Lesson 1 | Quadratics | 11th Grade Mathematics

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

The core standards covered in this lesson

G.GPE.A.2 — Derive the equation of a parabola given a focus and directrix.

Expressing Geometric Properties with Equations

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.

Interpreting Functions

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.

Interpreting Functions

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

The foundational standards covered in this lesson

A.APR.A.1

Arithmetic with Polynomials and Rational Expressions

A.APR.A.1 — Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
F.BF.B.3

Building Functions

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.
G.GPE.B.7

Expressing Geometric Properties with Equations

G.GPE.B.7 — Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. 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.
A.SSE.B.3

Seeing Structure in Expressions

A.SSE.B.3 — Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. 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.

Criteria for Success

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

Identify the roots of a quadratic function from the intercept form.
Identify the y-intercept from the standard form of the quadratic equation.
Identify the maximum or minimum of a quadratic function from the vertex form.
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.
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.
Using a point (focus) and a line (directrix), determine the equation for a parabola.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

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.

Fishtank Plus

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

Anchor Problems

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

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

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

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

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

Problem 3

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

Find the point on the $$y$$-axis equidistant from $$F$$ and $$l$$. Label this point $$P_0$$.
Find the point on the line $$x = 1$$ which is equidistant from $$F$$ and $$l$$. Label this point $$P_1$$.
Find the point on the line $$x = 2$$ which is equidistant from $$F$$ and $$l$$. Label this point $$P_2$$.
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

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

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

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

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…)

Open Middle Identical Quadratics
Illustrative Mathematics Graphs of Quadratic Functions
Open Middle Maximum Value of a Quadratic in Vertex Form

Lesson 2

Lesson Map

Topic A: Features of Quadratic Functions

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

F.IF.C.7.A F.IF.C.8.A G.GPE.A.2

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.

F.IF.C.7.A F.IF.C.8.A

Write a quadratic equation in intercept form by factoring. Describe the features of a quadratic function written in intercept form.

A.SSE.A.2 A.SSE.B.3.A F.IF.C.8.A

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

F.BF.B.3 F.IF.B.4 F.IF.C.9

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

A.REI.A.1 A.REI.B.4.B

Complete the square to convert an equation written in standard form to an equation written in vertex form.

A.REI.B.4.B A.SSE.B.3.B

Topic B: Imaginary Solutions and Operating with Complex Numbers

Identify solutions that are non-real from a graph and an equation using the discriminant. Define imaginary and complex numbers.

A.REI.B.4.B N.CN.A.1 N.CN.C.7

Add, subtract, and multiply complex numbers.

N.CN.A.1 N.CN.A.2

Topic C: Applications, Systems, and Inverse with Quadratics

Compare, analyze, and solve quadratic functions in projectile motion application problems.

F.BF.A.1 F.BF.B.3 F.IF.B.6 F.IF.C.8.A F.IF.C.9

Describe features of quadratic functions in the context of a word problem. Evaluate and use function notation to describe contextual situations.

F.BF.A.1 F.IF.B.6 F.IF.C.8.A F.IF.C.9

Identify solutions to a system of a quadratic function and a linear function graphically and algebraically.

A.REI.A.1 A.REI.C.7 A.REI.D.11

Quadratics

Objective

Common Core Standards

Core Standards

Expressing Geometric Properties with Equations

Interpreting Functions

Interpreting Functions

Foundational Standards

Arithmetic with Polynomials and Rational Expressions

Building Functions

Expressing Geometric Properties with Equations

Seeing Structure in Expressions

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Problem 3

Guiding Questions

References

Target Task

Additional Practice

Lesson Map

Request a Demo

Contact Information

School Information

What courses are you interested in?

ELA

Math

Are you interested in onboarding professional learning for your teachers and instructional leaders?

Any other information you would like to provide about your school?

Effective Instruction Made Easy