12.1 Transformations of Functions

12.1 Transformations of Functions

Quick Answer: On the NBT MAT paper, a transformed graph is the parent y = f(x) after translations, reflections, and stretches. Horizontal moves live inside the function and run opposite the sign you see: y = f(x − p) shifts p units right. Vertical moves sit outside: +q shifts up. Inverses swap coordinates (reflect in the line y = x). Reconstruct graphs from equations, word descriptions, and tables — there is no calculator, and this chapter uses no printed pictures.

Why transformations sit in the MAT functions block

The official MAT booklet asks you to interpret transformations of functions represented algebraically or graphically, including inverses. The parent families are the ones already taught: linear, quadratic, hyperbolic, exponential, and logarithmic. MAT is a calculator-free multiple-choice paper, written in the afternoon by candidates who also sit AQL that morning when a programme requires both. Independent OpenExamPrep teaching for this chapter uses original tables and verbal sketches so you practise rebuilding a graph the way a stem describes it, not by copying confidential NBT items.

NSC papers often scaffold: sketch first, then a later part that reuses the sketch. MAT does not. One item may give a table and four equations; another may name a reflection and a stretch in a single sentence. You have to name every move yourself.

Translation, reflection, stretch

Keep one parent y = f(x) in mind and apply one change at a time.

Vertical translation. y = f(x) + q moves every point (x, y) to (x, y + q). Positive q is up. A horizontal asymptote y = k becomes y = k + q. x-intercepts usually change.

Horizontal translation. y = f(x − p) moves points to (x + p, y): right if p > 0. y = f(x + p) moves left. The input that used to be 0 is now the input that makes the inside zero. For y = f(x − 4) that input is x = 4. This opposite-sign rule is the most common transformation trap on a timed multiple-choice item.

Reflections. y = −f(x) reflects in the x-axis. y = f(−x) reflects in the y-axis. Both together give y = −f(−x), a 180° rotation about the origin. Swapping the two axes is a designed distractor.

Stretches. y = a f(x) multiplies heights by a. If |a| > 1 the graph is taller; if 0 < |a| < 1 it is flatter. Negative a includes an x-axis flip. y = f(bx) scales horizontally: |b| > 1 compresses toward the y-axis. Trigonometric period in the next section uses this same b.

The quadratic you already know is a transformation

y = a(x − h)² + k is y = x² shifted h right, k up, and stretched by a. If a < 0 it also reflects in the x-axis and opens downward.

Worked example. Move y = x² two units right, three units down, then reflect in the x-axis.

1. Right 2: y = (x − 2)²
2. Down 3: y = (x − 2)² − 3
3. Reflect in the x-axis: y = −[(x − 2)² − 3] = −(x − 2)² + 3

The vertex (2, −3) after the translations becomes (2, 3). The equation y = −(x − 2)² − 3 only puts a minus on the squared bracket and leaves the vertex below the axis — a typical wrong option.

Hyperbola described by a table

The parent y = 1/x has asymptotes x = 0 and y = 0. A transformed graph g is given by this table of selected values:

Undefined at x = 2 means the vertical asymptote is x = 2. The later values 4, 7/2 drift toward 3, so the horizontal asymptote is y = 3. Try g(x) = 1/(x − 2) + 3:

• g(0) = 1/(−2) + 3 = 5/2
• g(1) = −1 + 3 = 2
• g(3) = 1 + 3 = 4
• g(4) = 1/2 + 3 = 7/2

The table matches. The competitor 1/(x + 2) + 3 would be undefined at x = −2, which the table never shows. The competitor 1/(x − 2) − 3 has the wrong horizontal asymptote.

Exponential and log, moved

y = 2^x passes through (0, 1) and (1, 2) and has horizontal asymptote y = 0. For h(x) = −2^{x+1} + 4 the piece x + 1 shifts the graph 1 unit left, the leading minus reflects in the x-axis, and +4 lifts the asymptote to y = 4. Hand checks: h(−1) = −1 + 4 = 3; h(0) = −2 + 4 = 2; h(1) = −4 + 4 = 0. A stem can list those three points and the asymptote in words and ask which equation is consistent.

For y = log₃(x − 1) + 2 the vertical asymptote is x = 1, the domain is x > 1, and the point (1, 0) on y = log₃ x is replaced by (2, 2). After a horizontal shift, demand that the log argument stay positive: x − 1 > 0, not x > 0.

Inverses as the MAT booklet treats them

Algebra: write y = f(x), swap x and y, and solve for y. Graph: reflect in the line y = x. The domain of f becomes the range of f^{-1}, and the range of f becomes the domain of f^{-1}.

Linear. f(x) = 2x − 6. Swap: x = 2y − 6, so f^{-1}(x) = (x + 6)/2. Check a pair: f(5) = 4 and f^{-1}(4) = 5.

Exponential–log pair. f(x) = 3^x has inverse log₃ x. The point (2, 9) becomes (9, 2). Domain and range swap: all real inputs with range (0, ∞) become domain (0, ∞) with all real outputs.

When the inverse is not a function. y = x² fails the horizontal-line test. Reflecting in y = x yields a sideways parabola. Restricting to x ≥ 0 gives inverse y = √x. If a table repeats the same y-value at two different x-values, an inverse function does not exist unless a domain restriction is stated.

Translated hyperbola. y = 1/x is self-inverse; y = 1/(x − 2) + 3 is not. Swap and rearrange:

x = 1/(y − 2) + 3

x − 3 = 1/(y − 2)

y = 1/(x − 3) + 2

A line in words, then inverted or stretched

A stem says the graph of f is a straight line through (0, 4) and (2, 0). Slope = (0 − 4)/(2 − 0) = −2, so f(x) = −2x + 4.

If g is the reflection of f in y = x, swap: x = −2y + 4, hence g(x) = (4 − x)/2 = 2 − x/2. Then g must pass through the swapped intercepts (4, 0) and (0, 2).

If instead g is a vertical stretch of f by 2, then g(x) = −4x + 8. The x-intercept stays at 2; the y-intercept doubles to 8. Vertical stretches never move x-intercepts — use that to eliminate options quickly.

Method on the afternoon paper

1. Name the parent family.
2. Read the algebra from the inside out: horizontal moves live with x; vertical moves sit outside.
3. Track one clear point and every asymptote.
4. For inverses, swap every given point and test which equation contains the new points.
5. Discard any option that slides the graph the wrong way horizontally. You cannot rescue that error with a calculator, because calculators are not allowed.