Transformations of a Trig Graph

In $y = a\sin x + c$, the $a$ sets the amplitude (vertical stretch) and the $c$ shifts the midline up or down. Nothing multiplies $x$ here, so the period stays $360^\circ$.

Sketch the graph of $y = 2\sin x + 1$ for $0^\circ \le x \le 360^\circ$, stating the amplitude, the period, and the range. [5]

The working

Step 1, read the amplitude and midline from $y = 2\sin x + 1$:

• amplitude $= 2$,
• midline $y = 1$ (the graph oscillates about $y = 1$). (B1, B1)

Step 2, find the range. The curve rises $2$ above and falls $2$ below the midline: $\text{max} = 1 + 2 = 3, \qquad \text{min} = 1 - 2 = -1, \qquad \text{so } -1 \le y \le 3. \quad \text{(A1)}$

Step 3, period. Since $x$ has coefficient $1$ inside the sine: $\text{period} = 360^\circ. \quad \text{(B1)}$

Step 4, key points for the sketch:

• maximum $3$ at $x = 90^\circ$,
• minimum $-1$ at $x = 270^\circ$,
• crosses the midline $y = 1$ at $x = 0^\circ, 180^\circ, 360^\circ$. (A1)

One full wave, shifted up so it sits between $-1$ and $3$.

Where the marks are won and lost

• The $+1$ moves the whole curve up, so the range is $-1 \le y \le 3$, not $-2 \le y \le 2$.
• The period is still $360^\circ$: only a coefficient of $x$ (like $\sin 2x$) would change it.
• Mark the max at $90^\circ$ and min at $270^\circ$; a sine graph starts by rising from its midline.

Common mistakes

• Giving the range as $-2 \le y \le 2$ (ignoring the $+1$ shift).
• Changing the period even though $x$ is unmodified.
• Drawing a cosine shape (starting at a maximum) instead of a sine.

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