the line. It is commonly represented by the letter $m$.
Slope-intercept form: Slope-intercept form is a way to express the equation of a straight line using the formula $y = mx + b$. In this formula, $m$ represents the slope and $b$ represents the y-intercept.
Standard form: Standard form of a linear equation in one variable is written as $Ax + B = 0$, where $A$ and $B$ are constants and $x$ is the variable. The coefficient $A$ should not be zero.
Transformation: Transformation refers to any operation that moves or changes a function in some way. Common transformations include translations, dilations, reflections, and rotations.
Vertical line: A vertical line is a straight line that goes up and down and has an undefined slope. It is represented by the equation $x = a$ where $a$ is a constant.
Vertical line test: The vertical line test is a method used to determine if a graph represents a function. If any vertical line intersects the graph at more than one point, the graph does not represent a function.
X-intercept: The x-intercept is the point where a graph crosses the x-axis, where the y-coordinate is zero. It represents the solution(s) to an equation when $y = 0$.
B: The variable 'b' is a commonly used term in various mathematical contexts, including linear functions, sum-to-product and product-to-sum formulas, the ellipse, and the hyperbola. It often represents a constant or a coefficient that provides important information about the behavior and characteristics of these mathematical concepts.
Constant Function: A constant function is a function where the output value is the same for any input value. Regardless of the input, the function always returns the same constant value, making it a special type of linear function and polynomial function.
Dependent Variable: The dependent variable is the variable in a mathematical relationship or scientific experiment that is observed or measured to determine the effect of the independent variable. It is the output or response variable that changes in value as the independent variable is manipulated.
Domain: The domain of a function refers to the set of input values for which the function is defined. It represents the range of values that the independent variable can take on, and it is the set of all possible values that can be plugged into the function to produce a meaningful output.
Direct Variation: Direct variation is a mathematical relationship between two variables where one variable is directly proportional to the other. This means that as one variable increases, the other variable increases at the same rate, and vice versa. Direct variation is a fundamental concept in understanding the behavior of linear functions and modeling real-world situations involving proportional relationships.
F(x): f(x) is a function notation that represents a relationship between an independent variable, x, and a dependent variable, f. It is a fundamental concept in mathematics that underpins the study of functions, their properties, and their applications across various mathematical topics.
Function Notation: Function notation is a way of representing and working with functions, where the function is denoted by a letter or symbol, and the input values are placed within parentheses after the function name. This notation allows for the clear and concise representation of functional relationships, which is essential in understanding and manipulating functions in various mathematical contexts.
Graphing: Graphing is the visual representation of mathematical relationships, typically using a coordinate system to plot points, lines, curves, or other geometric shapes. It is a fundamental skill in mathematics that allows for the interpretation, analysis, and communication of quantitative information.
Identity Function: The identity function is a special type of function where the output value is always equal to the input value. It is a fundamental concept in mathematics that is particularly relevant in the study of function composition and linear functions.
Independent Variable: The independent variable is the variable that is manipulated or changed in an experiment or study to observe the effect on the dependent variable. It is the variable that is intentionally varied or controlled in order to measure its impact on the outcome.
Linear Equation: A linear equation is a mathematical expression that represents a straight line on a coordinate plane. It is an equation where the variables are raised to the power of one and the variables are connected by addition or subtraction operations.
Linear Interpolation: Linear interpolation is a method used to estimate the value of a function between two known data points by assuming a linear relationship between them. It involves finding the equation of the line that passes through the two points and using it to calculate the unknown value.
Linearity: Linearity is a fundamental property of functions that describes a direct, proportional relationship between the independent and dependent variables. It is a key concept in the study of linear functions, where the output changes at a constant rate as the input changes.
Parallel Lines: Parallel lines are two or more lines that lie in the same plane and never intersect, maintaining a constant distance between them. This concept is fundamental in understanding linear equations and functions, as parallel lines share important geometric properties.
Perpendicular Lines: Perpendicular lines are a pair of lines that intersect at a right angle, forming a 90-degree angle between them. This geometric relationship is an important concept in the study of linear equations and functions.
Point-Slope Form: The point-slope form is an equation that represents a linear function by specifying a point on the line and the slope of the line. It is a useful way to write the equation of a line when you know a point it passes through and the slope of the line.
Rate of Change: The rate of change is a measure of how a dependent variable changes in relation to changes in an independent variable. It describes the slope or steepness of a line or curve, indicating the speed at which one quantity is changing with respect to another.
Slope: Slope is a measure of the steepness or incline of a line or a surface. It represents the rate of change between two variables, typically the change in the vertical direction (y-coordinate) with respect to the change in the horizontal direction (x-coordinate).
Slope-Intercept Form: Slope-intercept form is a way of representing a linear equation in two variables, typically written as $y = mx + b$, where $m$ represents the slope of the line and $b$ represents the $y$-intercept. This form allows for easy interpretation of the line's characteristics and is widely used in the study of linear functions and their applications.
Standard Form: Standard form is a way of expressing mathematical equations or functions in a specific, organized format. It provides a consistent structure that allows for easier manipulation, comparison, and analysis of these mathematical representations across various topics in algebra and beyond.
Vertical Line Test: The vertical line test is a graphical method used to determine whether a relation represents a function. It involves drawing vertical lines on the coordinate plane to check if each vertical line intersects the graph at no more than one point.
X-Intercept: The x-intercept of a graph is the point where the graph of a function or equation intersects the x-axis, indicating the value of x when the function's output or the equation's value is zero. The x-intercept is a crucial concept in understanding the behavior and properties of various mathematical functions and equations.
Y-intercept: The y-intercept is the point at which a line or curve intersects the y-axis, representing the value of the dependent variable (y) when the independent variable (x) is zero. It is a crucial concept in understanding the behavior and properties of various mathematical functions and their graphical representations.
Range: In mathematics, the range refers to the set of all possible output values (dependent variable values) that a function can produce based on its input values (independent variable values). Understanding the range helps in analyzing how a function behaves and what values it can take, connecting it to various concepts like transformations, compositions, and types of functions.