How Do You Graph f(x) = a sin(b(x+c)) + d?

To graph $f(x)=a\sin(b(x+c))+d$, identify the amplitude, period, horizontal translation, and midline, then plot five key points one quarter-period apart across a complete cycle. This method belongs to IB Mathematics: Analysis and Approaches SL 3.7: circular functions and works analytically on Paper 1, with GDC checking available on Paper 2.

Transformation Features

Begin with the parent graph $y=\sin x$, whose values over one cycle follow $0,1,0,-1,0$. These correspond to the midline, maximum, midline, minimum, and midline.

The maximum is $d+|a|$, the minimum is (d-|a|), and the range is

$$d-|a|\leq f(x)\leq d+|a|.$$

The starting point of an unreflected sine cycle occurs where the inner angle is zero, so $x=-c$.

Worked Example

Consider

$$f(x)=2\sin\left(3\left(x+\frac{\pi}{6}\right)\right)-1.$$

Here, the amplitude is $2$, the period is $\frac{2\pi}{3}$, and the quarter-period is $\frac{\pi}{6}$. The graph shifts left by and has midline , maximum , and minimum .

These points reproduce the transformed sine pattern. Plot them, join them with a smooth curve, and extend periodically if a wider domain is requested. Quarter-period spacing is useful because each step changes the parent sine value to the next value in $0,1,0,-1,0$.

Exam Technique

For a sketch, label both axes, the midline, maximum, minimum, and enough $x$-coordinates to demonstrate the period. Show your calculation of $T$ before plotting. A common misconception is that $x+c$ shifts right; it shifts left because $x+c=0$ gives . Also avoid using without modulus when : a period must be positive.