Transformations of Functions: Fresh Take

  • Shift graphs up, down, left, or right to understand how functions move on the coordinate plane
  • Flip graphs across the x-axis or y-axis to see how functions mirror themselves
  • Look at a graph to decide if a function is symmetrical around the y-axis (even), the origin (odd), or not symmetrical at all
  • Apply compressions and stretches to function graphs
  • Use different moves and changes like shifting, flipping, squishing, and stretching on graphs

Identifying Vertical Shifts

The Main Idea

  • Definition:
    • A vertical shift moves a function’s graph up or down without changing its shape
    • Achieved by adding or subtracting a constant to the function
  • Upward Shift:
    • [latex]g(x) = f(x) + c[/latex], where [latex]c > 0[/latex]
    • Moves the graph up by [latex]c[/latex] units
  • Downward Shift:
    • [latex]h(x) = f(x) - c[/latex], where [latex]c > 0[/latex]
    • Moves the graph down by [latex]c[/latex] units
  • Effect on Function Values:
    • Each [latex]y[/latex]-coordinate is increased or decreased by [latex]c[/latex]
    • [latex]x[/latex]-coordinates remain unchanged
  • Effect on Key Points:
    • y-intercept shifts vertically by [latex]c[/latex] units
    • Zeros of the function shift vertically (may change number of zeros)
Given [latex]f(x) = x^2[/latex], describe and graph [latex]g(x) = f(x) - 3[/latex] and [latex]h(x) = f(x) + 1[/latex].

Identifying Horizontal Shifts

The Main Idea

  • Definition:
    • A horizontal shift moves a function’s graph left or right without changing its shape
    • Achieved by adding or subtracting a constant inside the function
  • Rightward Shift:
    • [latex]g(x) = f(x - c)[/latex], where [latex]c > 0[/latex]
    • Moves the graph right by [latex]c[/latex] units
  • Leftward Shift:
    • [latex]h(x) = f(x + c)[/latex], where [latex]c > 0[/latex]
    • Moves the graph left by [latex]c[/latex] units
  • Effect on Function Values:
    • Each [latex]x[/latex]-coordinate is increased or decreased by [latex]c[/latex]
    • [latex]y[/latex]-coordinates remain unchanged
  • Effect on Key Points:
    • [latex]x[/latex]-intercepts shift horizontally by [latex]c[/latex] units
    • [latex]y[/latex]-intercept may change or disappear
Given [latex]f(x) = |x|[/latex], describe and graph [latex]g(x) = f(x - 2)[/latex] and [latex]h(x) = f(x + 1)[/latex].

Given the function [latex]f\left(x\right)=\sqrt{x}[/latex], graph the original function [latex]f\left(x\right)[/latex] and the transformation [latex]g\left(x\right)=f\left(x+2\right)[/latex] on the same axes. Is this a horizontal or a vertical shift? Which way is the graph shifted and by how many units?

Online graphing calculators can graph transformations using function notation. Use an online graphing calculator to graph the toolkit function [latex]f(x) = x^2[/latex]
Now, enter [latex]f(x+5)[/latex], and [latex]f(x)+5[/latex] in the next two lines.Now have the online graphing calculator make a table of values for the original function. Include integer values on the interval [latex][-5,5][/latex]. Replace the column labeled [latex]y_{1}[/latex] with [latex]f(x_{1})[/latex].Now replace [latex]f(x_{1})[/latex] with [latex]f(x_{1}+3)[/latex], and [latex]f(x_{1})+3[/latex].What are the corresponding functions associated with the transformations you have graphed?

Graphing Functions Using Reflections about the Axes

The Main Idea

  • Vertical Reflection:
    • Reflects graph across the [latex]x[/latex]-axis
    • Changes sign of output: [latex]g(x) = -f(x)[/latex]
  • Horizontal Reflection:
    • Reflects graph across the [latex]y[/latex]-axis
    • Changes sign of input: [latex]g(x) = f(-x)[/latex]
  • Effect on Graph:
    • Vertical reflection: Mirror image about [latex]x[/latex]-axis
    • Horizontal reflection: Mirror image about [latex]y[/latex]-axis
  • Domain and Range:
    • Reflections can affect the domain and range of functions
    • Vertical reflection may change the range
    • Horizontal reflection may change the domain
  • Composition of Reflections:
    • Can be combined with other transformations
    • Order of operations matters
Use an online graphing calculator to reflect the graph of [latex]f\left(x\right)=|x - 1|[/latex] (a) vertically and (b) horizontally.

