Unit 7

Introduction to Quadratic Functions

Unit Narrative

Prior to this unit, students have studied what it means for a relationship to be a function, used function notation, and investigated linear and exponential functions. In this unit, they look at some patterns that grow quadratically and contrast this growth with linear and exponential growth. They further observe that eventually these quadratic patterns grow more quickly than do linear patterns but more slowly than exponential patterns grow.

Students examine the important example of free-falling objects whose height over time can be modeled with quadratic functions. They use tables, graphs,...

to access Pre-Unit Assessments.

Section A

Section Goals

Section Narrative

In this section, students are introduced to patterns that grow quadratically. They compare these new patterns to linear and exponential patterns from earlier units and recognize that a new type of relationship is needed. They work with rectangular areas that have a given perimeter and visual patterns that grow quadratically with each step and use tables, graphs, and expressions to...

Let’s find the rectangle with the greatest area.

Let’s describe some patterns of change.

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Section A

Let’s find the rectangle with the greatest area.

Let’s describe some patterns of change.

Section B

Section Goals

Section Narrative

In this section, students examine quadratic relationships in the context of functions. They compare quadratic-function growth to that of linear and exponential functions. They notice that quadratic functions with a positive quadratic coefficient grow faster than linear functions and grow slower than exponential functions that have a growth factor greater than 1.

Then, students explore quadratic functions in contexts such...

Let’s describe some other geometric patterns.

Let’s compare quadratic and exponential changes and see which one grows faster.

Let’s measure falling objects.

Let’s look at the objects being launched in the air.

Let’s look at how to maximize revenue.

to access Section Checkpoints.

Section B

Let’s describe some other geometric patterns.

Let’s compare quadratic and exponential changes and see which one grows faster.

Let’s measure falling objects.

Let’s look at the objects being launched in the air.

Let’s look at how to maximize revenue.

Section C

Section Goals

Section Narrative

In this section, students formally recognize that there are multiple useful forms for quadratic expressions and functions. They examine standard and factored forms and use the distributive property to recognize the same relationship in both forms. At first, they use an area model, familiar from earlier grades, and similar diagrams to do the distribution, but are encouraged later to work...

Let’s use diagrams to help us rewrite quadratic expressions.

Let’s write quadratic expressions in different forms.

Let’s find out what quadratic expressions in standard and factored forms can reveal about the properties of their graphs.

to access Section Checkpoints.

to access Mid-Unit Assessments.

Section C

Let’s use diagrams to help us rewrite quadratic expressions.

Let’s write quadratic expressions in different forms.

Let’s find out what quadratic expressions in standard and factored forms can reveal about the properties of their graphs.

Section D

Section Goals

Section Narrative

In this section, students begin by exploring the effect on their graphs of the different parameters in factored and standard forms of quadratic functions. They recognize the importance of the vertex of a parabola and are introduced to the vertex form of a quadratic expression. Then they use the vertex form to examine how modifying a function can lead to...

Let’s graph some quadratic functions in factored form.

Let’s see how the numbers in expressions like \(\text-3x^2+4\) affect their graph.

Let’s change some other parts of a quadratic expression and see how they affect its graph.

Let’s examine graphs that represent the paths of objects being launched into the air.

Let’s find out about the vertex form.

Let’s graph equations in vertex form.

Let’s write new quadratic equations in vertex form to produce certain graphs.

to access Section Checkpoints.

Section D

Let’s graph some quadratic functions in factored form.

Let’s see how the numbers in expressions like \(\text-3x^2+4\) affect their graph.

Let’s change some other parts of a quadratic expression and see how they affect its graph.

Let’s examine graphs that represent the paths of objects being launched into the air.

Let’s find out about the vertex form.

Let’s graph equations in vertex form.

Let’s write new quadratic equations in vertex form to produce certain graphs.

to access End-of-Unit Assessments.

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Introduction to Quadratic Functions

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