Parent Graphs & Graphing Adjustments

Vocabulary flashcards covering common parent graphs and the eight standard graphing adjustments used in algebra and precalculus.

Parent Graph y = x (Linear)

A straight line passing through the origin with slope 1; the simplest linear function.

Parent Graph y = x² (Quadratic)

A parabola opening upward with vertex at the origin; displays even symmetry.

Parent Graph y = √x (Square-Root)

Starts at (0,0) and increases slowly to the right; defined only for x ≥ 0.

Parent Graph y = x³ (Cubic)

An S-shaped curve that passes through the origin; odd and symmetric about the origin.

Parent Graph y = |x| (Absolute-Value)

A V-shaped graph opening upward with vertex at the origin; outputs are always non-negative.

Parent Graph x² + y² = r² (Circle)

A circle centered at the origin with radius r; not a function but a common parent relation.

Reflection Across the x-Axis (y = –f(x))

Multiplies all y-values by –1, flipping the graph vertically over the x-axis.

Reflection Across the y-Axis (y = f(–x))

Replaces x with –x, flipping the graph horizontally over the y-axis.

Vertical Shift (y = f(x) + d)

Moves the graph up by d units if d > 0 and down by |d| units if d < 0 without altering its shape.

Horizontal Shift (y = f(x + c))

Moves the graph left by c units if c > 0 and right by |c| units if c < 0.

Vertical Stretch / Compression (y = a·f(x))

Stretches the graph away from the x-axis when a > 1 and compresses it toward the x-axis when 0 < a < 1; a < 0 also produces an x-axis reflection.

Horizontal Stretch / Compression (y = f(bx))

Compresses the graph toward the y-axis when b > 1 and stretches it away when 0 < b < 1; b < 0 also produces a y-axis reflection.

Absolute-Value Output (y = |f(x)|)

All portions of the graph below the x-axis are reflected above it; outputs become non-negative.

Absolute-Value Input (y = f(|x|))

The left half of the graph is replaced by a mirror image of the right half, producing y-axis symmetry.

Function Preservation Under Transformations

Any of the listed adjustments applied to a function f(x) will still produce a valid function.