Transformations (Shifts, Stretches, Reflections)

Vocabulary flashcards covering shifts, stretches/shrinks, reflections, and key parent functions from the video notes.

Absolute value function

The parent function y = |x|, V-shaped; used as a basis for transforming graphs (shifts, stretches, reflections).

Horizontal shift (translation)

Moving a graph left or right by inside-the-function changes: f(x − h) shifts right by h; f(x + h) shifts left by h.

Vertical shift (translation)

Moving a graph up or down by outside-the-function changes: f(x) + k shifts up by k; f(x) − k shifts down by k.

Horizontal stretch

Widening the graph by a factor b > 1; achieved by replacing x with x/b in the function (y = f(x/b)).

Horizontal compression

Narrowing the graph by a factor b > 1; achieved by replacing x with b x in the function (y = f(bx)).

Vertical stretch

Increasing the height of the graph by a factor a > 1; y = a·f(x).

Vertical compression

Reducing the height of the graph by a factor 0 < a < 1; y = a·f(x).

Reflection across the x-axis

Flipping the graph over the x-axis by multiplying output by −1: y = −f(x).

Reflection across the y-axis

Flipping the graph left-right by replacing x with −x: y = f(−x).

Inside vs outside transformations

Shifts and horizontal changes come from inside the function (affect x); vertical shifts and stretches come from outside (affect y).

Quadratic function (parent)

f(x) = x^2; a parabola opening upward; a base function for quadratic transformations.

Scaled input for even-power functions

For f(x) = x^2, f(a x) = (a x)^2 = a^2 x^2, indicating a vertical stretch by a^2 for this case (and related horizontal effects).

Cubic function (parent)

f(x) = x^3; an odd-symmetric curve used as the base for cubic transformations.

Composite transformation example

An example from the notes: g(x) = (−3x)^3 − 2 shows a horizontal reflection over the y-axis (−), horizontal compression by factor 1/3 (inside the input), and a vertical shift down by 2.