IB Maths AA HLTopic 2 — FunctionsPaper 1 & 2~7 min read
Stretches of Graphs
A stretch resizes a graph along one axis without rotating or flipping it. There are two kinds: vertical (y = af(x)) and horizontal (y = f(x/a)). The vertical one behaves how you’d expect; the horizontal one has a sneaky reciprocal in the scale factor that catches almost everyone the first time.
📘 What you need to know
Vertical stretch: y = af(x) is a stretch of scale factor a parallel to the y-axis. Every y-coordinate is multiplied by a.
Horizontal stretch: y = f(x/a) is a stretch of scale factor a parallel to the x-axis. Every x-coordinate is multiplied by a.
Watch out for y = f(ax): this is a horizontal stretch with scale factor 1/a, not a. The reciprocal flip catches most students.
Scale factor > 1 → graph gets bigger in that direction; 0 < SF < 1 → graph gets smaller (a “compression”).
Vertical stretch changes horizontal asymptotes (y = k → y = ak); vertical asymptotes stay.
Horizontal stretch changes vertical asymptotes (x = k → x = ak); horizontal asymptotes stay.
Points on the axis of stretch are fixed: x-intercepts don’t move under a vertical stretch; the y-intercept doesn’t move under a horizontal stretch.
Vertical stretch — y = af(x)
Multiply every y-value by a
Every point (p, q) on y = f(x) maps to (p, aq) on y = af(x)
Changes
y-coordinates ×a horizontal asymptote y = k → y = ak
vertical world is scaled
Stays the same
x-coordinates vertical asymptotes x-intercepts
anything with y = 0 doesn’t move
Horizontal stretch — y = f(x/a)
Multiply every x-value by a
Every point (p, q) on y = f(x) maps to (ap, q) on y = f(x/a)
🤔 Why divide x by a for a horizontal stretch of factor a?
The new graph at x = ap should give the same y-value as the old graph at x = p. So we need f((ap)/a) = f(p). The “÷a” inside the bracket undoes the stretch to get back to the original input.
Changes
x-coordinates ×a vertical asymptote x = k → x = ak
horizontal world is scaled
Stays the same
y-coordinates horizontal asymptotes y-intercept
anything with x = 0 doesn’t move
The reciprocal trap — y = f(ax)
If you see y = f(ax) instead of f(x/a), that’s a horizontal stretch of scale factor 1/a. So y = f(3x) compresses the graph horizontally by factor 3 (SF = 1/3); y = f(x/3) stretches the graph by factor 3.
Quick rule: whatever number is multiplying x inside the bracket, the scale factor is its reciprocal. f(2x) → SF 1/2; f(x/5) → SF 5; f(x/4) → SF 4. Always reciprocal.
Vertical vs horizontal stretch of the same parabola
🧭 Recipe — stretching a graph
Identify the type: number outside f → vertical stretch; coefficient or division of x inside f → horizontal stretch.
Read off the scale factor. For af(x), SF = a. For f(x/a), SF = a. For f(ax), SF = 1/a.
Apply to each labelled point: multiply y-coords by SF for vertical, x-coords by SF for horizontal.
Scale the matching asymptotes: HAs scale under vertical stretch; VAs scale under horizontal stretch.
Sketch the new graph with the same overall shape, just scaled. Label the transformed features.
Worked examples
WE 1
Find the equation after a vertical stretch
The graph of f(x) = 2x2 − x + 5 is stretched vertically with scale factor 3. Find the equation of the new graph.
Vertical stretch SF 3 → multiply f(x) by 3y = 3f(x) = 3(2x² − x + 5)y = 6x² − 3x + 15y = 6x² − 3x + 15every term gets multiplied — including the constant
WE 2
Stretch the key points of a graph
The graph of y = f(x) has a maximum at P(−4, 6) and a minimum at Q(2, −3). State the new coordinates after:
(a) a vertical stretch with scale factor 5 (b) a horizontal stretch with scale factor 3
(a) vertical stretch SF 5: multiply y-coords by 5P(−4, 6) → (−4, 30)Q(2, −3) → (2, −15)(a) max (−4, 30); min (2, −15)(b) horizontal stretch SF 3: multiply x-coords by 3P(−4, 6) → (−12, 6)Q(2, −3) → (6, −3)(b) max (−12, 6); min (6, −3)vertical stretch leaves x-coords alone; horizontal stretch leaves y-coords alone
WE 3
Describe the transformation — the reciprocal trap
The graph of y = f(x) is transformed to give y = f(4x). Describe the transformation fully.
