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NSWMaths AdvancedQuick questions
Year 12: Functions
Quick questions on Graph transformations: translations, reflections and dilations for HSC Maths Advanced functions
14short 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 vertical transformations (act on $y$, outside the function)?Show answer
These do what they say: a point $(x_0, y_0)$ on $y = f(x)$ becomes $(x_0, a y_0 + k)$ on $y = a f(x) + k$.
What is horizontal transformations (act on $x$, inside the function)?Show answer
A point $(x_0, y_0)$ on $y = f(x)$ becomes $\left(\frac{x_0 + h}{1}, y_0\right) = \left(x_0 + h, y_0\right)$ after a shift, and $\left(\frac{x_0}{b}, y_0\right)$ after a horizontal dilation.
What is the general form?Show answer
The most general single-variable transformation is
What is order of operations?Show answer
Inside the function: apply the dilation $b$ first, then the translation $h$. Outside: apply the dilation $a$ first, then the translation $k$. Mixing the order changes the answer.
What is effect on key features?Show answer
Translations move features without changing them. Dilations rescale distances. Reflections flip orientation.
What is a composite transformation?Show answer
Start with $y = x^2$. Find the equation after stretching vertically by $3$, reflecting in the $x$-axis, shifting right by $2$, and shifting up by $4$.
What is inside and outside dilations?Show answer
Sketch $y = 2 \sin(3 x)$ starting from $y = \sin x$.
What is working backwards?Show answer
The graph of $y = 4 - (x + 1)^2$ is the parabola $y = x^2$ reflected in the $x$-axis (giving $y = -x^2$), shifted left by $1$ (giving $y = -(x + 1)^2$), and shifted up by $4$. Vertex at $(-1, 4)$, opens downward.
What is tracking a single point?Show answer
The point $(0, 0)$ on $y = \sin x$ becomes which point on $y = 5 \sin(2(x - \pi / 4)) + 3$?
What is sign flip on horizontal translations?Show answer
$y = f(x - h)$ moves the graph right by $h$, not left. Drawing one specific point through the transformation is a fast check.
What is applying inside operations in the wrong order?Show answer
$f(2x - 4)$ is not the same as $f(2(x - 4))$. Factor first: $f(2x - 4) = f(2(x - 2))$, so the horizontal compression by $\frac{1}{2}$ is followed by a shift right by $2$, not $4$.
What is treating a horizontal dilation as a vertical effect?Show answer
$y = f(2 x)$ rescales the $x$-axis, not the $y$-axis. The $y$-values of any single point are unchanged.
What is confusing $-f $ with $f $?Show answer
The first reflects in the $x$-axis (flip top to bottom), the second in the $y$-axis (flip left to right). For an even function they look the same; for an odd function $f(-x) = -f(x)$, so they coincide there too.
What is missing the impact on the domain?Show answer
Reflecting or dilating horizontally changes the natural domain of a function that has a restricted domain (such as $\sqrt{x}$, $\ln x$, or $\arccos x$). Track the new domain along with the new equation.