Transformations of Functions [H]
Transformations of Functions
Any function \( y = f(x) \) can be transformed by modifying the equation in systematic ways. Each type of modification produces a predictable geometric change to the graph — a translation, reflection or stretch. Knowing these rules allows a transformed graph to be sketched from the original without re-plotting every point.
| Transformation | Equation | Effect on graph |
|---|---|---|
| Vertical translation up by \(a\) | \( y = f(x) + a \) | Every point shifts \(a\) units upward |
| Vertical translation down by \(a\) | \( y = f(x) - a \) | Every point shifts \(a\) units downward |
| Horizontal translation right by \(a\) | \( y = f(x - a) \) | Every point shifts \(a\) units to the right |
| Horizontal translation left by \(a\) | \( y = f(x + a) \) | Every point shifts \(a\) units to the left |
| Reflection in the x-axis | \( y = -f(x) \) | Every y-coordinate negated |
| Reflection in the y-axis | \( y = f(-x) \) | Every x-coordinate negated |
Tracking Key Points
The most reliable method is to track what happens to the coordinates of key points. For any point \( (a, b) \) on the original graph \( y = f(x) \):
- \( y = f(x) + k \): the point maps to \( (a,; b+k) \).
- \( y = f(x-k) \): the point maps to \( (a+k,; b) \).
- \( y = -f(x) \): the point maps to \( (a,; -b) \).
- \( y = f(-x) \): the point maps to \( (-a,; b) \).
Worked Examples
The graph of \( y = f(x) \) passes through \( (1, 3) \), \( (0, -1) \) and \( (4, 0) \). Write down the coordinates of the corresponding points on \( y = f(x-2) \) and on \( y = f(x) + 5 \).
\( y = f(x-2) \) shifts right by 2 — add 2 to every x-coordinate:
\( (1,3) o (3,3) \), \( (0,-1) o (2,-1) \), \( (4,0) o (6,0) \).
\( y = f(x)+5 \) shifts up by 5 — add 5 to every y-coordinate:
\( (1,3) o (1,8) \), \( (0,-1) o (0,4) \), \( (4,0) o (4,5) \).
Sketch \( y = -\sin x \) and \( y = \sin(-x) \) for \( 0° \leq x \leq 360° \).
\( y = -\sin x \): reflection in x-axis. The wave is flipped vertically — starts at 0, dips to \( -1 \) at 90°, back to 0 at 180°, up to 1 at 270°, returns to 0 at 360°.
\( y = \sin(-x) \): reflection in y-axis. For the sine function specifically, \( \sin(-x) = -\sin x \), so this produces the same graph as \( y=-\sin x \). (Sine is an odd function.)
\( y = -\cos x \): flip the cosine wave vertically — starts at \( -1 \), rises to 0 at 90°, peaks at 1 at 180°, back to 0 at 270°, returns to \(-1\) at 360°.
The curve \( y = f(x) \) has a minimum at \( (2, -3) \). State the coordinates of the minimum after each transformation: (i) \( y = f(x+1) \), (ii) \( y = f(x) - 4 \), (iii) \( y = -f(x) \).
(i) \( f(x+1) \): left 1 → minimum at \( (1, -3) \).
(ii) \( f(x)-4 \): down 4 → minimum at \( (2, -7) \).
(iii) \( -f(x) \): reflect in x-axis — minimum becomes maximum, y-coordinate negated → \( (2, 3) \) is now a maximum.
Key Takeaways
- \( f(x)+a \): vertical shift up by \(a\). \( f(x)-a \): vertical shift down by \(a\). Changes the y-coordinates.
- \( f(x-a) \): horizontal shift right by \(a\). \( f(x+a) \): horizontal shift left by \(a\). The sign inside the bracket is opposite to the direction of shift.
- \( -f(x) \): reflection in the x-axis — negate all y-coordinates. \( f(-x) \): reflection in the y-axis — negate all x-coordinates.
- Minima become maxima (and vice versa) under reflection in the x-axis. The x-coordinate is unchanged.
- Track key points individually — the most reliable way to sketch a transformed graph is to map the original's notable points through the transformation rules.