A.6(A)·Algebra II·Quadratic FunctionsReadiness

Quadratic Domain and Range

Determine reasonable domains and ranges of quadratic functions and interpret their solutions.

Students distinguish the unrestricted domain of a quadratic rule from a reasonable contextual domain, then use the vertex and direction of opening to determine range.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Algebra II key. Algebra II is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutor All Algebra II released practice50 minutes

Learn the concept

Start with the context

A polynomial quadratic is defined for every real input. A model can have a smaller reasonable domain. For a rectangle with perimeter 60 units, $l+w=30$ , so $l=30-w$ . Both $w>0$ and $30-w>0$ must hold.

Use the vertex for range

For $y=a(x-h)^2+k$ , the squared term is nonnegative. With unrestricted inputs, $a>0$ gives $y\ge k$ , while $a<0$ gives $y\le k$ . The vertex output k is included.

Check the endpoints

A rectangle with a zero side length is collapsed, so its width endpoints are excluded. The maximum area can still be included if the vertex input lies inside the reasonable domain. Restricting a domain may also change the range.

Worked example

A rectangle has perimeter 60 meters. Write its area in terms of width w and determine the reasonable domain and range.

From $2l+2w=60$ , obtain $l=30-w$ .
Area is $A(w)=w(30-w)=30w-w^2$ .
Positive dimensions require $w>0$ and $30-w>0$ , so $0<w<30$ .
Rewrite $A(w)=225-(w-15)^2$ . The maximum is 225 at $w=15$ ; every positive area up to 225 occurs.

$A(w)=30w-w^2$ ; domain $0<w<30$ meters; range $0<A\le 225$ square meters.

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Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

Algebra 2 Analyzing Attributes of Graphs of Quadratic Functions

UT IPSI · Kid2Kid / Texas Education Agency

Use the vertex and direction of opening to determine a quadratic function's range.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

1. Find the range of $g(x)=-(x-4)^2+9$ for all real x.
Show answer and reasoning
$y\le 9$
The parabola opens downward and has maximum output 9 at its vertex.
2. A rectangle has perimeter 84 inches. What widths are reasonable?
Show answer and reasoning
$0<w<42$ inches
The length is $42-w$ . Requiring both dimensions to be positive excludes 0 and 42.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Algebra II EOC · 2013 · Item 20 · 1 point

The function $A = 36w - w^2$ can be used to model the area of a rectangle in square inches, $A$ , if the rectangle has a perimeter of 72 inches and a width of $w$ inches. In this situation, which of the following best describes the domain of the function?

$0 < w < 6$
$0 < w < 72$
$0 < w < 18$
$0 < w < 36$

Show STAAR answer and explanation

$0 < w < 36$

Half the perimeter is 36, so the length is $36-w$ . Both dimensions must be positive: $w>0$ and $36-w>0$ , so $0<w<36$ .

$0 < w < 6$ : 6 is the square root of 36. Widths between 6 and 36 still give positive lengths.

$0 < w < 72$ : 72 is the full perimeter. A width over 36 inches would make the length negative.

$0 < w < 18$ : A width of 18 gives the maximum area, but wider rectangles up to 36 inches are still possible.

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Archived released STAAR Algebra II EOC · 2013 · Item 43 · 1 point

What is the range of $f(x) = x^2 + 1$ ?

All real numbers
All real numbers greater than or equal to 0
All real numbers greater than or equal to 1
All real numbers less than or equal to 1

Show STAAR answer and explanation

All real numbers greater than or equal to 1

Since $x^2\ge0$ for every x, $f(x)=x^2+1\ge1$ . The vertex $(0,1)$ is the minimum, and the parabola opens upward.

All real numbers: A parabola that opens upward has a minimum, so it cannot reach every real number.

All real numbers greater than or equal to 0: 0 is the range of $x^2$ ; adding 1 shifts every output up 1.

All real numbers less than or equal to 1: The parabola opens upward, so its outputs are at least 1, not at most 1.

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Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

Determine a reasonable domain from positive physical dimensions.
Identify a quadratic function's minimum or maximum output.
Write domain and range with correct endpoint inclusion.

Instructional sequence

1
Engage
5 min
Ask whether a rectangle can have a negative side length even when an area formula accepts that input.
2
Direct instruction
12 min
Separate algebraic domain from contextual restrictions. Connect range to the vertex and opening.
3
Guided practice
13 min
Write both dimensions of fixed-perimeter rectangles and require each dimension to be positive.
4
Graph connection
13 min
Compare ranges of unrestricted parabolas with ranges after contextual restrictions.
5
Closure
7 min
Students justify open and closed endpoints in the exit ticket.

Misconceptions to address

““Every quadratic has a contextual domain of all real numbers.””

Teacher move: The polynomial accepts every real input, but lengths, times, and counts may impose restrictions.

““The vertex x-coordinate is the greatest possible input.””

Teacher move: The vertex locates an extreme output. Inputs can often continue on both sides of it.

Materials

•Rectangle diagrams
•Quadratic graph cards
•Graphing calculator

Teaching tips

Label domain as inputs and range as outputs before introducing notation.
Write an inequality for every physical dimension.
Ask whether each endpoint represents an actual object or a collapsed one.

Exit ticket · 3–5 min

Find the domain and range of $f(x)=2(x+3)^2-5$ with no contextual restrictions.

Show expected answer

Domain: all real numbers. Range: $y\ge -5$ , because the squared term is nonnegative and the minimum occurs at $x=-3$ .

Sources & alignment

TEA 2013 Algebra II released test TEA 2013 Algebra II answer key and standards Texas Gateway, home of the UT IPSI Kid2Kid lessons

Alignment follows the standard TEA assigned to each 2013 released item. Video playback depends on YouTube access; written instruction remains available here.

Learn the concept

Start with the context

Use the vertex for range

Check the endpoints

Worked example

Watch the lesson

Algebra 2 Analyzing Attributes of Graphs of Quadratic Functions

Try it yourself

1. Find the range of g(x)=−(x−4)2+9g(x)=-(x-4)^2+9g(x)=−(x−4)2+9 for all real x.

2. A rectangle has perimeter 84 inches. What widths are reasonable?

Released STAAR practice

Archived released STAAR Algebra II EOC · 2013 · Item 20 · 1 point

Archived released STAAR Algebra II EOC · 2013 · Item 43 · 1 point

Learning objectives

Instructional sequence

Engage

Direct instruction

Guided practice

Graph connection

Closure

Misconceptions to address

1. Find the range of $g(x)=-(x-4)^2+9$ for all real x.