Curriculum  /  Math  /  11th Grade  /  Unit 2: Quadratics  /  Lesson 1

Quadratics

Lesson 1

Math

Unit 2

11th Grade

Lesson 1 of 11

Objective

Identify features of quadratic functions from equations and use these features to graph quadratic functions. Derive the equation of a parabola.

Common Core Standards

Core Standards

  • G.GPE.A.2 — Derive the equation of a parabola given a focus and directrix.
  • F.IF.C.7.A — Graph linear and quadratic functions and show intercepts, maxima, and minima.
  • F.IF.C.8.A — Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

Foundational Standards

  • A.APR.A.1
  • F.BF.B.3
  • G.GPE.B.7
  • A.SSE.B.3

Criteria for Success

  1. Identify the roots of a quadratic function from the intercept form.
  2. Identify the y-intercept from the standard form of the quadratic equation. 
  3. Identify the maximum or minimum of a quadratic function from the vertex form. 
  4. Describe, from the sign of the leading coefficient, how you know the function has a maximum or minimum without graphing from each of the forms of intercept, vertex, and standard form. 
  5. Graph a quadratic function by using the features of quadratic equations and by strategically choosing and plugging in domain values to define points that are solutions to the quadratic equation.
  6. Using a point (focus) and a line (directrix), determine the equation for a parabola.

Tips for Teachers

  • This lesson is the introduction to multiple equation forms for quadratic equations. In the subsequent lessons, students will work with each of these forms in greater depth. 
  • Anchor Problem 3 covers the derivation of the equation of a parabola. Depending on time, this lesson may be split over 2 days, covering Anchor Problem 3 on day 2.
  • Factoring and completing the square are very important skills in this unit. The teacher should consider giving some sort of pre-assessment to determine students’ fluency with factoring and completing the square. Factoring will be covered in greater depth in Lesson 3, and completing the square will be covered in greater depth in Lesson 6. 
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Anchor Problems

25-30 minutes

Problem 1

Below are three forms of the same quadratic function. 

$${f(x)=-4(x-3)^2 +4}$$

$${f(x)=-4(x-2)(x-4)}$$

$${f(x)=-4x^2 +24x -32}$$

Without graphing, describe the features of the quadratic function. 

Guiding Questions

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Problem 2

Given the points below that represent the intercepts, vertex, and y-intercept, write the equation of the quadratic function in: 

  • Standard form
  • Vertex form
  • Intercept form

$${(-5,1), (-4,0), (0,-24), (-6,0)}$$

Guiding Questions

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Problem 3

Suppose $$F = (0, 2)$$ and $$l$$ is the the $$x$$-axis.

  1. Find the point on the $$y$$-axis equidistant from $$F$$ and $$l$$. Label this point $$P_0$$.
  2. Find the point on the line $$x = 1$$ which is equidistant from $$F$$ and $$l$$. Label this point $$P_1$$.
  3. Find the point on the line $$x = 2$$ which is equidistant from $$F$$ and $$l$$. Label this point $$P_2$$. 
  4. If this process is repeated for the vertical lines $$x = a$$ for all real numbers $$a$$, what curve do the points $$P_a$$ make?

Guiding Questions

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References

Illustrative Mathematics Defining Parabolas Geometrically

Defining Parabolas Geometrically, accessed on Feb. 12, 2021, 5:06 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Target Task

5-10 minutes

Below is a quadratic equation with two missing values and a sketch of the graph of $${f(x)}$$.

$${f(x)}=r(x+2)^2 + m$$

  • What do you know about the value of $$r$$ and $$m$$ based on the graph shown? 

  • What do you know about the intercept form of the quadratic function? 

  • Why can’t you write a unique quadratic equation based on this information? 

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 that require error analysis of a mismatched graph and equation. 
  • Include problems where a verbal description of the features of the quadratic function is given, and students need to write the quadratic equation in the most appropriate form. 
  • Include problems where an equation is given and one value in the equation is changed. Answer: what effect does this change have on the graph? 
  • Include problems where the equation of the parabola is given but not in a recognizable form. Ask students to graph the quadratic equation by plugging in values, then write the equation in the forms that reveal features of the function. 
  • Include problems that require students to match the graph with the most likely quadratic equation and describe reasoning as well as additional information necessary to confirm the match. 
  • Include problems where students need to write multiple equations in one of the three forms that meet particular requirements. (For example, write three quadratic equations in intercept form that have roots of…; write three quadratic equations in intercept form that have a y-intercept of…)

Next

Identify the y-intercept and vertex of a quadratic function written in standard form through inspection and finding the axis of symmetry. Graph quadratic equations on the graphing calculator.

Lesson 2
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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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