Functions, Graphs and Features

Lesson 8

Math

Unit 1

9th Grade

Lesson 8 of 11

Objective


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

Common Core Standards


Core Standards

  • 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.
  • 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.A — Graph linear and quadratic functions and show intercepts, maxima, and minima.
  • 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

  • 8.F.B.5

Criteria for Success


  1. Describe that a contextual situation is nonlinear, and identify the features that make the function nonlinear.
  2. Describe that a parabola is a shape used to describe quadratic functions. 
  3. Describe the rate of change over intervals of a quadratic function. Compare the rate of change to a linear function. 
  4. Identify contexts that could be represented by a quadratic function, such as profit, falling objects, thrown objects, and area relationships.
  5. Describe the domain and range of quadratic functions as well as features of quadratic functions represented graphically. 

Tips for Teachers


This lesson introduces the idea of a quadratic 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 Units 7 and 8. 

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Anchor Problems


Problem 1

As seen in this video, a ball is placed at the top of a ramp and let go. Sketch a graph of the distance the ball travels as a function of time. 

Guiding Questions

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References

EngageNY Mathematics Algebra I > Module 1 > Topic A > Lesson 2Ball Rolling Down Ramp

Algebra I > Module 1 > Topic A > Lesson 2 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..

Problem 2

The table below gives the area of a square with sides of whole number lengths. Plot the points in the table on a graph and draw the curve that goes through the points. 

 

Side (cm) 0 1 2 3 4
Area $${(cm^2)}$$ 0 1 4 9 16

On the same graph, reflect the curve across the y-axis. 

Guiding Questions

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References

EngageNY Mathematics Algebra I > Module 1 > Topic A > Lesson 2Example 2

Algebra I > Module 1 > Topic A > Lesson 2 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..

Problem Set


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

Target Task


Below is a graph of the height of a toy rocket over time. This is modeled by a quadratic function. 

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

Additional Practice


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.

  • For the following problem, ask students to identify the variables they will use to graph against. This is a review of earlier in the unit. 

“Below is a stop-action shot of a basketball being shot toward the hoop. The pictures were taken at regular intervals. 
What are the variables in this situation? How can you tell that the rate of change is decreasing as the ball reaches the maximum height? Where in the picture is the rate of change the greatest? How do you know?”

  • Include problems that ask students to interpret graphs, such as this one: Below is a graph that models the relationship of the average speed $${(x)}$$, in miles per hour, to the cost, in cents $${(y)}$$, of driving my car. 
  1. What is the meaning of the point $${(20,22)}$$ in this scenario? 
  2. Describe this scenario by describing what is happening over particular intervals. 
  3. How fast am I driving if driving at that speed costs $0.31 a mile?

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Lesson 7

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Lesson 9

Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Features of Functions

Topic B: Nonlinear Functions

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