Transformation Of Graph Dse Exercise Today

Mastering Graph Transformations: The Ultimate DSE Exercise Guide

If you are preparing for the HKDSE Core Mathematics exam, you know that Functions and Graphs is a heavyweight topic. Within that topic, nothing causes quite as much confusion—or appears as frequently—as Graph Transformations.

Every year, students lose valuable marks because they confuse a "translation" with a "reflection" or forget the golden rules of scaling.

This post serves as a complete exercise guide. We will briefly recap the concepts, run through the must-know formulas, and then tackle three common types of DSE-style transformation questions. transformation of graph dse exercise

Question 1 (Basic Shifts)

The figure shows the graph of ( y = f(x) ).
(Sketch: a parabola with vertex at ((0,0)) passing through ((1,1)) and ((-1,1)).)

Write the equations of the following transformed graphs: Question 1 (Basic Shifts) The figure shows the

(a) Shift up by 3 units.
(b) Shift right by 2 units.
(c) Shift left by 1 unit and down by 4 units.

3. Worked Examples (DSE Style)

5. DSE Common Question Types & Examples

Part 1: The Four Pillars of Graph Transformation (DSE Core)

Before tackling complex exercises, let’s establish the foundational rules. Assume the original graph is ( y = f(x) ). ⚠️ Common Pitfall in DSE: Horizontal transformations are

| Transformation | Algebraic Change | Effect on Graph | DSE Common Example | |----------------|------------------|----------------|--------------------| | Translation (Horizontal) | ( y = f(x - h) ) | Shift RIGHT by ( h ) (if ( h>0 )) | Quadratic vertex shift | | Translation (Vertical) | ( y = f(x) + k ) | Shift UP by ( k ) (if ( k>0 )) | Sine/cosine vertical shift | | Reflection (x-axis) | ( y = -f(x) ) | Flip over x-axis | Exponential decay reflection | | Reflection (y-axis) | ( y = f(-x) ) | Flip over y-axis | Even/odd function tests | | Scaling (Vertical) | ( y = a f(x) ) | Stretch/compress vertically | Amplitude change in trig graphs | | Scaling (Horizontal) | ( y = f(bx) ) | Compress/stretch horizontally | Period change in sin/cos |

⚠️ Common Pitfall in DSE: Horizontal transformations are counter-intuitive.
( y = f(x - 2) ) moves the graph right, not left.
( y = f(2x) ) compresses horizontally (period halves), not expands.

Part 4: Intensive Exercise Bank (Without Solutions – For Practice)

Use these to drill before exams.

1. Translation: If ( y = x^2 + 1 ) is shifted 2 units right and 3 units down, write the new equation.
2. Reflection: The point ( (3, 5) ) lies on ( y = g(x) ). Find its image on ( y = -g(-x) ).
3. Scaling: Explain how the graph of ( y = \cos x ) changes to ( y = 3\cos(\fracx2) ).
4. Combined: Starting from ( y = \ln x ), apply: reflect over y-axis, then shift left 1, then stretch vertically by 2. Write final equation.
5. Graph Sketching (DSE Paper 1): Given ( f(x) = (x-1)^2 ), sketch ( y = -f(x+2) + 3 ). Label vertex and intercepts.
6. Multiple Choice (Tricky): Which transformation maps ( y = e^x ) to ( y = e^2-x )?
   (A) Reflect over y-axis, then shift right 2
   (B) Shift right 2, then reflect over y-axis
   (C) Reflect over x-axis, then shift left 2
   (D) Shift left 2, then reflect over y-axis