Transformations of Function Graphs
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Direct answer
Moving a graph around the plane without changing its shape obeys a small grammar: y = f(x) + c shifts it up by c, while y = f(x + c) shifts it left by c — inside changes act opposite to their sign. Multiplication stretches: y = a f(x) stretches vertically by |a| (and flips across the x-axis if a < 0), while y = f(ax) compresses horizontally by factor |a|. Reflections read y = -f(x) (x-axis) and y = f(-x) (y-axis). The modulus pair is the classic discriminator: y = |f(x)| keeps the graph where f ≥ 0 and reflects the below-axis part upward, whereas y = f(|x|) keeps the right half and mirrors it left, discarding the original left half. When transformations compose, order matters — f(2x + 3) means shift left 3 first, then compress by 2, because f(2x + 3) = f(2(x + 3/2)).
What you must remember
- Vertical and horizontal shifts: y = f(x) + c moves up (down for c < 0); y = f(x + c) moves left by |c| when c > 0 — inside changes counter the sign.
- Stretches: y = a f(x) scales all outputs by a (vertical stretch |a| > 1, compression |a| < 1); y = f(ax) squeezes the x-axis by factor a.
- Reflections: y = -f(x) flips across the x-axis; y = f(-x) flips across the y-axis; y = -f(-x) does both (half-turn about the origin).
- Modulus pair: y = |f(x)| reflects negative parts upward (graph never dips below the axis); y = f(|x|) is even, right half kept and copied to the left.
- The other modulus: |y| = f(x) keeps only the portion with f(x) ≥ 0 and reflects it below the x-axis — the graph becomes symmetric about the x-axis.
- Order of operations: for f(2x + 3), first shift left 3/2 — wait: factor as f(2(x + 3/2)) and read inside-out: shift left 3/2, then compress horizontally by 2; reversing the order gives a different graph.
- Domain tracking: transformations carry the domain with them — f(x + 3) uses inputs shifted three left, and range changes track the vertical moves only.
Building a graph step by step
Start from y = x^2 and build y = -2(x + 1)^2 + 3 in four labelled moves. Shift left 1: y = (x + 1)^2 puts the vertex at (-1, 0). Stretch vertically by 2: y = 2(x + 1)^2 narrows the parabola. Reflect across the x-axis: y = -2(x + 1)^2 opens downward. Shift up 3: y = -2(x + 1)^2 + 3 sets the vertex at (-1, 3), a downward parabola with x-intercepts where -2(x + 1)^2 + 3 = 0, i.e., x = -1 ± √(3/2). Now contrast the modulus pair on the sine curve: y = |sin x| turns every trough into a crest (period visually halves to π), while y = sin|x| coincides with sin x for x ≥ 0 and mirrors it to the left — smooth at 0, but not periodic. And |y| = sin x keeps only the arcs above the axis, each reflected below as well, producing a chain of loops. Sketching all four from one parent graph is the single best revision exercise for this chapter.
Where students slip
JEE Main asks which equation matches a given sketch (parabola and modulus sketches dominate) and the number of solutions of pairs like |x| = f(x) read graphically. Advanced leans on the modulus dichotomy — solving |f(x)| = g(x) by splitting at f's zeros — and on order-of-composition questions where f(2x + 3) is compared with f(2x) + 3 and f(2(x + 3)). The dependable errors: shifting f(x + 3) right instead of left (the inside sign reversal); applying the horizontal compression before the shift in f(2x + 3) — correct order is shift then compress, since the +3 lives inside the argument as 2(x + 3/2); believing |f(x)| and f(|x|) are the same thing (they agree only when f is already even and nonnegative); and forgetting that |y| = f(x) requires f(x) ≥ 0 before reflection can even begin. A quiet but examined point: y = f(|x|) is always even regardless of f, which instantly settles symmetry questions. Functions and their graphs anchor the sets-relations-functions unit in both syllabi, and graph reading runs through roots, maxima and area questions everywhere else.
Frequently asked questions
Which way does y = f(x + 3) move the graph?
Three units left — inside (argument) changes act opposite to their sign, so +3 shifts toward negative x.
What is the difference between |f(x)| and f(|x|)?
|f(x)| reflects the below-axis portion upward, changing the range; f(|x|) keeps the right half of the graph and mirrors it to the left, making the function even and changing the domain.
In what order are the transformations of f(2x + 3) applied?
Rewrite as f(2(x + 3/2)): shift left 3/2 first, then compress horizontally by factor 2 — reversing the order produces a genuinely different graph.
What does the equation |y| = f(x) represent?
Only the part of the curve with f(x) ≥ 0 survives, and it is reflected both above and below the x-axis, yielding x-axis symmetry.
How do reflections y = -f(x) and y = f(-x) differ?
The first flips the graph across the x-axis (outputs negated); the second flips across the y-axis (inputs negated), the move that turns any function into its mirror image.