GCSE Edexcel Maths Transformations of Graphs: Here is the graph of $y = sin \

Here is the graph of $y = sin \, x^2$ for $-180 \leq x \leq 180$ - Edexcel - GCSE Maths - Question 18 - 2018 - Paper 1

Question 18

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Here is the graph of $y = sin \, x^2$ for $-180 \leq x \leq 180$. On the grid, sketch the graph of $y = sin \, x^2 - 2$ for $-180 \leq x \leq 180$.

Worked Solution & Example Answer:Here is the graph of $y = sin \, x^2$ for $-180 \leq x \leq 180$ - Edexcel - GCSE Maths - Question 18 - 2018 - Paper 1

Step 1

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Answer

Understanding the Transformation: The equation $y = sin \, x^2 - 2$ indicates a vertical shift of the graph of $y = sin \, x^2$ by -2 units. This implies that every point on the original graph will be lowered by 2 units.
Determine Key Points: Start by noting some key values of the original graph. Since $sin \, x^2$ oscillates between -1 and 1, the function $y = sin \, x^2$ achieves its maximum value of 1 at certain points and minimum value of -1 at others. Those transitions will occur periodically based on $x^2$ .
Shifting the Graph: By lowering the entire graph of $y = sin \, x^2$ by 2 units, the maximum point will now be at $y = 1 - 2 = -1$ and the minimum will be at $y = -1 - 2 = -3$ .
Plotting the New Points: Begin plotting the new key points across the interval $-180 \leq x \leq 180$ . Use known values of $x$ for which $x^2$ is a multiple of $\frac{\pi}{2}$ to find the corresponding points on the original and transformed graphs.
Final Sketch: Finally, connect these key points smoothly, maintaining the sinusoidal nature of the graph while ensuring that the entire graph is presented below the y-axis since the maximum is at -1 and the minimum at -3.