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3.5 Transformation of Functions

3 min read•june 24, 2024

Function transformations are powerful tools for manipulating graphs. They let you shift, reflect, stretch, and compress functions without changing their core shape. These techniques are crucial for understanding how functions behave and relate to each other.

By mastering transformations, you can quickly sketch complex functions and solve equations. This skill is essential for analyzing real-world data and modeling various phenomena in fields like physics, economics, and engineering. It's a fundamental building block for advanced math.

Function Transformations

Vertical and horizontal shifts

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  • Vertical shifts move the graph of a function up or down along the without changing its shape
    • Add a positive constant kk to the function [f(x)](https://www.fiveableKeyTerm:f(x))[f(x)](https://www.fiveableKeyTerm:f(x)) to shift the graph up by kk units ([f(x) + k](https://www.fiveableKeyTerm:f(x)_+_k))
    • Subtract a positive constant kk from the function f(x)f(x) to shift the graph down by kk units (f(x)−kf(x) - k)
  • Horizontal shifts move the graph of a function left or right along the without changing its shape
    • Replace xx with x−hx - h in the function f(x)f(x) to shift the graph right by hh units ([f(x - h)](https://www.fiveableKeyTerm:f(x_-_h)))
    • Replace xx with x+hx + h in the function f(x)f(x) to shift the graph left by hh units ([f(x + h)](https://www.fiveableKeyTerm:f(x_+_h)))
  • These shifts can affect the and of the function

Reflections across axes

  • Reflecting a function across the x-axis produces a mirror image of the graph above or below the x-axis
    • Multiply the function f(x)f(x) by -1 to reflect the graph across the x-axis ([−f(x)](https://www.fiveableKeyTerm:−f(x))[-f(x)](https://www.fiveableKeyTerm:-f(x)))
  • Reflecting a function across the y-axis produces a mirror image of the graph on the opposite side of the y-axis
    • Replace xx with −x-x in the function f(x)f(x) to reflect the graph across the y-axis ([f(−x)](https://www.fiveableKeyTerm:f(−x))[f(-x)](https://www.fiveableKeyTerm:f(-x)))

Even and odd functions

  • Even functions have graphs that are symmetric about the y-axis
    • A function f(x)f(x) is even if f(−x)=f(x)f(-x) = f(x) for all xx in the
    • Examples of even functions: f(x)=x2f(x) = x^2, f(x)=cos⁡(x)f(x) = \cos(x)
  • Odd functions have graphs that are symmetric about the
    • A function f(x)f(x) is odd if f(−x)=−f(x)f(-x) = -f(x) for all xx in the domain
    • Examples of odd functions: f(x)=x3f(x) = x^3, f(x)=sin⁡(x)f(x) = \sin(x)
  • Functions that are neither even nor odd do not have about the y-axis or
    • Example of a neither even nor : f(x)=2xf(x) = 2^x

Compressions and stretches

  • Vertical compressions and stretches change the height of the graph without changing its width
    • Multiply the function f(x)f(x) by a constant 0<∣a∣<10 < |a| < 1 to vertically compress the graph by a factor of 1∣a∣\frac{1}{|a|} ([af(x)](https://www.fiveableKeyTerm:af(x))[af(x)](https://www.fiveableKeyTerm:af(x)))
    • Multiply the function f(x)f(x) by a constant ∣a∣>1|a| > 1 to vertically stretch the graph by a factor of ∣a∣|a| (af(x)af(x))
  • Horizontal compressions and stretches change the width of the graph without changing its height
    • Replace xx with xb\frac{x}{b} in the function f(x)f(x) where ∣b∣>1|b| > 1 to horizontally compress the graph by a factor of 1∣b∣\frac{1}{|b|} (f(xb)f(\frac{x}{b}))
    • Replace xx with xb\frac{x}{b} in the function f(x)f(x) where 0<∣b∣<10 < |b| < 1 to horizontally stretch the graph by a factor of 1∣b∣\frac{1}{|b|} (f(xb)f(\frac{x}{b}))

