4.2 Modeling with Linear Functions
4 min read•june 24, 2024
are essential tools for understanding real-world relationships. They help us analyze how one variable affects another, like how time impacts distance traveled or quantity influences price. These models simplify complex scenarios into easy-to-understand equations.
By identifying key components like and , we can interpret and predict outcomes. This skill is crucial for making informed decisions in various fields, from economics to engineering. Linear models provide a foundation for more advanced mathematical concepts and problem-solving techniques.
Linear Function Models
Linear models from real-world scenarios
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- Identify dependent (y) and independent (x) variables in a given scenario
- (y) output or response variable changes based on
- Independent variable (x) input or explanatory variable can be controlled or changed
- Determine slope () and y-intercept from verbal description
- Slope change in dependent variable per unit change in independent variable (miles per hour, cost per item)
- y-intercept initial value or starting point of dependent variable when independent variable is zero (starting distance, base cost)
- Use of linear equation to create model:
- slope (2 miles per hour, $0.50 per item)
- y-intercept (10 miles starting distance, $25 base cost)
Data analysis for linear functions
- Identify variables and relationships within data set
- Determine which variable depends on the other (price depends on quantity, distance depends on time)
- Plot data points on to visualize relationship between variables
- Use for independent variable and for dependent variable (time on x-axis, distance on y-axis)
- Create a to observe the overall pattern and potential outliers
- Determine if relationship between variables is linear by observing pattern of plotted points
- Points following straight line indicate (constant )
- Curved pattern indicates (, )
- Calculate slope using two distinct points from data set:
- Choose points far apart for more accurate slope ((6, 12) gives slope of 2)
- Identify y-intercept by extending line to intersect y-axis or using known point and slope to solve for in
- y-intercept is y-value when x is zero (line crossing y-axis at (0, 3) means y-intercept is 3)
- Substitute known point and slope into and solve for (point (1, 5) with slope 2 gives $b = 3)
- Construct linear function model using slope-intercept form:
- Plug in calculated slope and y-intercept ($y = 2x + 3)
Interpretation of linear model features
- Slope interpretation
- Rate of change of dependent variable with respect to independent variable (2 miles per hour, $0.50 per item)
- Direction of relationship (positive slope increasing, negative slope decreasing)
- Change in dependent variable for one-unit increase in independent variable ($2 increase in price for each additional item)
- y-intercept interpretation
- Value of dependent variable when independent variable is zero (starting distance of 10 miles, base cost of $25)
- Starting point or initial value of relationship (car begins trip at 10 miles, cost begins at $25 before adding items)
- interpretation
- Value of independent variable when dependent variable is zero (0 items sold, 0 hours elapsed)
- Found by setting and solving for in linear function model ( gives $x = -1.5)
- Contextual meaning
- Relate slope, y-intercept, and to real-world scenario (slope of 2 miles per hour, starting distance of 10 miles, reaches destination in -1.5 hours)
- Discuss implications and predictions based on (each additional item increases price by $0.50, can predict cost for any number of items)
- Consider limitations of (predicting within the range of data) and (predicting beyond the range of data)
Analyzing relationships and model fit
- : Measure of the strength and direction of the between variables
- Strong positive correlation: As x increases, y tends to increase
- Strong negative correlation: As x increases, y tends to decrease
- Weak correlation: Little to no consistent pattern between x and y
- : Determining whether changes in one variable directly cause changes in another
- Correlation does not imply causation; other factors may influence the relationship
- : Differences between observed y-values and predicted y-values from the model
- Used to assess how well the linear model fits the data
- Small, randomly distributed residuals indicate a good fit
Key Terms to Review (38)
$b$: $b$ is a variable that represents an unknown or changing quantity in the context of modeling with linear functions. It is a key component of the general linear function equation, $y = mx + b$, where $b$ represents the y-intercept of the line.
$m = \frac{y_2 - y_1}{x_2 - x_1}$: The term $m = \frac{y_2 - y_1}{x_2 - x_1}$ represents the slope of a linear function. It quantifies the rate of change between two points on a linear graph, describing how the dependent variable (y) changes with respect to the independent variable (x).
$m$: $m$ is a variable that represents the slope of a linear function. The slope is a measure of the steepness or incline of a line, and it describes the rate of change between the dependent and independent variables in a linear relationship.
$y = mx + b$: $y = mx + b$ is the equation for a linear function, where $y$ represents the dependent variable, $x$ represents the independent variable, $m$ represents the slope or rate of change, and $b$ represents the $y$-intercept, or the value of $y$ when $x = 0$. This equation is fundamental to understanding and modeling linear relationships in the context of 4.2 Modeling with Linear Functions.
