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MathematicsRelations and FunctionsJEE
  1. 1. Relations
  2. 2. Reflexive · Symmetric · Transitive
  3. 3. Equivalence
  4. 4. Functions
  5. 5. Composition
  6. 6. One-One & Onto
  7. 7. Bijections & Counting
  8. 8. Inverse
  9. 9. Wavy Curve
  10. 10. Domain & Range
  11. 11. Modulus
  12. 12. Trig Graphs
  13. 13. Trig Domains & Ranges
  14. 14. Inverse Trig
  15. 15. Exp & Log
  16. 16. [x], {x}, sgn
  17. 17. Even & Odd
  18. 18. Periodic
  19. 19. Functional Equations
  20. 20. Transformations
Relations and Functions · Part 20 of 20

Transformation of Graphs

Sketch new graphs from known ones: shifts, stretches and reflections, the three modulus transformations, and greatest-integer transformations, ending with combined transformations step by step.

Builds on: Part 11 · Modulus Function, Part 16 · Greatest Integer, Fractional Part, Signum and Max-Min.

Transformation of GraphsVideo coming soon
Modulus

|f(x)|, f(|x|), |y| = f(x)

For f(x) = x² − 2x: |f(x)| flips the dip up; f(|x|) mirrors the right half (always even); |y| = f(x) keeps the parts above the axis and mirrors them down (not a function).

Three modulus transformations.
Three different rules.
Stepwise transformation of |x|.
Dashed: the intermediate steps.
Combined

2|x − 1| − 3 from |x|

Inside first: right 1; then ×2 (steeper); then down 3. Vertex (1, −3), roots −1/2 and 5/2.

Greatest integer

[x²]

[f(x)] jumps wherever f crosses an integer: [x²] jumps at ±1, ±√2, ±√3, ±2.

Step graph of [x²].
Dashed: y = x².
Step graph of [x]².
Rounded down first, then squared.
Greatest integer

[x]²

f([x]) is constant on each [n, n + 1): [x]² gives the steps 4, 1, 0, 1, 4.

Worked examples

From the video

1. Graph of y = f(x + 3) − 2 from y = f(x)?

3 left, 2 down.

2. Vertex and roots of y = 2|x − 1| − 3.

(1, −3); x = −1/2 and 5/2.

JEE-style question

Your turn

The graph of y = f(x + 3) − 2 is obtained from y = f(x) by shifting it:

(a)3 left, 2 down
(b)3 right, 2 down
(c)3 left, 2 up
(d)3 right, 2 up
Show the answer and the traps

+3 inside: 3 units left. −2 outside: 2 units down. Option (a).

(b) and (d) move right: the classic sign trap for horizontal shifts.

(c) moves up: outside the bracket, the sign means what it says.

Watch out

Common mistakes

Shifting f(x + k) to the rightInside changes act in reverse: f(x + k) moves left.
Confusing |f(x)| with f(|x|)|f(x)| flips the part below the x-axis; f(|x|) mirrors the right half.
Treating |y| = f(x) as a functionIt mirrors the upper part down: not a function.
Practice

Try these

1. How is y = 3 − 2f(x) obtained from y = f(x)?

Stretch vertically by 2, reflect in the x-axis, then shift up 3.

2. Sketch y = |x² − 4| and give its range.

Parabola with the part between −2 and 2 flipped up; range [0, ∞).

3. How is y = f(2x − 4) obtained from y = f(x)?

f(2(x − 2)): squeeze by 2 towards the y-axis, then shift right 2.

4. Number of solutions of |x² − 4| = 3.

4 (x² = 7 or x² = 1).

All 20 partsChapter hub