x is multiplied by 4 inside the bracketthat’s the form y = f(ax) with a = 4Apply the reciprocal ruleSF = 1/a = 1/4horizontal stretch with scale factor 1/4 (parallel to the x-axis)SF less than 1 means the graph gets squashed toward the y-axis — every x-coordinate is divided by 4
WE 4
Vertical stretch of a rational function
The graph of y = 1x + 3 is stretched vertically with scale factor 2.
(a) Find the equation of the new graph. (b) State its asymptotes.
(a) Multiply the entire function by 2y = 2(1/x + 3) = 2/x + 6(a) y = 2/x + 6(b) Asymptotes — HA scales, VA staysoriginal VA: x = 0 → unchangedoriginal HA: y = 3 → scaled by 2 → y = 6(b) VA: x = 0; HA: y = 6vertical stretch only touches y-values — including the y-value of the horizontal asymptote
WE 5
Horizontal stretch of a rational function
The graph of y = 2x − 6 is stretched horizontally with scale factor 2.
(a) Find the equation of the new graph. (b) State its vertical asymptote.
(a) Replace x with x/2y = 2/((x/2) − 6)simplify denominator: (x/2) − 6 = (x − 12)/2y = 2 ÷ (x − 12)/2 = 4/(x − 12)(a) y = 4/(x − 12)(b) VA scales by 2original VA: x = 6 → scaled by 2 → x = 12(b) VA: x = 12double-check: setting denominator to zero in the new equation gives x = 12, matching the scaled VA
WE 6
Find an unknown scale factor
The graph of y = f(x) passes through the point (5, 12). After a vertical stretch with scale factor a, this point maps to (5, 30). Find the value of a.
Vertical stretch SF a multiplies the y-coordinate by a12 × a = 30a = 30/12 = 5/2a = 5/2x-coordinate didn’t change — confirms it’s a vertical stretch and not a horizontal one
💡 Top tips
Outside the bracket → vertical, inside the bracket → horizontal. Same rule that worked for translations and reflections.
Vertical stretch is intuitive: y = 4f(x) is a stretch of factor 4 — no surprises.
Horizontal stretch needs the reciprocal: y = f(4x) is a stretch of factor 1/4, not 4.
Anchor with a fixed feature: under a vertical stretch, x-intercepts stay; under a horizontal stretch, the y-intercept stays.
SF > 1 makes the graph bigger; 0 < SF < 1 makes it smaller. Use the right wording — examiners want “stretch with scale factor 1/3”, not “compression”.
Combine asymptote rules: vertical stretch scales HAs; horizontal stretch scales VAs. The one matching the direction of the stretch.
Verify with a test point: pick one labelled point, apply the rule, check it matches the GDC plot.
⚠ Common mistakes
Treating y = f(2x) as a horizontal stretch with SF 2. It’s actually SF 1/2. The reciprocal flip is the most common slip.
Stretching in the wrong direction: y = af(x) is vertical (outside), y = f(x/a) is horizontal (inside).
Scaling the wrong asymptote: under a vertical stretch, only HAs change; under a horizontal stretch, only VAs change.
Forgetting that x-intercepts are fixed under a vertical stretch — multiplying 0 by anything still gives 0.
Multiplying only some terms when applying af(x). Every term in the expression gets scaled, including constants.
Forgetting “parallel to the x-axis” or “parallel to the y-axis” when stating a transformation. Examiners want the full description.
Confusing a stretch with a translation when the graph “moves”. A stretch resizes; only points off the axis appear to “move”.
Three transformations down, one to go. The next note covers composite transformations — when two or more of these get applied together. The order matters: get it wrong and a graph that should be 4 units up ends up 8 units up. The rules sound technical, but they boil down to a single observation about which side of the bracket each operation lives on.
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