What does y = f(x − h) do to the graph of y = f(x)?

It shifts the graph h units to the right if h > 0, and |h| units to the left if h < 0.

What does y = f(x) + k do?

It shifts the graph k units up if k > 0, and |k| units down if k < 0.

What is the difference between y = af(x) and y = f(ax)?

y = af(x) scales vertically: a stretch if a > 1, a compression if 0 1, a stretch if 0 < a < 1.

Which board sets my paper if I study in Gujrat?

Gujrat district is listed under the Board of Intermediate and Secondary Education, Gujranwala. PECTAA writes the books. Your board sets the paper and the dates, so follow its notices.

Transformation of Graphs: PECTAA Class 12 Maths, Punjab Board

Q: What does y = f(x) + k do?

It shifts the graph k units up if k > 0, and |k| units down if k < 0.

Q: What is the difference between y = af(x) and y = f(ax)?

y = af(x) scales vertically: a stretch if a > 1, a compression if 0 1, a stretch if 0 < a < 1.

Q: What happens if a is negative?

The graph is also reflected: in the x-axis for y = af(x), and in the y-axis for y = f(ax). The book keeps a > 0 to study scaling alone.

Q: Which board sets my paper if I study in Gujrat?

Gujrat district is listed under the Board of Intermediate and Secondary Education, Gujranwala. PECTAA writes the books. Your board sets the paper and the dates, so follow its notices.

You already know the graph of y = x². Do you need a new table of values for y = (x − 2)² − 2? No. Slide the old parabola 2 right and 2 down. Transformations let you draw new graphs from ones you know, and the paper tests exactly that skill.

Unit 1 of the PECTAA Mathematics 12 book is Graphical Representation of Functions. Section 1.9 covers the transformation of graphs. The book names reflection, rotation, stretching, compression and shifting, then focuses on three: horizontal shift, vertical shift and scaling. This note is for FSc Part 2 (2nd year, Class 12) students in Lahore, Faisalabad, Sargodha, Rawalpindi, Multan, Gujranwala and the other Punjab boards. Gujrat district comes under the Gujranwala board. PECTAA writes the books. Your own board sets your paper.

What to know

Shifting. The graph moves without changing its shape or size.

Horizontal shift: replace x by x − h, giving y = f(x − h). If h > 0 the graph moves h units right; if h < 0 it moves |h| units left.
Vertical shift: add k to the output, giving y = f(x) + k. If k > 0 the graph moves k units up; if k < 0 it moves |k| units down.
Both together: y = f(x − h) + k.

The book's parabola example. y = x² has vertex (0, 0). y = (x − 2)² has vertex (2, 0): the same curve moved 2 right. y = x² − 2 has vertex (0, −2): moved 2 down.

Scaling. The graph is stretched or compressed but keeps its general shape. Here a > 0.

Vertical scaling, y = af(x): a > 1 stretches the graph vertically by factor a; 0 < a < 1 compresses it by factor 1/a. Only y-values change.
Horizontal scaling, y = f(ax): a > 1 compresses the graph horizontally by factor a; 0 < a < 1 stretches it by factor 1/a. Only x-values change.
Both together: y = af(bx), with a, b > 0.

Negative a. Then the change includes a reflection: in the x-axis for y = af(x), and in the y-axis for y = f(ax).

The book's scaling examples. y = (1/50)(x − 2)² is y = (x − 2)² compressed vertically by factor 50. y = (3x − 2)² is y = (x − 2)² with x replaced by 3x, a horizontal compression by factor 3.

Key points

Inside the bracket acts on x and feels "backwards". x − 2 moves right; 3x squeezes sideways.
Outside the function acts on y and feels "normal". +k moves up; a > 1 stretches up.
Order matters when several steps are combined. Apply them one at a time, as the book does.

Worked examples

Example 1 (book Example 13). f(x) = ln(x² + 1). Apply (i) a vertical stretch by 2, then (ii) a shift 1 unit left.

Vertical stretch, a = 2: g(x) = 2 ln(x² + 1).
Shift left 1, so h = −1: y = g(x + 1) = 2 ln[(x + 1)² + 1].
Simplify: y = 2 ln(x² + 2x + 2).

Example 2 (book Example 14). f(x) = x/(x + 1). Apply in order: vertical stretch by 2, horizontal stretch by 3, shift 2 right, shift 1 down.

Vertical stretch by 2: g(x) = 2x/(x + 1).
Horizontal stretch by 3 (b = 1/3): s(x) = g(x/3) = (2x/3)/(x/3 + 1) = 2x/(x + 3).
Shift 2 right: t(x) = s(x − 2) = (2x − 4)/(x + 1).
Shift 1 down: y = (2x − 4)/(x + 1) − 1 = (x − 5)/(x + 1).

Example 3. Describe how to get y = (x + 3)² + 4 from y = x². Move 3 units left (h = −3) and 4 units up (k = 4). The vertex moves from (0, 0) to (−3, 4).

Example 4. Start from y = x². Write the function after a vertical compression by factor 2 and then a shift 3 units left.

Compression by 2 means a = 1/2: y = (1/2)x².
Shift 3 left means h = −3: y = (1/2)(x + 3)².

Common mistakes in the exam

Moving the wrong way. f(x − 2) moves right, not left.
Mixing the two scalings. af(x) with a > 1 stretches vertically; f(ax) with a > 1 compresses horizontally.
Writing a horizontal stretch by 3 as f(3x). A stretch by 3 is f(x/3); f(3x) is a compression by 3.
Changing the order of steps. In Example 14, shifting before stretching gives a different function.
Forgetting the reflection when a < 0.

Exam technique

Name each step and write the new function after every step, as in the book's examples.
Track one point, such as the vertex, to check your shifts.
Simplify the final expression fully.
Check your board's notice for the sections in your paper.

Practice questions

Find the vertex of y = (x − 5)² − 1. Answer: (5, −1).
Which transformation turns y = f(x) into y = f(x) − 3? Answer: A shift 3 units down.
Describe y = 4f(x) compared with y = f(x). Answer: A vertical stretch by factor 4.
Describe y = f(2x) compared with y = f(x). Answer: A horizontal compression by factor 2.
f(x) = x². Write the function after a shift 1 unit right and then 2 units up. Answer: y = (x − 1)² + 2.
Which reflection does y = f(−x) give? Answer: A reflection in the y-axis.

Ready to practise this? Continue in 2nd Year Mathematics.