We will build from simple recognition to complex composite transformations, mimicking DSE question difficulty.
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:
(a) Shift up by 3 units.
(b) Shift right by 2 units.
(c) Shift left by 1 unit and down by 4 units.
The graph of ( y = h(x) ) is transformed by a reflection in the x-axis, followed by a horizontal translation of 3 units left, giving ( y = \frac1x+2 ). Find ( h(x) ).
For ( y = 3f(2x) ):
The transformation of graphs is not a topic to memorize—it is a skill to internalize through structured, repetitive exercise. DSE examiners frequently disguise transformations within function notation, composite functions, or trigonometric modeling. By mastering the exercise blueprint outlined above—starting with basic shifts, progressing to composites, and practicing reverse logic—you will turn graph transformations into a reliable scoring zone.
Final Exercise for You (Answer below):
The graph of ( y = 2^x ) is reflected in the line ( y = x ), then stretched vertically by factor 3, then translated 2 units down. Find the equation of the resulting curve.
Answer: Reflection in ( y=x ) gives inverse: ( y = \log_2 x ).
Then vertical stretch ×3: ( y = 3 \log_2 x ).
Then down 2: ( y = 3 \log_2 x - 2 ).
Now go forth and transform every graph the DSE throws at you! transformation of graph dse exercise
The transformation of graphs in the Hong Kong Diploma of Secondary Education (HKDSE) curriculum involves modifying the function
through translation, reflection, and scaling (enlargement or contraction). Quick Summary of Transformations
Transformations can be categorized based on whether they affect the coordinates: Transformation Algebraic Change Visual Effect Vertical Translation Shift up/down by Horizontal Translation Shift left ( +kpositive k ) or right ( −knegative k Reflection (x-axis) Flips the graph vertically. Reflection (y-axis) Flips the graph horizontally. Vertical Scaling ) or shrink ( ) vertically. Horizontal Scaling ) or stretch ( ) horizontally. Step-by-Step Exercise
Question:
The graph of ( y = f(x) ) passes through (2, 3). It is transformed as follows:
Step 1: Reflect in y-axis.
Step 2: Stretch vertically by factor 3.
Step 3: Shift left 1 unit and up 2 units.
Find the coordinates of the image of the point (2, 3) after all transformations, and express the final transformation in the form ( y = a f(bx + c) + d ). We will build from simple recognition to complex
Solution:
Final point: ( (-3, 11) )
Equation form:
Start: ( y = f(x) )
Reflect y-axis: ( y = f(-x) )
Vert stretch ×3: ( y = 3f(-x) )
Shift left 1: replace x with ( x+1 ) inside f: ( y = 3f(-(x+1)) = 3f(-x - 1) )
Shift up 2: ( y = 3f(-x - 1) + 2 )
Thus: ( a=3, b=-1, c=-1, d=2 ) → ( y = 3f(-x - 1) + 2 )