Learn on PengienVision, Algebra 1Chapter 5: Piecewise Functions

Lesson 3: Step Functions

In this Grade 11 Algebra 1 lesson from enVision Chapter 5, students learn to graph and apply step functions, including the ceiling function and floor function, which round inputs up or down to the nearest integer. Students explore how step functions are a type of piecewise-defined function with constant pieces, using interval notation such as ⌈x⌉ and ⌊x⌋. Real-world applications include modeling field trip transportation and rental costs to reinforce how step functions represent situations where output values change in discrete jumps.

Section 1

Step Functions as Piecewise Constant Functions

Property

A step function is a piecewise-defined function where each piece has a constant value over an interval. The general form is:

f(x) = \begin{cases} c_1 & \text{if } x \in I_1 \\ c_2 & \text{if } x \in I_2 \\ \vdots & \vdots \\ c_n & \text{if } x \in I_n \end{cases}

where $c_1, c_2, \ldots, c_n$ are constants and $I_1, I_2, \ldots, I_n$ are intervals.

Examples

Section 2

Rounding Numbers

Property

How to round a number to a specific place value:

Locate the given place value.
Look at the digit immediately to the right of the given place value.
Determine if this digit is greater than or equal to 5.
- Yes: add 1 to the digit in the given place value. Handle any regrouping if necessary.
- No: keep the digit in the given place value unchanged.
For whole numbers: replace all digits to the right with zeros. For decimals: drop all digits to the right.

Section 3

Ceiling Function Definition and Notation

Property

The ceiling function $f(x) = \lceil x \rceil$ rounds any real number $x$ up to the nearest integer.
If $x$ is already an integer, then $\lceil x \rceil = x$ .

Examples

Section 4

Floor Function Definition and Notation

Property

The floor function $f(x) = \lfloor x \rfloor$ rounds any real number down to the nearest integer.
The floor function always returns the greatest integer that is less than or equal to the input value.

Examples

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Chapter 5: Piecewise Functions

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Lesson 1
Lesson 1: The Absolute Value Function
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Lesson 2
Lesson 2: Piecewise-Defined Functions
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Lesson 3Current
Lesson 3: Step Functions
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Lesson 4
Lesson 4: Transformations of Piecewise-Defined Functions
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Section 1

Step Functions as Piecewise Constant Functions

Property

A step function is a piecewise-defined function where each piece has a constant value over an interval. The general form is:

f(x) = \begin{cases} c_1 & \text{if } x \in I_1 \\ c_2 & \text{if } x \in I_2 \\ \vdots & \vdots \\ c_n & \text{if } x \in I_n \end{cases}

where $c_1, c_2, \ldots, c_n$ are constants and $I_1, I_2, \ldots, I_n$ are intervals.

Examples

Section 2

Rounding Numbers

Property

How to round a number to a specific place value:

Locate the given place value.
Look at the digit immediately to the right of the given place value.
Determine if this digit is greater than or equal to 5.
- Yes: add 1 to the digit in the given place value. Handle any regrouping if necessary.
- No: keep the digit in the given place value unchanged.
For whole numbers: replace all digits to the right with zeros. For decimals: drop all digits to the right.

Section 3

Ceiling Function Definition and Notation

Property

The ceiling function $f(x) = \lceil x \rceil$ rounds any real number $x$ up to the nearest integer.
If $x$ is already an integer, then $\lceil x \rceil = x$ .

Examples

Section 4

Floor Function Definition and Notation

Property

The floor function $f(x) = \lfloor x \rfloor$ rounds any real number down to the nearest integer.
The floor function always returns the greatest integer that is less than or equal to the input value.

Examples

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 5: Piecewise Functions

Back to enVision, Algebra 1

Lesson 1
Lesson 1: The Absolute Value Function
Learn Practice
Lesson 2
Lesson 2: Piecewise-Defined Functions
Learn Practice
Lesson 3Current
Lesson 3: Step Functions
Learn Practice
Lesson 4
Lesson 4: Transformations of Piecewise-Defined Functions
Learn Practice

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Step Functions as Piecewise Constant Functions

Property

A step function is a piecewise-defined function where each piece has a constant value over an interval. The general form is:

f(x) = \begin{cases} c_1 & \text{if } x \in I_1 \\ c_2 & \text{if } x \in I_2 \\ \vdots & \vdots \\ c_n & \text{if } x \in I_n \end{cases}

where $c_1, c_2, \ldots, c_n$ are constants and $I_1, I_2, \ldots, I_n$ are intervals.

Examples

Section 2

Rounding Numbers

Property

How to round a number to a specific place value:

Locate the given place value.
Look at the digit immediately to the right of the given place value.
Determine if this digit is greater than or equal to 5.
- Yes: add 1 to the digit in the given place value. Handle any regrouping if necessary.
- No: keep the digit in the given place value unchanged.
For whole numbers: replace all digits to the right with zeros. For decimals: drop all digits to the right.

Section 3

Ceiling Function Definition and Notation

Property

The ceiling function $f(x) = \lceil x \rceil$ rounds any real number $x$ up to the nearest integer.
If $x$ is already an integer, then $\lceil x \rceil = x$ .

Examples

Section 4

Floor Function Definition and Notation

Property

The floor function $f(x) = \lfloor x \rfloor$ rounds any real number down to the nearest integer.
The floor function always returns the greatest integer that is less than or equal to the input value.

Examples

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 5: Piecewise Functions

Back to enVision, Algebra 1

Lesson 1
Lesson 1: The Absolute Value Function
Learn Practice
Lesson 2
Lesson 2: Piecewise-Defined Functions
Learn Practice
Lesson 3Current
Lesson 3: Step Functions
Learn Practice
Lesson 4
Lesson 4: Transformations of Piecewise-Defined Functions
Learn Practice

Lesson 3: Step Functions

Property

Examples

Property

Property

Examples

Property

Examples

Chapter 5: Piecewise Functions

Lesson 1: The Absolute Value Function

Lesson 2: Piecewise-Defined Functions

Lesson 3: Step Functions

Lesson 4: Transformations of Piecewise-Defined Functions

Lesson overview

Property

Examples

Property

Property

Examples

Property

Examples

Chapter 5: Piecewise Functions

Lesson 1: The Absolute Value Function

Lesson 2: Piecewise-Defined Functions

Lesson 3: Step Functions

Lesson 4: Transformations of Piecewise-Defined Functions

Lesson 1: The Absolute Value Function

Lesson 2: Piecewise-Defined Functions

Lesson 3: Step Functions

Lesson 4: Transformations of Piecewise-Defined Functions