Combining multiple transformations

  • When applying multiple transformations to a function, follow this order:
  1. Compressions and stretches
  2. Reflections
  3. Horizontal shifts
  4. Vertical shifts
  • Combine transformations by applying them to the function in the correct order
    • Example: −2f(3x−1)+4-2f(3x - 1) + 4 represents a function with these transformations:
      1. by a factor of 13\frac{1}{3} (3x3x)
      2. right by 13\frac{1}{3} units (x−1x - 1)
      3. by a factor of 2 (−2f-2f)
      4. across the x-axis (−2f-2f)
      5. up by 4 units (+4+ 4)

Function Composition and Inverse Functions

  • involves applying one function to the output of another function
  • The reverses the effect of a function, effectively "undoing" its operation
  • Composition and inverse functions are closely related to transformations, as they can change the domain and range of the original function

Key Terms to Review (41)

-f(x): -f(x) represents the reflection of the function f(x) across the x-axis. When you apply this transformation to a function, every point on the graph of f(x) is mirrored over the x-axis, resulting in the output values changing their signs. This transformation is significant as it directly affects the behavior and shape of the graph, allowing for a deeper understanding of how functions can be manipulated and visualized through algebraic expressions and graphical representations.
Af(x): The term 'af(x)' refers to the transformation of a function f(x) by a constant factor a. This transformation scales the function vertically, either expanding or contracting the graph of the function depending on the value of a.
Axes of symmetry: Axes of symmetry are lines that divide a figure into two mirror-image halves. In hyperbolas, these axes typically refer to the transverse and conjugate axes.
Domain: The domain of a function is the complete set of possible input values (x-values) that allow the function to work within its constraints. It specifies the range of x-values for which the function is defined.
Domain: The domain of a function refers to the set of input values for which the function is defined. It represents the range of values that the independent variable can take on, and it is the set of all possible values that can be plugged into the function to produce a meaningful output.
Even function: An even function is a function $f(x)$ where $f(x) = f(-x)$ for all $x$ in its domain. This symmetry means the graph of an even function is mirrored across the y-axis.
Even Function: An even function is a mathematical function where the output value is the same regardless of whether the input value is positive or negative. In other words, the function satisfies the equation f(x) = f(-x) for all values of x in the function's domain.
F(-x): The function f(-x) is a transformation of the original function f(x) where the input variable x is negated, or multiplied by -1. This transformation reflects the function across the y-axis, resulting in a mirrored or inverted version of the original function.
F(x - h): The expression f(x - h) represents a horizontal shift of the function f(x). The function is shifted h units to the right if h is positive, or h units to the left if h is negative. This transformation is known as a horizontal shift and is an important concept in the study of function transformations.
F(x + h): The expression f(x + h) represents a transformation of the original function f(x) by shifting the input variable x by a constant value h. This transformation is known as a horizontal shift, as it moves the function left or right on the coordinate plane without changing its shape or orientation.
F(x): f(x) is a function notation that represents a relationship between an independent variable, x, and a dependent variable, f. It is a fundamental concept in mathematics that underpins the study of functions, their properties, and their applications across various mathematical topics.
F(x) + k: The expression 'f(x) + k' represents a vertical shift or translation of a function f(x) by a constant value k. This transformation affects the graph of the function by moving it up or down on the y-axis, without changing the shape or orientation of the original function.
F(x/b): The term f(x/b) represents a transformation of a function f(x) where the input variable x is divided by a constant b. This transformation is known as a horizontal scaling or stretching of the original function.
Function Composition: Function composition is the process of combining two or more functions to create a new function. The output of one function becomes the input of the next function, allowing for the creation of more complex mathematical relationships.
Horizontal compression: Horizontal compression transforms a function by reducing its width. It is achieved by multiplying the input variable by a factor greater than 1.
Horizontal Compression: Horizontal compression is a transformation of a function that causes the graph of the function to be compressed or squeezed along the horizontal (x) axis, effectively reducing the width or period of the function. This transformation affects the independent variable (x) of the function.
Horizontal reflection: A horizontal reflection is a transformation that flips a function's graph over the y-axis. It changes the sign of the x-coordinates of all points on the graph.
Horizontal Shift: A horizontal shift is a transformation of a function that moves the graph of the function left or right along the x-axis, without changing the shape or orientation of the graph. This concept is applicable to various types of functions, including transformations of functions, absolute value functions, exponential functions, trigonometric functions, and the parabola.
Horizontal stretch: A horizontal stretch is a transformation that scales a function's graph horizontally by multiplying the input values by a constant factor. If $0 < k < 1$, the graph stretches away from the y-axis.
