Vertical stretching and compressing functions homework answers rely on one central idea: multiplying a function’s output by a constant changes its graph’s height without normally changing its input values. If a function is transformed as g(x) = a·f(x), every original y-value is multiplied by a. This simple rule makes it possible to identify transformations, complete tables, sketch graphs, and write transformed equations confidently.
Introduction to Vertical Stretching and Compressing
A function can be transformed by changing its inputs, its outputs, or both. A vertical transformation changes the outputs, which are the y-coordinates on a graph. The coefficient outside the function determines how far each point moves away from or toward the x-axis.
For the transformed function
g(x) = a f(x),
each point follows this mapping:
(x, y) → (x, ay) Easy to understand, harder to ignore. Surprisingly effective..
The x-coordinate remains unchanged because the input has not been altered. Only the output is multiplied by the constant a.
The Main Transformation Rule
The value of |a| determines the type and size of the vertical transformation:
- If |a| > 1, the graph of f is vertically stretched by a factor of |a|.
- If 0 < |a| < 1, the graph is vertically compressed by a factor of |a|.
- If a < 0, the graph is also reflected across the x-axis.
- If a = 1, the graph remains unchanged.
- If a = 0, every output becomes zero, producing the horizontal line y = 0. This is a constant function rather than an ordinary stretch or compression.
Some textbooks describe g(x) = ½f(x) as a compression by a factor of 2, while others call it a vertical compression with multiplier ½. Both descriptions refer to the same graph: every y-coordinate becomes half as far from the x-axis.
Vertical Stretch Examples
Suppose the parent function is:
f(x) = x²
A vertical stretch by a factor of 3 produces:
g(x) = 3f(x) = 3x².
Compare several corresponding points:
| Point on f(x) = x² | Point on g(x) = 3x² |
|---|
