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NSWMaths AdvancedQuick questions

Year 12: Functions

Quick questions on Graph transformations: translations, reflections and dilations for HSC Maths Advanced functions

4short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is horizontal transformations (act on xx, inside the function)?
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A point (x0,y0)(x_0, y_0) on y=f(x)y = f(x) becomes (x0+h1,y0)=(x0+h,y0)\left(\frac{x_0 + h}{1}, y_0\right) = \left(x_0 + h, y_0\right) after a shift, and (x0b,y0)\left(\frac{x_0}{b}, y_0\right) after a horizontal dilation.
What is the general form?
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The most general single-variable transformation is
What are effect on key features?
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Translations move features without changing them. Dilations rescale distances. Reflections flip orientation.
What is build a transformed graph one step at a time?
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The reliable way to sketch a combined transformation is to apply one operation at a time and redraw, rather than trying to jump to the final picture. Below, the curve y=−3(x−2)2+4y = -3(x - 2)^2 + 4 is built from y=x2y = x^2 in the exact order set by the general form: dilations and reflections first, then translations, with outside acting on yy and inside acting on xx. Each stage keeps the previous curve as a dashed ghost so you can see what moved. (This is the same worked Step 1 below, drawn out.)

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