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Mathematics - Secondary Curriculum Secondary Mathematics I
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Strand: FUNCTIONS - Interpreting Linear and Exponential Functions (F.IF)

Understand the concept of a linear or exponential function and use function notation. Recognize arithmetic and geometric sequences as examples of linear and exponential functions (Standards F.IF.13). Interpret linear or exponential functions that arise in applications in terms of a context (Standards F.IF.46). Analyze linear or exponential functions using different representations (Standards F.IF.7, 9).

Standard F.IF.5

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

  • Average Cost
    For a function that models a relationship between two quantities, students will interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
  • Derivate
    Students may use the applet in this lesson to graph a function and a tangent line and view its equation.
  • Domain and Range video
    This video introduces the concepts of domain and range.
  • Function Flyer
    The applet on this site allows the students to manipulate the graph of a function by changing the value of exponents, coefficients and constants.
  • FUNCTIONS - Interpreting Linear and Exponential Functions (F.IF) - Sec Math I Core Guide
    The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I Interpreting Linear and Exponential Functions (F.IF).
  • Functions and the Vertical Line Test
    The vertical line test for functions is the focus of this lesson plan.
  • Graphing Calculator
    A free online graphing calculator.
  • Graphing Stories
    The purpose of this task is to have students represent each indicated relationship of a given variable vs. time graphically with special attention to representing key features of increasing and decreasing intervals, maximums and minimums, intercepts, and constant and variable rates of change.
  • Graphit
    With this interactive applet students are able to create graphs of functions and sets of ordered pairs on the same coordinate plane.
  • Hoisting the Flag 1
    In this task, students are given a scenario and different graphs that could describe the relationship of the quantities in the situation, the height of a flag as a function of time.
  • Hoisting the Flag 2
    In this task, students are given a scenario and different graphs that could describe the relationship of the quantities in the situation, the height of a flag as a function of time.
  • How is the Weather?
    This task can be used as a quick assessment to see if students can make sense of a graph in the context of a real world situation. The task could also be used to generate a group discussion on interpreting functions given by graphs.
  • Influenza Epidemic
    The principal purpose of this task is to probe students' ability to correlate symbolic statements about a function using function notation with a graph of the function, and to interpret their answers in terms of the quantities between which the function describes a relationship.
  • Introduction to the Materials (Math 1)
    Introduction to the Materials in the Mathematics One of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems.
  • IXL Game: Linear Functions: Standard Form
    This game will help the student understand linear functions, specifically the standard form by finding x- and y-intercepts. This is just one of many online games that supports the Utah Math core. Note: The IXL site requires subscription for unlimited use.
  • Lake Sonoma
    This task asks students to describe features of a graph. It provides an opportunity to introduce (or use) mathematical terminology that makes communication easier and more precise, such as: periodic behavior, maxima, minima, outliers, increasing, decreasing, slope.
  • Laptop Battery Charge 2
    This task uses a situation that is familiar to students to solve a problem they probably have all encountered before: How long will it take until an electronic device has a fully charged battery? A linear model can be used to solve this problem. The task combines ideas from statistics, functions and modeling. It is a nice combination of ideas in different domains in the high school curriculum.
  • Logistic Growth Model, Abstract Version
    This task is for instructional purposes only and students should already be familiar with some specific examples of logistic growth functions such as that given in ''Logistic growth model, concrete case.''
  • Logistic Growth Model, Explicit Version
    This problem introduces a logistic growth model in the concrete setting of estimating the population of the U.S.
  • Model air plane acrobatics
    This task could serve as an introduction to periodic functions and as a lead-in to sinusoidal functions. By visualizing the height of a plane that is moving along the circumference of a circle several times, students get the idea that output values of the height functions will repeat themselves after each complete revolution. They also connect the situation with key features on the graph, for example they interpret the midline and amplitude of the function as the height of the center of the circle and its radius.
  • Modeling London's Population
    The purpose of this task is to model the population data for London with a variety of different functions. In addition to the linear, quadratic, and exponential models, this task introduces an additional model, namely the logistic model.
  • Module 3: Features of Functions - Student Edition (Math 1)