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.
Causation: Causation refers to the relationship between two events or variables where one event or variable directly causes or influences the other. It is a fundamental concept in understanding the nature of relationships and the underlying mechanisms that drive various phenomena.
Coordinate plane: A coordinate plane is a two-dimensional surface formed by the intersection of a vertical line (y-axis) and a horizontal line (x-axis). These axes divide the plane into four quadrants used for graphing equations and geometric shapes.
Coordinate Plane: The coordinate plane, also known as the Cartesian coordinate system, is a two-dimensional plane that uses a horizontal (x-axis) and a vertical (y-axis) line to define the position of a point. It provides a way to represent and analyze relationships between variables in a visual and mathematical manner.
Correlation: Correlation is a statistical measure that describes the strength and direction of the linear relationship between two variables. It quantifies how changes in one variable are associated with changes in another variable.
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.
Equation in quadratic form: An equation is in quadratic form if it can be written as $a(x^2 + bx + c) = 0$ where $a, b,$ and $c$ are constants. It often involves substituting a variable to simplify into a standard quadratic equation.
Exponential growth: Exponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value. This results in the function increasing rapidly over time.
Exponential Growth: Exponential growth is a pattern of change where a quantity increases at a rate proportional to its current value. This means the quantity grows by a consistent percentage over equal intervals of time, leading to rapid, accelerating growth. Exponential growth is a fundamental concept in mathematics and has applications across various fields, including biology, economics, and technology.
Extrapolation: Extrapolation involves predicting values outside the range of your data set by extending the trend line. It assumes that the existing pattern continues beyond known data points.
Extrapolation: Extrapolation is the process of estimating or predicting a value or trend outside the known range of a variable or data set. It involves using an established pattern or relationship to make inferences about future or unobserved values.
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.
Interpolation: Interpolation is the process of estimating the value of a function or data point between known data points. It is a mathematical technique used to approximate unknown values based on the surrounding data, allowing for the prediction or estimation of intermediate values within a given range of known data points.
Linear Function Models: A linear function model is a mathematical representation of a real-world relationship where the dependent variable changes at a constant rate with respect to the independent variable. These models are widely used in various fields to analyze and predict linear trends in data.
Linear model: A linear model is a mathematical representation of a relationship between two variables that can be expressed with a linear equation, typically in the form $y = mx + b$. It is used to predict the value of one variable based on the value of another.
Linear relationship: A linear relationship is a relationship between two variables where the change in one variable is proportional to the change in the other. This relationship can be represented by a straight line on a graph using the equation $y = mx + b$.
Linear Relationship: A linear relationship is a mathematical association between two variables where the change in one variable is proportional to the change in the other. This type of relationship is characterized by a constant rate of change, resulting in a straight-line graph when plotted.
Non-linear Relationship: A non-linear relationship is a type of relationship between two variables where the change in one variable is not proportional to the change in the other variable. This means that the relationship between the variables cannot be represented by a straight line, as it exhibits a curved or complex pattern.
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.
Quadratic: A quadratic is a polynomial function of the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$. Quadratics are characterized by a parabolic shape and have many important applications in mathematics and the sciences.
Rate of change: Rate of change measures how one quantity changes in relation to another. In algebra, it often refers to the slope of a line, indicating how the dependent variable changes with respect to the independent variable.
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.
Residuals: Residuals refer to the differences between the observed values and the predicted values in a statistical model. They represent the portion of the observed data that is not explained by the model, providing insights into the model's accuracy and the presence of unaccounted factors.
Scatter plot: A scatter plot is a type of graph used to display and assess the relationship between two numerical variables. Each point on the scatter plot represents an individual data point with its coordinates corresponding to the values of the two variables.
Scatter Plot: A scatter plot is a type of data visualization that displays the relationship between two numerical variables by plotting individual data points on a coordinate plane. It allows for the identification of patterns, trends, and potential relationships between the variables.
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 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.
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.
System of linear equations: A system of linear equations consists of two or more linear equations involving the same set of variables. The solutions to the system are the points where the graphs of these equations intersect.
X-Axis: The x-axis is the horizontal axis on a coordinate plane, typically running left to right. It is used to represent the independent variable in a graph and helps visualize the relationship between two or more variables.
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 $y = 0$.
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-axis: The y-axis is the vertical axis in a rectangular coordinate system, which represents the dependent variable and is typically used to plot the values or outcomes of a function. It is perpendicular to the x-axis and provides a visual reference for the range of values a function can take on.
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.