Transformation Of Graph Dse Exercise
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Before tackling complex exercises, let’s establish the foundational rules. Assume the original graph is ( y = f(x) ).
| 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.
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.
Graph transformation is a fundamental topic in analytic geometry and function analysis. For DSE candidates, mastering graph shifts, reflections, stretches, and compressions is essential for solving complex function problems quickly without plotting every point.
This report provides:
Final: ( y = \frac1x+2 )
Step 2 (last applied): Horizontal shift left 3 → reverse: shift right 3.
y = 1/( (x-3) + 2 ) = 1/(x - 1)
Step 1 (first applied): Reflection in x-axis → reverse: reflect again in x-axis (multiply by -1).
y = - 1/(x - 1)
Therefore: ( h(x) = -\frac1x-1 ).
When ( y = f(2x) ), the domain is halved. When ( y = \sqrt-x ), domain becomes ( x \le 0 ). Always check domain in DSE long questions.
Simon Bates, BBC Radio Devon
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Before tackling complex exercises, let’s establish the foundational rules. Assume the original graph is ( y = f(x) ).
| 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.
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.
Graph transformation is a fundamental topic in analytic geometry and function analysis. For DSE candidates, mastering graph shifts, reflections, stretches, and compressions is essential for solving complex function problems quickly without plotting every point.
This report provides:
Final: ( y = \frac1x+2 )
Step 2 (last applied): Horizontal shift left 3 → reverse: shift right 3.
y = 1/( (x-3) + 2 ) = 1/(x - 1)
Step 1 (first applied): Reflection in x-axis → reverse: reflect again in x-axis (multiply by -1).
y = - 1/(x - 1)
Therefore: ( h(x) = -\frac1x-1 ).
When ( y = f(2x) ), the domain is halved. When ( y = \sqrt-x ), domain becomes ( x \le 0 ). Always check domain in DSE long questions.
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