General Function Transformations
General function transformations is a Grade 11 Algebra 1 topic from enVision Chapter 10 that applies universal transformation rules to any function f(x). Vertical stretches use g(x) = af(x): |a| > 1 stretches, 0 < |a| < 1 compresses. Horizontal compressions use g(x) = f(bx): |b| > 1 compresses the graph, 0 < |b| < 1 stretches it. Reflections across the x-axis use g(x) = -f(x). The key distinction: multiplying the output f(x) affects vertical changes; multiplying the input x affects horizontal changes, but in the opposite direction from the coefficient.
Key Concepts
For any function $f(x)$, transformations follow these patterns: Vertical: $g(x) = af(x)$ where $|a| 1$ stretches, $0 < |a| < 1$ compresses Horizontal: $g(x) = f(bx)$ where $|b| 1$ compresses, $0 < |b| < 1$ stretches Reflection: $g(x) = f(x)$ reflects across x axis.
Common Questions
What does g(x) = 3f(x) do to the graph of f(x)?
It stretches the graph vertically by a factor of 3. Every y-value is tripled while x-values remain unchanged.
What does g(x) = f(2x) do to the graph of f(x)?
It compresses the graph horizontally by a factor of 1/2. The graph appears twice as narrow because each x-value must be halved to get the original outputs.
What does g(x) = -f(x) do?
It reflects the graph across the x-axis. Every point (x, y) maps to (x, -y).
What does g(x) = -(1/2)f(x) do?
It both reflects across the x-axis (negative sign) and compresses vertically by 1/2. All y-values are negated and halved.
Why does multiplying inside f(bx) compress horizontally rather than stretch?
To get a given output, the input must be b times smaller. This means the graph fits the same outputs into a narrower x-range, compressing horizontally.
Do these transformation rules apply only to quadratic functions?
No. These rules apply universally to any function type: linear, quadratic, square root, absolute value, exponential, etc.