Curriculum  /  Math  /  11th Grade  /  Unit 2: Quadratics  /  Lesson 4

Quadratics

Lesson 4

Math

Unit 2

11th Grade

Lesson 4 of 11

Objective

Transform a quadratic function in vertex form. Describe the domain, range, and intervals where the function is increasing and decreasing.

Common Core Standards

Core Standards

  • F.BF.B.3 — Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
  • F.IF.B.4 — For a function that models a relationship between two quantities, 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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. 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.9 — Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Criteria for Success

  1. Identify the transformations of a quadratic function from the vertex form of the function.
  2. Compare graphs and equations of quadratic functions based on the features and transformations. 
  3. Identify intervals where the function is increasing and decreasing. Describe that a function that is increasing has a slope that is positive, whereas a function that is decreasing has a slope that is negative. 
  4. Identify intervals where the value of the function is positive and where the value of the function is negative.
  5. Describe transformed quadratic functions by describing the domain, range, intercepts, vertex, and movement/scaling from the parent function.
  6. Check solutions to problems using a graphing calculator. 

Tips for Teachers

  • The anchor problems and target task target fluency; however, the problems noted in the problem set guidance provide additional practice on the conceptual thinking required with transformations.
  • The anchor problems should be used to ensure that students have the basics down, and the problem set should be used to look at transformations from multiple perspectives.
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Anchor Problems

25-30 minutes

Problem 1

Below is the parent function of a quadratic function: 

Given the equation in vertex form and the corresponding graph, identify the transformations to each function.

Equation Graph Transformation(s)
$${g(x)=(x+1)^2+3}$$  
$${r(x)=-2(x-1)^2}$$  
$${t(x)=(2x-1)^2}$$  

 

Guiding Questions

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Problem 2

A quadratic equation is 

  • decreasing over the interval $${-\infty < x < 3}$$
  • increasing over the interval $${3 < x < \infty}$$
  • and has a rate of change of zero at $${x=3}$$. 

Describe how each of the three graphs below fit this description.

 

Guiding Questions

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Target Task

5-10 minutes

Problem 1

Given $${f(x)=x^2}$$ shown on the graph below, perform the necessary transformations and graph such that $${ g(x)=-2(x-1)^2+2}$$.

Problem 2

Analyze the function below in terms of: 

  • Interval(s) where the function is increasing
  • Interval(s) where the function is decreasing
  • Zeros 
  • Maximum value
  • Domain
  • Range

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.

  • Include problems where students need to transform the graph according to the equation.
  • Include problems where students need to identify which quadratic function has the greater maximum, given an equation and a graph.
  • Include problems where students are given a graph with no number or tick marks on the $$x$$- and $$y$$-axis and an equation written in vertex form. Ask students, "Is it possible for this equation to describe this graph? Why or why not?"
  • Include problems where students need to generate a function in vertex form that meets constraints such as: "Sketch the quadratic function indicated by the following description. Then write an equation that describes this function. Assume the function is not dilated from the parent function. 
    • Decreasing rate of change over the interval $${{-\infty < x < -1}}$$
    • Increasing function over the interval $${{-\infty < x < -1}}$$
    • Increasing rate of change over the interval $${{-1 < x < \infty}}$$
    • Decreasing function over the interval $${{-1 < x < \infty}}$$
    • The maximum of this function is $$9$$.

Next

Solve quadratic equations written in vertex form and describe graphical features from vertex form.

Lesson 5
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Lesson Map

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

Topic A: Features of Quadratic Functions

Topic B: Imaginary Solutions and Operating with Complex Numbers

Topic C: Applications, Systems, and Inverse with Quadratics

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