Graphs and Transformations of Functions: Notes and FAQs

The graph of a function is the set of all points (x, f(x)) — a picture of how the output changes as the input changes. A transformation takes a known "parent" graph, like y = x², and shifts, stretches or flips it to produce a new graph without plotting dozens of points.

Transformations matter because they turn one memorised shape into an entire family of graphs. Instead of plotting y = −2(x − 3)² + 5 point by point, you start from the parabola y = x² and apply four quick moves: shift right 3, stretch by 2, flip over the x-axis, shift up 5. This is the foundation of curve sketching in calculus.

Key Concepts

• Parent function: the simplest form of a family, e.g. y = x², y = x³, y = √x, y = |x|, y = 1/x. Every other member is a transformed version.
• Translation (shift): adding a constant inside the function moves the graph horizontally; adding outside moves it vertically.
• Stretch/compression: y = a·f(x) stretches vertically by |a|; y = f(bx) compresses horizontally by 1/|b|.

How It Works

To graph y = a·f(b(x − h)) + k, work from the inside out in this order:

1. Start with the parent: sketch y = f(x) — for example the V-shape of y = |x| with its corner at (0, 0).
2. Horizontal shift: (x − h) moves every point h units right (h > 0) or left (h < 0). The corner of y = |x − 2| moves to (2, 0).
3. Horizontal stretch: f(bx) squeezes the graph toward the y-axis by factor 1/|b|. If b is negative, also reflect across the y-axis.
4. Vertical stretch and reflection: multiply all y-values by a. If a is negative, the graph flips over the x-axis too. For a = −2 on the V-shape, the V opens downward and is twice as steep.
5. Vertical shift: add k to every y-value — the whole graph slides up (k > 0) or down (k < 0).

Worked Example

Sketch y = −2|x − 3| + 4 using transformations of y = |x|.

1. Parent: y = |x| is a V with corner at (0, 0), opening upward.
2. Shift right 3: y = |x − 3| has its corner at (3, 0).
3. Vertical stretch by 2 and flip: y = −2|x − 3| opens downward; at x = 4 the value is −2|1| = −2, so the slope is twice as steep.
4. Shift up 4: the corner moves to (3, 4), and the value at x = 4 becomes −2 + 4 = 2.
5. Verify intercepts: set −2|x − 3| + 4 = 0 → |x − 3| = 2 → x = 1 or x = 5. The downward V crosses the x-axis at (1, 0) and (5, 0) with its peak at (3, 4). Correct.

Common Misconceptions

• "f(x − 3) shifts left." It shifts RIGHT. The graph moves so that what used to happen at x now happens at x = 3 — inside changes act in reverse.
• "f(2x) stretches the graph wider." It compresses it horizontally by half. Multiplying x by 2 makes the function reach each value twice as fast.

Real-World Applications

• Physics experiments: shifting a position-time graph accounts for a delayed start; scaling the time axis converts between seconds and hours.
• Signal processing: delaying, amplifying and inverting waveforms are exactly the transformations f(t − d), a·f(t) and −f(t).

Every transformed graph is a parent shape moved and resized: inside changes shift and squeeze horizontally (sign reversed from intuition), outside changes stretch, flip and shift vertically. Master this inside-out order and you can sketch whole curve families without plotting tables of values.