1.56 Quadratic Graph Definitions

Quadratic Graph Definitions establishes the vocabulary for the ways a parabola's position, orientation, and width can be altered relative to a basic reference parabola, describing the overall category of such alterations and the three specific types—horizontal repositioning, vertical repositioning, and vertical stretching or compressing—that account for every such alteration.

Quadratic Graph Transformation

A quadratic graph transformation is any systematic change applied to the basic parabola y = x² that repositions, stretches, compresses, or reflects the resulting graph while preserving its overall parabolic shape. Every quadratic function's graph can be understood as the basic parabola having undergone some specific combination of quadratic graph transformations, making the basic parabola a useful reference point for describing any other parabola's appearance.

Quadratic Horizontal Shift

A quadratic horizontal shift is a transformation that moves the entire parabola left or right along the horizontal axis without altering its shape, produced by replacing x with (x − h) inside the quadratic expression, where a positive h shifts the graph to the right and a negative h shifts it to the left. The transformed function y = (x − 3)² produces a parabola identical in shape to y = x², but shifted 3 units to the right, with its vertex relocated accordingly.

Quadratic Vertical Shift

A quadratic vertical shift is a transformation that moves the entire parabola up or down along the vertical axis without altering its shape, produced by adding a constant k to the quadratic expression, where a positive k shifts the graph upward and a negative k shifts it downward. The transformed function y = x² + 4 produces a parabola identical in shape to y = x², but shifted 4 units upward, with its vertex relocated accordingly.

Quadratic Vertical Scale

A quadratic vertical scale is a transformation that stretches or compresses the parabola vertically, and may also reflect it, produced by multiplying the quadratic expression by a constant a, where a value of a greater than 1 in absolute value stretches the parabola into a narrower shape, a value between 0 and 1 in absolute value compresses it into a wider shape, and a negative value of a reflects the parabola so that it opens in the opposite direction.

Together, these definitions describe every possible transformation of a parabola as some combination of a quadratic horizontal shift, a quadratic vertical shift, and a quadratic vertical scale, each altering the basic parabola y = x² in a specific, predictable way that can be read directly from the constants h, k, and a appearing in the transformed equation.