    The Mathematics Vision Project, Secondary Math One Module 3, Features of Functions, is the culminating functions module in Secondary Math I. In this module, students broaden their thinking about functions to relationships that are not either linear or exponential.
  • Module 3: Features of Functions - Teacher Notes (Math 1)
    The Mathematics Vision Project, Secondary Math One Module 3 Teacher Notes, Features of Functions, is the culminating functions module in Secondary Math I. In this module, students broaden their thinking about functions to relationships that are not either linear or exponential.
  • Multi-Function Data Flyer
    The applet in this lesson allows students to plot ordered pairs and then change the values in order to observe the effects of those changes.
  • Oakland Coliseum
    This deceptively simple task asks students to find the domain and range of a function from a given context. The function is linear and if simply looked at from a formulaic point of view, students might find the formula for the line and say that the domain and range are all real numbers. However, in the context of this problem, this answer does not make sense, as the context requires that all input and output values are non-negative integers, and imposes additional restrictions.
  • Playing Catch
    This task gives the graph of the height of a ball over time and asks for a story that could be represented by this graph. The graph is the mathematical representation of a situation and features of the graph correspond to specific moments in the story the graph tells. The purpose of the task is to get away from plotting graphs by focusing on coordinate points and instead looking at the bigger picture a qualitative view.
  • Polynomials video
    The video introduces and explains the topic.
  • Possible or Not
    Students can look at graphed functions from real-life examples and determine whether the graph makes sense or not in this activity.
  • Rate of Change and Slope
    This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review.
  • Rate of Change and Slope video
    This video introduces the concepts.
  • Solar Radiation Model
    The task is a seemingly straightforward modeling task that can lead to more involved tasks if the instructor expands on it. In this task, students also have to interpret the units of the input and output variables of the solar radiation function.
  • Student Task: Interpreting Functions
    This task consists of a set of 2 short questions.
  • Telling a Story With Graphs
    In this task students are given graphs of quantities related to weather. The purpose of the task is to show that graphs are more than a collection of coordinate points, that they can tell a story about the variables that are involved and together they can paint a very complete picture of a situation, in this case the weather.
  • Temperature Change
    This task gives an easy context to introduce the idea of average rate of change. This problem could be done as a Think-Pair-Share activity. After posing the question, students can decide what they think and why and then discuss their answer with their neighbor.
  • The Aquarium
    The purpose of this task is to connect graphs with real life situations. Graphs tell a story. Specific features of a graph connect to specific features of a story. A point on a graph captures a specific instant in the story.
  • The Canoe Trip, Variation 1
    The purpose of this task is to give students practice constructing functions that represent a quantity of interest in a context, and then interpreting features of the function in the light of that context. It can be used as either an assessment or a teaching task.
  • The Canoe Trip, Variation 2
    The primary purpose of this task is to lead students to a numerical and graphical understanding of the behavior of a rational function near a vertical asymptote, in terms of the expression defining the function. The canoe context focuses attention on the variables as numbers, rather than as abstract symbols.
  • The High School Gym
    In this task, students will calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval.
  • The Restaurant
    The purpose of this task is to get students thinking about the domain and range of a function representing a particular context.
  • The story of a flight
    This task uses data from an actual flight computer.
  • Warming and Cooling
    This task is meant to be a straight-forward assessment task of graph reading and interpreting skills. This task helps reinforce the idea that when a variable represents time, t=0 is chosen as an arbitrary point in time and positive times are interpreted as times that happen after that.
  • Words - Tables - Graphs
    The purpose of the task is to show that graphs can tell a story about the variables that are involved.

UEN logo - in partnership with Utah State Board of Education (USBE) and Utah System of Higher Education (USHE).  Send questions or comments to USBE Specialist - Lindsey  Henderson and see the Mathematics - Secondary website. For general questions about Utah's Core Standards contact the Director - Jennifer  Throndsen.

These materials have been produced by and for the teachers of the State of Utah. Copies of these materials may be freely reproduced for teacher and classroom use. When distributing these materials, credit should be given to Utah State Board of Education. These materials may not be published, in whole or part, or in any other format, without the written permission of the Utah State Board of Education, 250 East 500 South, PO Box 144200, Salt Lake City, Utah 84114-4200.