Trig Graphs and Transformations (AA SL)
Every graph of \(f(x) = a\sin(b(x+c)) + d\) or the cosine equivalent is the parent \(\sin x\) or \(\cos x\) curve stretched, compressed and shifted by the four constants \(a\), \(b\), \(c\) and \(d\). Once you can read off what each constant does, you can sketch or interpret the graph of a Ferris wheel, a tide, or any other periodic context without plotting a single point. It's part of the broader Trig Equations & Graphs topic.
20 questions on this sub-topic.
Amplitude and period
Covered under IB syllabus reference SL3.7. For \(f(x)=a\sin(b(x+c))+d\), two quantities let you sketch the graph's overall shape before worrying about the fine detail.
Amplitude
\(\text{amplitude} = |a|\)
How far the curve rises and falls above and below its midline \(y=d\). A negative \(a\) flips the graph vertically but doesn't change the amplitude itself.
Period
\(\text{period} = \dfrac{2\pi}{|b|}\)
How long the graph takes to complete one full cycle. A larger \(b\) compresses the graph horizontally, giving a shorter period.
Need the full syllabus wording and worked equations-and-graphs overview? See Trig Equations & Graphs.
Worked examples
For \(y = 4\sin(2x)\):
(a) State the amplitude.
(b) State the period.
Worked solution
(a) Amplitude \(=4.\) A1
(b) Using the period formula \(\dfrac{2\pi}{b}\): Period \(=\dfrac{2\pi}{2}\) M1 \(=\pi.\) A1
The graph of \(y=\cos x\) is transformed to \(y=\cos\!\left(x-\tfrac{\pi}{2}\right)\).
(a) Describe the transformation.
(b) State the simpler function this equals.
Worked solution
(a) Replacing \(x\) by \(x-\tfrac{\pi}{2}\) is a horizontal translation \(\tfrac{\pi}{2}\) to the right. A1
(b) A right-shift of cosine by \(\tfrac{\pi}{2}\) gives \(y\) R1 \(=\sin x.\) A1
A sinusoidal graph has a maximum at \((1, 9)\) and the next minimum at \((5, 1)\).
Find an equation of the form \(y = a\cos(b(x-c)) + d\).
Worked solution
\(d=\dfrac{9+1}{2}=5\), \(a\) M1 \(=4\). A1
Max to next min is half a period: \(5-1=4\), so period \(8\) and \(b=\dfrac{2\pi}{8}\) M1 \(=\dfrac{\pi}{4}.\) A1
Max at \(x=1\Rightarrow c=1\). So \(y\) A1 \(=4\cos\!\left(\tfrac{\pi}{4}(x-1)\right)+5.\) A1
Common mistakes
- Confusing amplitude with period. The amplitude is simply \(|a|\); the period needs the extra step of dividing \(2\pi\) by \(|b|\) - mixing these up gives a graph with the right height but the wrong width, or vice versa.
- Getting the phase shift backwards. In \(a\sin(b(x+c))+d\), a positive \(c\) shifts the graph left, not right - it's easy to assume "+c" always means moving in the positive direction.
- Ignoring the effect of a negative \(a\). When \(a<0\), the graph is reflected in its midline, so what would have been a maximum becomes a minimum and vice versa - forgetting this flips every turning point in a sketch.
Ready to practise properly?
20 trig-graph questions, marked instantly like the real exam.
Quick answers
How do you find the amplitude and period of a sine or cosine graph?
For \(f(x) = a\sin(b(x+c)) + d\), the amplitude is \(|a|\) and the period is \(\dfrac{2\pi}{|b|}\) (or \(\dfrac{360^\circ}{|b|}\) in degrees).
What does the c and d do in f(x) = a sin(b(x+c)) + d?
\(c\) shifts the graph horizontally: adding to \(x\) inside the bracket moves it left, subtracting moves it right. \(d\) shifts the whole graph vertically, raising or lowering the midline (and the maximum and minimum) by \(d\).