2.3 Transformations of functions

Functions are like building blocks that can be reshaped and moved around. In this section, we'll learn how to transform these blocks by shifting, stretching, and flipping them. These tricks let us create new functions from old ones, opening up a world of possibilities.

Understanding transformations is key to mastering functions and graphs. We'll explore how simple changes to a function's equation can dramatically alter its graph. This knowledge will help us model real-world situations and solve complex problems more easily.

Transformations of Functions

Vertical and Horizontal Shifts

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• Vertical shifts move the graph of a function up or down by a constant value
  • Represented by adding or subtracting a constant to the output of the function
    • [f(x) + k](https://www.fiveableKeyTerm:f(x)_+_k) or $f(x) - k$, where $k$ is the vertical shift
  • Example: $f(x) = x^2 + 3$ shifts the graph of $y = x^2$ up by 3 units
• Horizontal shifts move the graph of a function left or right by a constant value
  • Represented by adding or subtracting a constant to the input of the function
    • $f(x + h)$ or [f(x - h)](https://www.fiveableKeyTerm:f(x_-_h)), where $h$ is the horizontal shift
  • Example: $f(x) = (x - 2)^2$ shifts the graph of $y = x^2$ to the right by 2 units
• The order of operations for vertical and horizontal shifts is important
  • Horizontal shifts are applied first, followed by vertical shifts
  • Example: $f(x) = (x - 1)^2 + 2$ shifts the graph of $y = x^2$ to the right by 1 unit and then up by 2 units
• Vertical and horizontal shifts can be combined to create more complex transformations of functions

Combining Transformations

• Multiple transformations can be combined to create more complex functions and their graphs
  • The order of operations for combining transformations is: reflections, stretches, horizontal shifts, and then vertical shifts
  • Example: $f(x) = -2(x + 1)^2 - 3$ reflects $y = x^2$ across the x-axis, stretches it vertically by a factor of 2, shifts it to the left by 1 unit, and then shifts it down by 3 units
• When combining transformations, consider how each transformation affects the graph of the function and how they interact with each other
• The equation of a transformed function can be written by applying the transformations to the original function in the correct order
  • Example: A function $f(x)$ with a vertical stretch of 2, a horizontal shift of +3, and a vertical shift of -1 would be written as $2f(x-3) - 1$
• Combining transformations can be used to model real-world situations and create more complex mathematical models

Shifting Graphs

Identifying Shifts from Graphs

• To determine the equation of a transformed function given its graph, identify the original function and then analyze the transformations applied to it
  • Look for vertical and horizontal shifts, reflections, and stretches
  • Example: If the graph of $y = x^3$ is shifted 2 units to the left and 1 unit down, the transformed function would be $f(x) = (x + 2)^3 - 1$
• Sketch the graph of the original function and then apply the transformations step-by-step to visualize the transformed function and verify the equation
• Consider the domain and range of the original function and how the transformations affect them

Identifying Shifts from Descriptions

• When given a description of a transformed function, identify the original function and then apply the transformations in the correct order to write the equation
  • Pay attention to the specific values given for each transformation
  • Example: If $f(x) = \sqrt{x}$ is shifted 3 units to the right and 2 units up, the transformed function would be $f(x) = \sqrt{x - 3} + 2$
• Sketch the graph of the original function and then apply the transformations step-by-step to visualize the transformed function and verify the equation
• Consider the domain and range of the original function and how the transformations affect them

Reflections and Stretches

Reflections

• Reflections flip the graph of a function across an axis
  • A reflection across the x-axis is represented by negating the output of the function
    • $-f(x)$
    • Example: $f(x) = -x^2$ reflects the graph of $y = x^2$ across the x-axis
  • A reflection across the y-axis is represented by negating the input of the function
    • $f(-x)$
    • Example: $f(x) = (-x)^3$ reflects the graph of $y = x^3$ across the y-axis
• Reflections are applied first when combining transformations

Stretches

• Vertical stretches elongate or compress the graph of a function vertically by a constant factor
  • Represented by multiplying the output of the function by a constant
    • $a \cdot f(x)$, where $|a| > 1$ stretches the graph vertically and $0 < |a| < 1$ compresses the graph vertically
    • Example: $f(x) = 3x^2$ stretches the graph of $y = x^2$ vertically by a factor of 3
• Horizontal stretches elongate or compress the graph of a function horizontally by a constant factor
  • Represented by dividing the input of the function by a constant
    • $f(x/b)$, where $|b| > 1$ compresses the graph horizontally and $0 < |b| < 1$ stretches the graph horizontally
    • Example: $f(x) = \sqrt{x/2}$ stretches the graph of $y = \sqrt{x}$ horizontally by a factor of 2
• The order of operations for reflections and stretches is important
  • Reflections are applied first, followed by stretches

Equations of Transformed Functions

Writing Equations of Transformed Functions

• To write the equation of a transformed function, apply the transformations to the original function in the correct order
  • Reflections, stretches, horizontal shifts, and then vertical shifts
  • Example: If $f(x) = |x|$ is reflected across the x-axis, stretched vertically by a factor of 2, shifted 1 unit to the right, and shifted 3 units down, the transformed function would be $f(x) = -2|x - 1| - 3$
• Pay attention to the specific values given for each transformation
• Verify the equation by sketching the graph of the original function and applying the transformations step-by-step

Analyzing Transformed Functions

• Consider the domain and range of the original function and how the transformations affect them
  • Example: If $f(x) = \sqrt{x}$ is shifted 2 units to the left, the domain of the transformed function would be $x \geq -2$
• Analyze the key features of the transformed function, such as intercepts, asymptotes, and intervals of increasing or decreasing
  • Example: If $f(x) = -\frac{1}{x}$ is stretched vertically by a factor of 3 and shifted 1 unit up, the transformed function $f(x) = -\frac{3}{x} + 1$ would have a horizontal asymptote at $y = 1$ and a vertical asymptote at $x = 0$
• Use the equation of the transformed function to solve problems and model real-world situations