lesson 9 problem solving practice qualitative graphs

Curriculum  /  Math  /  9th Grade  /  Unit 1: Functions, Graphs and Features  /  Lesson 9

Functions, Graphs and Features

Lesson 9 of 11

Criteria for Success

Tips for teachers, anchor problems, problem set, target task, additional practice.

Sketch an exponential function that represents a situation. Identify key features of the graph and relate to context.

Common Core Standards

Core standards.

The core standards covered in this lesson

Creating Equations

A.CED.A.2 — Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Interpreting Functions

F.IF.B.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. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

F.IF.C.7.E — Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Linear, Quadratic, and Exponential Models

F.LE.A.3 — Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Foundational Standards

The foundational standards covered in this lesson

8.F.B.5 — Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Describe the shape that is used to model exponential functions.
• Describe the rate of change over intervals of an exponential function. Compare the rate of change to a linear function and to a quadratic function. 
• Identify contexts that could be represented by an exponential function, such as interest, population growth, car depreciation, and bacteria growth. 
• Describe the domain and range of exponential functions, as well as features of the exponential functions represented graphically.

Suggestions for teachers to help them teach this lesson

This lesson introduces the idea of an exponential function within the analysis that students have been doing on other functions in this unit. Students will need to be able to recognize the shape and describe the general features but will not need to go in-depth. This work will be done in Unit 6. 

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

The table below gives the number of bacteria over time, shown in this video .  Plot the points in the table on a graph and draw the curve that goes through the points. 

Based on the pattern you see in the table of values, extend the graph to 4 seconds. 

Guiding Questions

Below is a graph of an amount of money invested tripling each year. This is modeled by an exponential function. 

Describe the features of the function in terms of intervals where the function is increasing/decreasing, intervals where the rate of change is constant/increasing/decreasing, and intercepts. Describe what each of these features means in the context of the problem. 

A set of suggested resources or problem types that teachers can turn into a problem set

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Assume that a bacteria population doubles every hour. Which of the following three tables of data, with x representing time in hours and y the count of bacteria, could represent the bacteria population with respect to time? For the chosen table of data, plot the graph of data. Label the axes appropriately with units. 

Algebra I > Module 1 > Topic A > Lesson 3 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds . © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US  license. Accessed Dec. 2, 2016, 5:15 p.m..

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

• Include problems where students need to identify whether something is always, sometimes, or never true. Use domain, range, intercepts, and rate of change. For example, is the statement shown always, sometimes, or never true? Explain your reasoning. “The rate of change of an exponential function is constant.” 
• Yummy Math Harlem Shake
• EngageNY Mathematics Algebra I > Module 1 > Topic A > Lesson 3 — Problem Set
• EngageNY Mathematics Algebra I > Module 1 > Topic A > Lesson 4 — General function analysis (not tied to particular parent functions)

Topic A: Features of Functions

Model a contextual situation graphically using appropriate scales and features.

F.IF.B.4 F.IF.B.6 N.Q.A.1 N.Q.A.2 N.Q.A.3

Define functions and evaluate points in tables, graphs, and contextual situations using function notation.

F.IF.A.1 F.IF.A.2

Identify features of functions, including x-intercept and y-intercept, in context. Evaluate function notation in context.

F.IF.A.2 F.IF.C.9

Identify the domain and range through contextual situations, and explore domain and range on a graph. Represent domain and range with inequalities.

F.IF.A.1 F.IF.B.5

Calculate and interpret the rate of change from two points on a graph, in a situation, or in function notation.

F.IF.A.2 F.IF.B.4 F.IF.B.6

Describe and sketch functions using the features of domain and range, intercepts, function behavior, and the value of the function.

Analyze the key features of a contextual situation and model these graphically.

F.IF.B.4 F.IF.B.5 F.IF.B.6 N.Q.A.2

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Topic B: Nonlinear Functions

Draw quadratic functions represented contextually. Identify key features of the graph and relate to context.

A.CED.A.2 F.IF.B.5 F.IF.C.7.A F.LE.A.3

A.CED.A.2 F.IF.B.5 F.IF.C.7.E F.LE.A.3

Draw a graph to represent a system of functions. Identify the solution to a system represented graphically and in context.

A.REI.D.11 F.IF.A.2 F.IF.B.5

Analyze functions and identify parent functions of graphs. Identify variables of a situation and the scale of the associated graph. Represent a situation in a graph, table, and description.

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8th Grade - Qualitative Graph

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Interpret a qualitative graph representing a contextual situation. SPI 0806.1.2 Links verified on 7/23/2014 Qualitative Graphs - from Learning Math: Patterns, Functions, and Algebra presented by Annenberg Media Rate of Change: Connecting Slope to Real Life - a lesson on using slope to find the rate of change accompanied by graphs and explanations Rate of Change Practice Problems - nine practice problems (including a graph to interpret) followed by an answer key Using the X and Y Intercept to Graph Linear Equations - This follow up to the lesson on rate of change deals with linear equations written in standard form rather than slope intercept form Finding the X Intercept and Y Intercept to Graph Standard Form Equations - five practice problems to practice using what was learned in the lesson above, followed by an answer key Review Quiz - the first problem asks students to match graphs with statements describing some situation

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Lesson 9 Skills Practice Qualitative Graphs 13 The graph below displays the distance Bryan was from home as he ran in preparation for a marathon Describe the change in distance over time. =

Gauth ai solution.

Lesson 9 Homework Practice Qualitative Graphs

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Exercise 2 Page   350

Qualitative graphs are graphs used to represent situations that may not have numerical values or graphs in which numerical values are not included.

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Example Solution:

Practice exercises

Asses your skill.

Jamaal purchased the same number of action figures daily for one week, and then he sold most of them on the Internet. We want to sketch a qualitative graph to represent the situation. Let's remember the definition of this type of graph.

Notice that we do not know the initial number of the figures, or how many figures Jamaal purchased each day — the numerical values are not included. We only know how the total number of action figures changed over time.

• At first, Jamaal purchased the same number of action figures daily for one week. This means that the total number was growing daily by the same amount.
• Then he sold most of them on the Internet. As a result, the total number of figures decreased by a significant amount.

We can model this situation using a qualitative graph that very closely resembles a bar graph . There will not be any numbers on either of the axes because we are not given any numerical values.

Note that this is just one of many ways to represent the situation.