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Elementary Functions and Graph Transformations in College Algebra

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Basic Elementary Functions

Definition and Overview

Elementary functions are fundamental building blocks in algebra. Understanding their properties, graphs, and transformations is essential for analyzing more complex functions.

  • Identity Function:

  • Square Function:

  • Cube Function:

  • Square Root Function:

  • Cube Root Function:

  • Absolute Value Function:

Properties: Domain and Range

Function

Equation

Domain

Range

Identity

All real numbers ()

All real numbers ()

Square

All real numbers ()

Cube

All real numbers ()

All real numbers ()

Square Root

Cube Root

All real numbers ()

All real numbers ()

Absolute Value

All real numbers ()

Examples: Evaluating Elementary Functions

  • At :

  • At :

    • is not a real number (since square root of a negative is not real)

  • At :

  • At :

    • is not a real number

Transformations of Functions

Definition and Types

A transformation of a function is an operation that changes its graph in a predictable way. The main types are translations (shifts), stretches/shrinks, and reflections.

Vertical Transformations (Shifts)

The graph of is a vertical shift of .

  • If , the graph shifts upward by units.

  • If , the graph shifts downward by units.

Example: The graph of is the graph of shifted up 4 units.

Horizontal Transformations (Shifts)

The graph of is a horizontal shift of .

  • If , the graph shifts left by units.

  • If , the graph shifts right by units.

Note: The direction of horizontal shifts is opposite to the sign of .

Example: The graph of is the graph of shifted left 4 units.

Examples: Shifts of Square Root Functions

  • is shifted up 5 units.

  • is shifted down 4 units.

  • is shifted left 5 units.

  • is shifted right 4 units.

Vertical and Horizontal Translations (Shifts): Combined

Functions can be shifted both vertically and horizontally. For example, shifts the graph of right by units and up by units.

Example: If , then is the graph of shifted right 2 units and up 3 units.

Stretches, Shrinks, and Reflections

Vertical Stretches and Shrinks

The graph of is a vertical stretch or shrink of .

  • If , the graph is stretched vertically by a factor of .

  • If , the graph is shrunk vertically by a factor of .

Example: is a vertical stretch of by a factor of 2. is a vertical shrink by a factor of 0.5.

Reflections

  • reflects the graph of across the x-axis.

  • reflects the graph of across the y-axis.

Example: is the reflection of across the x-axis.

Summary Table: Effects of Transformations

Transformation

Equation

Effect

Vertical Shift

Up units if , down units if

Horizontal Shift

Left units if , right units if

Vertical Stretch

,

Stretches graph vertically by

Vertical Shrink

,

Shrinks graph vertically by

Reflection (x-axis)

Reflects graph across x-axis

Reflection (y-axis)

Reflects graph across y-axis

Combining Transformations

Multiple transformations can be applied to a function in sequence. The order of operations can affect the final graph.

  • Example: is the graph of reflected across the x-axis, shifted right 3 units, and up 1 unit.

  • Example: is the graph of reflected across the x-axis and shifted left 2 units.

Application: Graphing and Analysis

  • To analyze a transformed function, identify the sequence of shifts, stretches/shrinks, and reflections.

  • Graphing multiple related functions on the same axes helps visualize the effects of each transformation.

Additional info:

  • Understanding these transformations is foundational for graphing polynomial, rational, exponential, and logarithmic functions in College Algebra.

  • Practice by applying transformations to basic functions and predicting the resulting graphs.

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