2.11 Transformations of graphsIB Maths: Analysis and Approaches HL: Revision notes

• $y = f(x) + b$: vertical translation by $b$ (up if $b > 0$). Every point $(x, y) \to (x, y + b)$.
• $y = f(x - a)$: horizontal translation by $a$ (to the right if $a > 0$). Every point $(x, y) \to (x + a, y)$.

So $y = (x - 3)^{2} + 4$ is $y = x^{2}$ translated 3 right and 4 up; its vertex moves from $(0, 0)$ to $(3, 4)$. IB papers may write a translation as a vector with top entry $a$ and bottom entry $b$.

• $y = -f(x)$: reflection in the $x$-axis. $(x, y) \to (x, -y)$.
• $y = f(-x)$: reflection in the $y$-axis. $(x, y) \to (-x, y)$.

For example, $y = -x^{2}$ opens downwards, and $y = e^{-x}$ is the reflection of $y = e^{x}$ in the $y$-axis. Points on the mirror line stay fixed: $x$-intercepts are unchanged by $y = -f(x)$.

• $y = pf(x)$: vertical stretch with scale factor $p$. $(x, y) \to (x, py)$.
• $y = f(qx)$: horizontal stretch with scale factor $\frac{1}{q}$. $(x, y) \to \left(\frac{x}{q}, y\right)$.

So $y = f(2x)$ squashes the graph towards the $y$-axis (scale factor $\frac{1}{2}$), and $y = f\left(\frac{1}{2}x\right)$ stretches it away (scale factor 2). A vertical stretch leaves $x$-intercepts fixed; a horizontal stretch leaves $y$-intercepts fixed.

When several transformations are combined, apply them one at a time to the equation, in the stated order.

From $y = x^{2}$ to $y = 3x^{2} + 2$: a vertical stretch with scale factor 3, then a translation 2 up. In the other order you get $y = 3(x^{2} + 2) = 3x^{2} + 6$.

Similarly, reflecting $y = x^{2}$ in the $x$-axis and then translating up 4 gives $-x^{2} + 4$, but translating first and then reflecting gives $-x^{2} - 4$.

Two vertical changes (a stretch or reflection and a vertical translation) do not commute; nor do two horizontal ones. A vertical and a horizontal transformation can be done in either order.

Describe transformed graphs through their features rather than a sketch:

• Asymptotes: $y = e^{x}$ has asymptote $y = 0$; $y = 3 - e^{2x}$ has asymptote $y = 3$.
• Range: $e^{x} > 0$, so $-e^{2x} < 0$ and $3 - e^{2x} < 3$.
• Vertex / maximum: in context, a vertical translation of a profit function changes the maximum profit but not the number of items that gives it.
• Zeros: a vertical stretch keeps zeros fixed, so break-even points do not change when all profits are scaled.