Function Transformations to Know for Algebra and Trigonometry

Function transformations change the position and shape of graphs in various ways. Understanding vertical and horizontal shifts, stretches, compressions, and reflections helps in graphing functions accurately, connecting concepts across Algebra and Trigonometry, College Algebra, and Honors Algebra II.

1. Vertical shifts (f(x) + k)

   • Adding a constant ( k ) to the function shifts the graph vertically.
   • If ( k > 0 ), the graph moves up; if ( k < 0 ), it moves down.
   • The shape of the graph remains unchanged; only the position changes.

2. Horizontal shifts (f(x + h))

   • Adding a constant ( h ) inside the function shifts the graph horizontally.
   • If ( h > 0 ), the graph moves left; if ( h < 0 ), it moves right.
   • The shape of the graph remains unchanged; only the position changes.

3. Vertical stretches and compressions (af(x))

   • Multiplying the function by a constant ( a ) stretches or compresses the graph vertically.
   • If ( |a| > 1 ), the graph stretches; if ( 0 < |a| < 1 ), it compresses.
   • The x-coordinates remain the same, but the y-coordinates are scaled.

4. Horizontal stretches and compressions (f(bx))

   • Multiplying the input ( x ) by a constant ( b ) stretches or compresses the graph horizontally.
   • If ( |b| > 1 ), the graph compresses; if ( 0 < |b| < 1 ), it stretches.
   • The y-coordinates remain the same, but the x-coordinates are scaled.

5. Reflections over the x-axis (-f(x))

   • Negating the function reflects the graph over the x-axis.
   • Points that were above the x-axis move below it and vice versa.
   • The shape of the graph is preserved, but the orientation is inverted.

6. Reflections over the y-axis (f(-x))

   • Negating the input ( x ) reflects the graph over the y-axis.
   • Points on the right side of the y-axis move to the left side and vice versa.
   • The shape of the graph is preserved, but the orientation is inverted.

7. Absolute value transformations (|f(x)|)

   • Taking the absolute value of the function reflects any negative values above the x-axis.
   • The graph remains unchanged for positive values; negative values become positive.
   • This transformation can create a "V" shape in regions where the original function was below the x-axis.

8. Composite transformations (combining multiple transformations)

   • Multiple transformations can be applied in sequence to a function.
   • The order of transformations matters; for example, a vertical shift followed by a reflection will yield different results than the reverse.
   • Understanding how to combine transformations helps in accurately graphing complex functions.

9. Parent functions and their transformations

   • Parent functions are the simplest forms of functions (e.g., linear, quadratic, cubic).
   • Transformations can be applied to parent functions to create new functions.
   • Recognizing the parent function helps predict the effects of transformations on the graph.

10. Graphing transformed functions

    • Start with the parent function and apply transformations step-by-step.
    • Use key points from the parent function to determine the new positions after transformations.
    • Always check the effects of vertical and horizontal shifts, stretches, and reflections on the overall shape and position of the graph.