Identifying Vertical vs Horizontal Transformations
Vertical transformations modify a function's output (y-values), while horizontal transformations modify its input (x-values). In Grade 11 enVision Algebra 1 (Chapter 10: Working With Functions), students learn that constants placed outside the function — like g(x) = k · f(x) — cause vertical changes, while constants placed inside the function argument — like g(x) = f(k · x) — cause horizontal changes. This inside-versus-outside principle applies to stretches, compressions, and translations, making it the key to classifying any transformation.
Key Concepts
Vertical transformations modify the output: $g(x) = k \cdot f(x)$.
Horizontal transformations modify the input: $g(x) = f(k \cdot x)$.
Common Questions
What is the key to distinguishing vertical from horizontal transformations?
Constants outside the function notation (applied to the output) create vertical transformations; constants inside the function notation (applied to the input) create horizontal transformations.
How does g(x) = 3f(x) transform the graph?
The constant 3 is outside the function, affecting y-values (output). This creates a vertical stretch by factor 3.
How does g(x) = f(3x) transform the graph?
The constant 3 is inside the function, affecting x-values (input). This creates a horizontal compression by factor 1/3 (the graph narrows).
Why does multiplying the input by k > 1 compress the graph horizontally?
Multiplying x by k requires x to reach values k times faster to produce the same outputs, so the graph compresses toward the y-axis.
Does a vertical shift apply inside or outside the function?
Outside: g(x) = f(x) + k. The + k is applied to the output, shifting the graph vertically.
Does a horizontal shift apply inside or outside the function?
Inside: g(x) = f(x − h). The − h is applied to the input, shifting the graph horizontally.