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IB DP Maths AA · Topic 2 Functions · SL 2.11 · AHL 2.16 · ~6 min read

Transformations of graphs

Translations, reflections and stretches of y = f(x), combining transformations in the right order, mapping points, and (HL) the graphs of |f(x)|, f(|x|), 1/f(x), f(ax + b) and [f(x)]².

TranslationsReflectionsStretchesCombined transformationsMapping points|f(x)| & f(|x|) (HL)1/f(x) (HL)[f(x)]² (HL)
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No tag = SL and HLHL ONLY HL students onlyFORMULA BOOKLET given in the examM1A1 how marks are awardedhighlight = comes up again and againTest yourself at the end

In 30 seconds

  • Outside the bracket changes y (as you would expect): f(x) + b moves up b, pf(x) stretches vertically by p, −f(x) reflects in the x-axis.
  • Inside the bracket changes x (the opposite way): f(x − a) moves RIGHT a, f(qx) stretches horizontally by 1/q, f(−x) reflects in the y-axis.
  • Combined: do the inside changes in the order a point's x-value would be undone, and the outside changes in the order you would calculate them. Tracking one point is the safest check.
  • HL: |f(x)| flips negative parts up; f(|x|) mirrors the right half; 1/f(x) turns roots into asymptotes; [f(x)]2 is never negative.

By the end you can

  • Describe and sketch translations, reflections and stretches of y = f(x).
  • Apply combined transformations and find the image of a point.
  • Write down the equation of a transformed graph from a description.
  • Sketch y = |f(x)|, y = f(|x|), y = 1/f(x), y = f(ax + b) and y = [f(x)]2. HL

1The six basic transformations

f(x) + 1.5: up 1.5f(x − 2): right 2−f(x): reflect in x-axisf(−x): reflect in y-axis2f(x): vertical stretch × 2f(2x): horizontal stretch × ½Grey dashed: the original y = f(x).
Each panel changes the grey curve in one way.
New graphTransformationPoint (x, y) goes to
y = f(x) + btranslation by (0, b)(x, y + b)
y = f(x − a)translation by (a, 0)(x + a, y)
y = pf(x)vertical stretch, factor p(x, py)
y = f(qx)horizontal stretch, factor 1/q(x/q, y)
y = −f(x)reflection in the x-axis(x, −y)
y = f(−x)reflection in the y-axis(−x, y)
Common mistakeThinking f(x + 3) moves the graph right. f(x + 3) moves it left 3: the new graph reaches each y-value when x is 3 smaller.

2Combining transformations

Worked example

Describe the transformations that map y = f(x) onto y = 2f(x − 3) + 1.

  1. horizontal translation 3 right (inside) A1
  2. vertical stretch factor 2, then vertical translation 1 up (outside: stretch before shifting) A1A1
Worked example

The point (4, 6) lies on y = f(x). Find its image on y = 3f(2x) − 1.

  1. x: 2x = 4 ⇒ x = 2 M1
  2. y: 3 × 6 − 1 = 17 M1
  3. (2, 17) A1
Worked example

y = x2 is translated by (2, −3) and then reflected in the x-axis. Find the equation.

  1. translation: y = (x − 2)2 − 3 A1
  2. reflection: y = −(x − 2)2 + 3 A1
Exam tipThe order matters for outside changes: 2f(x) + 1 (stretch then add) is not 2(f(x) + 1) = 2f(x) + 2. Track one point through each step to check.
  • Asymptotes move too: if y = f(x) has asymptote x = 1, then y = f(x − 2) has x = 3; y = f(x) + 4 moves a horizontal asymptote up 4.

3Modulus graphs HL

xyxy|f(x)|: flip the parts below the x-axisf(|x|): keep x ≥ 0, mirror it
Left: reflect below-axis parts upwards. Right: throw away x < 0 and copy x ≥ 0 across the y-axis.
  • y = |f(x)|: keep parts with y ≥ 0; reflect parts with y < 0 in the x-axis. Points on the x-axis stay put and often become sharp corners.
  • y = f(|x|): for x ≥ 0 it is the same as f(x); for x < 0 it mirrors the x ≥ 0 part. The graph is always even.
  • y = f(ax + b): write it as f(a(x + b/a)): a horizontal stretch by 1/a, then a translation of −b/a.

4Reciprocal and squared graphs HL

xyy = f(x)y = 1/f(x)Roots of f become vertical asymptotes; a minimum of f becomesa maximum of 1/f; and 1/f → 0 where f → ±∞.
Every root of f gives a vertical asymptote of 1/f, and f and 1/f cross where f = ±1.
  • y = 1/f(x): roots of f → vertical asymptotes; as f → ±∞, 1/f → 0; a maximum of f becomes a minimum of 1/f (and vice versa) at the same x; 1/f has the same sign as f.
  • y = [f(x)]2: always ≥ 0; roots of f become touching points (minima); where |f| < 1 the graph lies closer to the axis, where |f| > 1 it rises above |f|.
Worked example

f(x) = x2 − 4. Sketch y = 1/f(x): give asymptotes and the turning point.

  1. vertical asymptotes at the roots: x = ±2 A1
  2. horizontal asymptote y = 0 A1
  3. f has a minimum at (0, −4), so 1/f has a maximum at (0, −¼) A1
GDC (calculator) methodCheck every sketch on the GDC with abs( ) or 1/( ), but in Paper 1 you must be able to draw them from the rules above.
Linking questionTrig graphs are transformations too: y = 3sin(2(x − π/4)) + 1 is a stretch, a stretch, a shift and a shift of sin x (Trig identities, equations and graphs).
Idea for your IA / explorationExplore how families of transformed graphs model real data, for example fitting y = a sin(b(x − c)) + d to daylight hours or tide heights.

Test yourself — cover the answers first

Quick check

  1. Describe y = f(x) + 5.
  2. Describe y = f(x + 2).
  3. Describe y = f(3x).
  4. (2, 5) is on y = f(x). Where is it on y = −f(x)?
  5. HL What is the image of the root x = 3 of f on y = 1/f(x)?

Exam-style

  1. The graph of y = f(x) has a maximum at (1, 4). Find the maximum of y = 2f(x + 3) − 5.
  2. HL f(x) = x(x − 3). Find the coordinates of the turning point of y = |f(x)| between x = 0 and x = 3.
Show answers
  1. translation up 5
  2. translation 2 left
  3. horizontal stretch, factor ⅓
  4. (2, −5)
  5. a vertical asymptote x = 3
  6. x: 1 − 3 = −2
    y: 2(4) − 5 = 3
    (−2, 3)
  7. f has a minimum at (1.5, −2.25)
    reflected upwards
    (1.5, 2.25), a maximum of |f|
Practise this chapter
IB Maths AA practice papers
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Independent notes written by Riyaz Sir. Not produced, endorsed or approved by the International Baccalaureate Organization. Syllabus codes refer to the IB DP subject guides.