[latex]x[/latex] −2 0 2 4
[latex]f\left(x\right)[/latex] 5 10 15 20

Using the function [latex]f\left(x\right)[/latex] given in the table above, create a table for the functions below.

a. [latex]g\left(x\right)=-f\left(x\right)[/latex]

b. [latex]h\left(x\right)=f\left(-x\right)[/latex]

Graphing Functions Using Stretches and Compressions

The Main Idea

  • Vertical Stretches and Compressions:
    • Affect output values: [latex]g(x) = a \cdot f(x)[/latex]
    • [latex]a > 1[/latex]: Vertical stretch
    • [latex]0 < a < 1[/latex]: Vertical compression
  • Horizontal Stretches and Compressions:
    • Affect input values: [latex]g(x) = f(b \cdot x)[/latex]
    • [latex]0 < b < 1[/latex]: Horizontal stretch by factor [latex]\frac{1}{b}[/latex]
    • [latex]b > 1[/latex]: Horizontal compression by factor [latex]\frac{1}{b}[/latex]
  • Effect on Graph:
    • Vertical: Changes height of graph
    • Horizontal: Changes width of graph
  • Negative Values:
    • [latex]a < 0[/latex]: Combine vertical stretch/compression with reflection over x-axis
    • [latex]b < 0[/latex]: Combine horizontal stretch/compression with reflection over y-axis
  • Domain and Range:
    • Vertical transformations may affect the range
    • Horizontal transformations may affect the domain
A function [latex]f[/latex] is given below. Create a table for the function [latex]g\left(x\right)=\frac{3}{4}f\left(x\right)[/latex].

[latex]x[/latex] 2 4 6 8
[latex]f\left(x\right)[/latex] 12 16 20 0

Write the formula for the function that we get when we vertically stretch (or scale) the identity toolkit function by a factor of 3, and then shift it down by 2 units.
Check your work with an online graphing calculator.

Write a formula for the toolkit square root function horizontally stretched by a factor of 3.
Use an online graphing calculator to check your work.

Performing a Sequence of Transformations

The Main Idea

  • Order of Transformations:
    • For [latex]y = a \cdot f(b(x-c))+d[/latex], the order is: a. Horizontal shift b. Horizontal stretch/compression c. Reflections d. Vertical stretch/compression e. Vertical shift
  • Horizontal Transformations:
    • Shift: [latex]f(x-c)[/latex] shifts right by [latex]c[/latex] units
    • Stretch/Compression: [latex]f(bx)[/latex] stretches by factor [latex]\frac{1}{b}[/latex]
  • Vertical Transformations:
    • Shift: [latex]f(x)+d[/latex] shifts up by [latex]d[/latex] units
    • Stretch/Compression: [latex]a \cdot f(x)[/latex] stretches by factor [latex]a[/latex]
  • Reflections:
    • Horizontal: [latex]f(-x)[/latex] reflects across y-axis
    • Vertical: [latex]-f(x)[/latex] reflects across x-axis
  • Composite Transformations:
    • Apply transformations from inside to outside
    • Pay attention to the order of operations
Given [latex]f\left(x\right)=|x|[/latex], sketch a graph of [latex]h\left(x\right)=f\left(x - 2\right)+4[/latex].
Check your work with an online graphing calculator.

Write a formula for a transformation of the toolkit reciprocal function [latex]f\left(x\right)=\dfrac{1}{x}[/latex] that shifts the function’s graph three units to the left and one unit down.

Given the toolkit function [latex]f\left(x\right)={x}^{2}[/latex], graph [latex]g\left(x\right)=-f\left(x\right)[/latex] and [latex]h\left(x\right)=f\left(-x\right)[/latex]. Take note of any surprising behavior for these functions.

You can view the transcript for “Functions Transformations: A Summary” here (opens in new window).

Determine Whether a Functions is Even, Odd, or Neither

The Main Idea

  • Even Functions:
    • Symmetrical about the y-axis
    • [latex]f(x) = f(-x)[/latex] for all x in the domain
    • Example: [latex]f(x) = x^2[/latex]
  • Odd Functions:
    • Symmetrical about the origin
    • [latex]f(x) = -f(-x)[/latex] for all x in the domain
    • Example: [latex]f(x) = x^3[/latex]
  • Neither Even nor Odd:
    • Functions that don’t satisfy either condition
    • Example: [latex]f(x) = 2^x[/latex]
  • Special Cases:
    • [latex]f(x) = 0[/latex] is both even and odd
    • Polynomial functions with only even powers are even
    • Polynomial functions with only odd powers are odd
  • Graphical Interpretation:
    • Even: Unchanged when reflected over y-axis
    • Odd: Unchanged when rotated 180° about the origin
Is the function [latex]f\left(s\right)={s}^{4}+3{s}^{2}+7[/latex] even, odd, or neither?

You can view the transcript for “Introduction to Odd and Even Functions” here (opens in new window).