Transformation Of Graph Dse Exercise Info

Answers:

Answers:

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.

Question: The graph of $y = x^2 - 4x$ is drawn. (a) Write down the coordinates of the vertex and the x-intercepts. (b) The graph is translated left by 3 units and down by 5 units. Find the equation of the new graph. (c) The graph is reflected about the y-axis. Find the new equation. transformation of graph dse exercise

👀 Think about it first!

. . .

Solution:

(a) Finding Key Features: Original equation: $y = x^2 - 4x$

(b) Translation:

Method 1 (Algebraic Substitution): $y = [(x + 3)^2 - 4(x + 3)] - 5$ $y = [x^2 + 6x + 9 - 4x - 12] - 5$ $y = x^2 + 2x + 2 - 5$ Answer: $y = x^2 + 2x - 3$

Method 2 (Using Vertex Coordinates): Original vertex $(2, -4)$. New vertex: $(2 - 3, -4 - 5) = (-1, -9)$. Equation form: $y = (x - h)^2 + k$ $y = (x - (-1))^2 - 9 \implies y = (x + 1)^2 - 9$. (Both methods yield the same result upon expansion).

(c) Reflection about y-axis:

Trig graphs test horizontal scaling (period change) and vertical scaling (amplitude) most intensely.

Question 6:
The graph of ( y = \cos x ) is transformed to ( y = 3\cos(2x - \pi) + 1 ). Describe the sequence. Answers:

Solution:
Rewrite as ( y = 3\cos[2(x - \frac\pi2)] + 1 )
Sequence from ( \cos x ):

Let ( g(x) = |f(x+2)| - 3 ). If ( f(x) = (x-1)^2 - 4 ),
(a) Find the x‑intercepts of ( g(x) ).
(b) Sketch ( y = g(x) ).

A progressive set of exercises (4 weeks) for secondary students preparing for Hong Kong DSE (or equivalent) on graph transformations: translations, reflections, stretches/compressions, and combinations. Each week has objectives, worked examples, practice questions, and answers.

Before we dive into the exercise, ensure you have this table memorized. Let the equation of the graph be $y = f(x)$.

| Transformation | New Equation | Effect on Graph | | :--- | :--- | :--- | | Vertical Translation | $y = f(x) + k$ | Shift up by $k$ units (if $k > 0$). | | | $y = f(x) - k$ | Shift down by $k$ units. | | Horizontal Translation | $y = f(x - k)$ | Shift right by $k$ units. | | | $y = f(x + k)$ | Shift left by $k$ units. | | Reflection | $y = -f(x)$ | Reflect about the x-axis. | | | $y = f(-x)$ | Reflect about the y-axis. | | Scaling (Stretch/Compress) | $y = k \cdot f(x)$ | Vertical stretch by factor $k$ (if $k > 1$). | | | $y = f(kx)$ | Horizontal compression by factor $\frac1k$. | The figure shows the graph of ( y = f(x) )

⚠️ The DSE Trap: The most common mistake in DSE exams is horizontal translation and scaling.