Horizontal Stretch: A horizontal stretch is a transformation of a function that alters the x-coordinate of the graph, causing it to appear wider or narrower. This transformation affects the domain of the function, rather than the range.
Inverse function: An inverse function reverses the operation of a given function. If $f(x)$ is a function, its inverse $f^{-1}(x)$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.
Inverse Function: An inverse function is a function that reverses the operation of another function. It undoes the original function, mapping the output back to the original input. Inverse functions are crucial in understanding the relationships between different mathematical concepts, such as domain and range, composition of functions, transformations, and exponential and logarithmic functions.
Magnitude: Magnitude refers to the size or quantity of a mathematical object, often represented as the absolute value of a number. It gives the distance of a number from zero on the number line.
Odd function: An odd function is a function $f(x)$ that satisfies the condition $f(-x) = -f(x)$ for all $x$ in its domain. Graphically, odd functions exhibit symmetry about the origin.
Odd Function: An odd function is a function that satisfies the property $f(-x) = -f(x)$ for all $x$ in the domain of the function. This means that the graph of an odd function is symmetric about the origin, with the graph being a reflection across both the $x$-axis and the $y$-axis.
Origin: In the rectangular coordinate system, the origin is the point where the x-axis and y-axis intersect. It is denoted by the coordinates (0, 0).
Origin: The origin is a specific point in a coordinate system that serves as the reference point for all other points. It is the intersection of the x-axis and y-axis, and is typically denoted as the point (0, 0). The origin is a fundamental concept in various mathematical and scientific contexts, as it provides a common starting point for measurement and analysis.
Parent function: A parent function is the simplest form of a function in a family of functions that preserves the shape and general characteristics of the entire family. For logarithmic functions, the parent function is $f(x) = \log_b(x)$ where $b$ is the base of the logarithm.
Parent Function: The parent function is the basic or original form of a function, from which other related functions are derived through transformations. It serves as the foundation for understanding how various functions can be modified and manipulated to create new functions with different characteristics.
Range: In mathematics, the range refers to the set of all possible output values (dependent variable values) that a function can produce based on its input values (independent variable values). Understanding the range helps in analyzing how a function behaves and what values it can take, connecting it to various concepts like transformations, compositions, and types of functions.
Reflection: Reflection is a transformation of a function that creates a mirror image of the original function across a specified axis. This concept is fundamental in understanding the behavior and properties of various mathematical functions.
Symmetry: Symmetry is the quality of being made up of exactly similar parts facing each other or around an axis. It is a fundamental concept in mathematics and geometry that describes the balanced and harmonious arrangement of elements in an object or function.
Transformation: Transformation refers to any operation that moves or changes a function in some way. Common transformations include translations, dilations, reflections, and rotations.
Vertical compression: A vertical compression is a transformation that scales a function's graph towards the x-axis. This is achieved by multiplying the function by a constant factor between 0 and 1.
Vertical Compression: Vertical compression is a transformation of a function that scales the function vertically, effectively shrinking or stretching the function along the y-axis. This transformation can impact the amplitude, range, and behavior of the function.
Vertical shift: A vertical shift is a transformation that moves a graph up or down in the coordinate plane by adding or subtracting a constant to the function's output. It does not affect the shape of the graph, only its position.
Vertical Shift: Vertical shift refers to the movement of a graph or function up or down the y-axis, without affecting the shape or orientation of the graph. This transformation changes the y-intercept of the function, but leaves the x-intercepts and the overall shape unchanged.
Vertical stretch: A vertical stretch is a transformation that scales a function's graph away from the x-axis by multiplying all y-values by a factor greater than 1. It does not affect the x-values of the function.
Vertical Stretch: Vertical stretch is a transformation of a function that involves scaling the function vertically, either by expanding or compressing the function along the y-axis. This transformation affects the amplitude or the range of the function, without changing its basic shape or period.
X-Axis: The x-axis is the horizontal axis on a coordinate plane, typically running left to right. It is used to represent the independent variable in a graph and helps visualize the relationship between two or more variables.
Y-axis: The y-axis is the vertical axis in a rectangular coordinate system, which represents the dependent variable and is typically used to plot the values or outcomes of a function. It is perpendicular to the x-axis and provides a visual reference for the range of values a function can take on.
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Glossary
Glossary
3.6 Absolute Value Functions