In y = a sin(bx) + c, the number a changes the amplitude (how tall the wave is) and the number c translates the whole graph up or down. They do different jobs, and you need both to write the maximum, minimum and range. This appears when a question asks for the greatest or least value of an expression, or asks you to find a, b and c from a graph.
It builds on tracking period and phase shift and on the graph shapes in trigonometric equations and graphs.
How do you separate the two changes?
- Midline first. The constant added at the end, c, sets the horizontal line y = c that the wave oscillates around.
- Amplitude next. The multiplier a stretches the wave vertically. The amplitude is |a|, a positive number.
- Maximum and minimum. Maximum = c + |a| and minimum = c − |a|. The range of y is every value from the minimum to the maximum.
- A negative a reflects the wave. It turns peaks into troughs but leaves the amplitude and the midline the same.
Horizontal translation is a different change again. It lives inside the bracket, as in the shift you read in the previous lesson, and it does not alter the maximum or minimum values.
Worked example
For y = 2 − 3 sin x with 0° ≤ x ≤ 360°, state the amplitude, the maximum and minimum values and where they occur.
Step 1, rewrite with the constant last: y = −3 sin x + 2.
Step 2, midline and amplitude: c = 2 gives the midline y = 2. The multiplier is −3, so the amplitude is |−3| = 3.
Step 3, maximum and minimum: maximum = 2 + 3 = 5, and minimum = 2 − 3 = −1. The range is −1 ≤ y ≤ 5.
Step 4, where they occur: the maximum needs sin x = −1, which is x = 270°. The minimum needs sin x = 1, which is x = 90°.
Check: at x = 270°, y = 2 − 3(−1) = 5. At x = 90°, y = 2 − 3(1) = −1. At x = 0°, y = 2, which lies on the midline.
The mistake to watch for
A common slip is to quote the maximum value as the amplitude, or to forget the sign.
Mistaken answer: “Amplitude is 5, because that is the highest y-value.”
The 5 is the maximum, not the amplitude. It includes the lift of 2 from the vertical shift.
The correction is to measure from the midline, not from zero. The amplitude is 5 − 2 = 3.
Another check is (maximum − minimum) ÷ 2 = (5 − (−1)) ÷ 2 = 3. If the graph were not moved upward, the amplitude and the maximum would match, which is why the confusion is easy.
Check yourself
1. State the maximum and minimum values of y = 4 sin x + 2.
Show answer
Midline y = 2, amplitude 4. Maximum = 2 + 4 = 6. Minimum = 2 − 4 = −2.
Maximum 6, minimum −2
2. A graph of y = a cos x + c, with a > 0, has a maximum value of 9 and a minimum value of 1. Find a and c.
Show answer
a = (9 − 1) ÷ 2 = 4. c = (9 + 1) ÷ 2 = 5. Check: 5 + 4 = 9 and 5 − 4 = 1.
a = 4, c = 5
3. For y = 1 − 2 cos 3x, state the amplitude, the period and the maximum value.
Show answer
The amplitude is |−2| = 2. The period is 360°/3 = 120°. The midline is y = 1. The maximum occurs when cos 3x = −1, giving y = 1 + 2 = 3.
Amplitude 2, period 120°, maximum 3
Where this leads next
With all five skills in place, test them together in the mixed practice set, and use the mistake log and retest queue where it fits. The quadratic structure explorer shows the turning point of a quadratic beside its graph, which is a helpful way to practise reading values from pictures. The non-calculator working trainer supports the arithmetic.
If graph questions still feel like a list of rules, online one-to-one Additional Mathematics tuition lets a teacher ask you to explain each change in your own words.