MLF - Modeling Linear Functions (Overview)
Modeling Linear Functions
Introduction
You pay $40 a month to get 5GB of data to use for your phone. Last month, you went over by 35 MB and you are charged $0.50 per MB. How much do you owe? If you can figure this problem out, then you can work with linear equations. The study of linear equations is one of the most important in all of Algebra and used to model many situations in the real world. In this unit, you will practice writing, solving, and graphing linear equations. We will explore these functions in three different forms: as a sequence, as a graph, and as an equation. Connecting each of these representations will be essential for you as we work on problem solving!
Essential Questions
- How do I use graphs to represent and solve real-world equations?
- What is a function and how do I use it to model real-world situations?
- How do I interpret the parts of a function in the context of the problem?
- How do I interpret key features of graphs in context?
- Why are sequences functions?
- How do I write recursive and explicit formulas for arithmetic sequences?
- How do I use graphs to represent and solve real-world equations?
- How do I interpret functions that arise in applications in terms of context?
- How do I use different representations to analyze linear functions?
- How do I build a linear function that models a relationship between two quantities?
- How can we use real-world situations to construct and compare linear models and solve problems?
- How do I interpret expressions for functions in terms of the situation they model?
- What are the specific features that distinguish the graphs of linear and non-linear functions from one another?
Key Terms
The following key terms will help you understand the content in this module.
Arithmetic Sequence - A sequence of numbers in which the difference between any two consecutive terms is the same.
Average Rate of Change - The change in the value of a quantity by the elapsed time. For a function, this is the change in the y-value divided by the change in the x-value for two distinct points on the graph.
Coefficient - A number multiplied by a variable in an algebraic expression.
Constant Rate of Change - With respect to the variable x of a linear function y = f(x), the constant rate of change is the slope of its graph.
Continuous - Describes a connected set of numbers, such as an interval.
Domain - The set of x-coordinates of the set of points on a graph. The set of x-coordinates of a given set of ordered pairs. The value that is the input in a function or relation.
End Behaviors - The appearance of a graph as it is followed farther and farther in either direction.
Explicit Expression - A formula that allows direct computation of any term for a sequence a 1 , a 2 , a 3 , . . . , a n , . . . .
Expression - Any mathematical calculation or formula combining numbers and/or variables using sums, differences, products, and quotients including fractions, exponents, roots, logarithms, functions, or other mathematical operations.
Interval Notation - A notation representing an interval as a pair of numbers. The numbers are the endpoints of the interval. Parentheses and/or brackets are used to show whether the endpoints are excluded or included.
Linear Function - A function with a constant rate of change and a straight line graph.
Linear Model - A linear function representing real-world phenomena. The model also represents patterns found in graphs and/or data.
Ordered Pair - A pair of numbers, (x, y), that indicates the position of a point on a Cartesian plane.
Range - The set of y-coordinates of the set of points on a graph. The set of y-coordinates of a given set of ordered pairs. The set of all possible outputs of a function or relation.
Recursive Formula - A formula that requires the computation of all previous terms to find the value of an .
Slope - The ratio of the vertical and horizontal changes between two points on a surface or a line.
Term - A value in a sequence--the first value in a sequence is the 1st term, the second value is the 2nd term, and so on. A term is also any of the monomials that make up a polynomial.
Variable - A letter or symbol used to represent a number.
x-intercept - The point where a line meets or crosses the x-axis.
y-intercept - The point where a line meets or crosses the y-axis.
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