Over $0^\circ$ to $360^\circ$:

The asymptotes of $\tan x$ exist because $\tan x=\dfrac{\sin x}{\cos x}$, which blows up wherever $\cos x=0$.

A vertical stretch, scale factor $a$ — every $y$-value is multiplied by $a$, so the amplitude becomes $a$. The period is unchanged, and points on the $x$-axis stay fixed.

A vertical translation by $a$ — the midline moves from $y=0$ to $y=a$. For $y=3\sin x+k$ with maximum value 10: the base max is 3, so $3+k=10 \Rightarrow k=7$, giving a new minimum of $-3+7=4$.

A horizontal stretch, scale factor $\tfrac1a$ — counter-intuitively, the graph gets squashed when $a>1$. The new period is $\dfrac{360°}{a}$. For $y=\sin(2x)$: period $=180°$, so it completes two full cycles in $360°$.

A horizontal translation, and the direction is the opposite of what the sign suggests: $y=f(x+a)$ shifts left by $a$; $y=f(x-a)$ shifts right by $a$.

This explains why $\cos x$ is just $\sin x$ shifted $90°$ left: $\cos x = \sin(x+90°)$.

State both transformations separately: vertical stretch scale factor 3 (parallel to $y$-axis), and horizontal stretch scale factor $\tfrac12$ (parallel to $x$-axis).

Find the calculator's principal value $x_1$, then use symmetry for the rest:

• $\sin x=k$: second solution is $180°-x_1$.
• $\cos x=k$: second solution is $360°-x_1$.
• $\tan x=k$: solutions repeat every $180°$ ($x_1, x_1+180°,\dots$).

Example: $\sin x=0.5$ for $0°\le x\le360°$: $x_1=30°$, so $x=30°$ or $180°-30°=150°$.

Rearrange the target equation until one side exactly matches the equation of the curve you plotted, then the solution is where that curve meets the resulting straight line.

Worked exam-style question: Having plotted $y=2(0.5)^x-1$, solve $2(0.5)^x+2x-3.5=0$.

1. Isolate the exponential term: $2(0.5)^x=-2x+3.5$.
2. Subtract 1 from both sides to match the plotted curve: $2(0.5)^x-1=-2x+2.5$.
3. Draw $y=-2x+2.5$ on the same axes; the $x$-coordinate where it crosses the curve is the solution.

• Assuming $y=\sin(2x)$ stretches the period (it halves it, to $180°$).
• Shifting $f(x+a)$ to the right instead of left.
• Forgetting to draw asymptotes for $\tan x$ or $a^x+k$ graphs.
• Stopping at the calculator's one solution for a trig equation instead of using symmetry to find the rest.