MathsMediumClass 11

Graphs — Standard function transformations

Sets, Relations & Functions

JEE Qs

Hard

min

Master the order of transformations for combined operations (scaling/reflection first, then shifting) and practice with a variety of base functions to build intuition.

🧮 Key Formulas

y = f(x) + c (vertical shift up by c units)

y = f(x) - c (vertical shift down by c units)

y = f(x + c) (horizontal shift left by c units)

y = f(x - c) (horizontal shift right by c units)

y = c * f(x) (vertical stretch by factor c if c>1, compression if 0<c<1)

y = f(c * x) (horizontal compression by factor c if c>1, stretch if 0<c<1)

y = -f(x) (reflection about the x-axis)

y = f(-x) (reflection about the y-axis)

y = |f(x)| (reflect the part of graph below x-axis about x-axis)

y = f(|x|) (for x>=0, graph is same as f(x); for x<0, graph is reflection of x>=0 part about y-axis)

y = [f(x)] (Greatest Integer Function applied to y-values)

y = {f(x)} (Fractional Part Function applied to y-values)

✅ Key Points for JEE

1Recognize the base function y=f(x) before applying transformations; visualize the parent graph first.
2For combined transformations (e.g., y = a * f(b * x + c) + d), the typical order is: horizontal scaling/reflection (1/b), then horizontal shifting (-c/b), then vertical scaling/reflection (a), and finally vertical shifting (d).
3Transformations affecting 'x' directly (inside f(x)) like f(x+c) or f(cx) operate on the x-coordinates and often behave counter-intuitively (e.g., +c shifts left).
4Transformations affecting 'f(x)' (outside f(x)) like f(x)+c or c*f(x) operate on the y-coordinates and behave intuitively.
5Be precise when applying absolute value transformations: `y=|f(x)|` changes negative y-values, `y=f(|x|)` changes the graph for x<0.

⚠️ Common Mistakes

✕Incorrectly applying horizontal shifts (e.g., y=f(x+2) shifts right instead of left).
✕Incorrect order of operations for multiple transformations, especially mixing scaling and shifting on the x-axis (e.g., transforming f(2x+4) as f(2(x+2)) first).
✕Confusing `y = |f(x)|` with `y = f(|x|)` and applying the wrong reflection rule.
✕Not understanding how domain and range are affected by various transformations, especially for functions like `sqrt(x)` or `1/x`.

NCERT Chapters

Class 11 Maths Ch 2: Relations and Functions (Introduction to functions and their basic graphs)
Class 11 Maths Ch 3: Trigonometric Functions (Graphs of trigonometric functions which are then transformed)