How do amplitude and period change the shape of a sine or cosine graph?
Quick answer
For y = A sin(Bx) or y = A cos(Bx), the amplitude is |A| and the period is 2π/|B|, assuming radians and no vertical shift. Amplitude sets how far the graph reaches above and below its midline; the period sets the horizontal length of one complete cycle. Find both before sketching, then mark key points across one cycle.
The question, the details, the next step.
When I look at y = A sin(Bx) or y = A cos(Bx), which coefficient changes the height and which changes the width?
The coefficient A controls vertical height: the amplitude is |A|, the distance from the midline to a peak or trough. The multiplier B inside the angle controls horizontal cycle length: in radians, the period is 2π/|B|. So a larger |A| makes the graph taller, while a larger |B| makes each cycle shorter. These formulas assume there is no vertical shift. [S1]
A negative A reflects the curve across its midline, but its amplitude stays positive because amplitude uses the absolute value. For the equations here, the midline is y = 0. [S1]
Can you compare y = 2 sin(x) and y = cos(2x) before I sketch them?
Yes. Read off A and B separately:
- For y = 2 sin(x), A = 2 and B = 1. Its amplitude is 2 and its period is 2π.
- For y = cos(2x), A = 1 and B = 2. Its amplitude is 1 and its period is π.
That matches the comparison given in OpenStax. [S1] The sine graph reaches 2 units above and below its midline, while the cosine graph reaches 1 unit above and below. The cosine completes a cycle in half the horizontal distance of the sine graph, so it repeats more frequently as x increases.
Before drawing, note the axis units: this comparison uses radians. It also has no added constant outside the sine or cosine, so neither graph is shifted vertically.
How do I turn those numbers into a sketch, rather than just writing amplitude and period?
First draw the midline y = 0 and mark the vertical extremes using the amplitude. Then divide one period into four equal horizontal intervals and use the function’s familiar cycle pattern to place the middle points.
For y = 2 sin(x), one cycle runs from x = 0 to x = 2π, with quarter-cycle spacing of π/2. Its five landmarks are (0, 0), (π/2, 2), (π, 0), (3π/2, −2), and (2π, 0).
For y = cos(2x), one cycle runs from x = 0 to x = π, with quarter-cycle spacing of π/4. Its landmarks are (0, 1), (π/4, 0), (π/2, −1), (3π/4, 0), and (π, 1). [S1]
These are plotting landmarks for the two example equations, not a substitute for checking that your equation has the same form and no extra shift. Join the points with a smooth wave, then repeat the cycle if you need a wider graph. The key contrast is visible: the first wave is taller; the second is compressed horizontally.
What should I check if my answer seems to have the right height but the wrong width?
Recheck the number multiplying x inside the parentheses or angle. Use its absolute value as B, then calculate 2π/|B|; do not use A in the period formula. Conversely, use |A| for amplitude, not B. For example, with y = −3 sin(4x), the amplitude is 3 and the period is π/2: the negative sign reflects the graph, and the angle multiplier shortens its cycle. [S1]
Also verify whether the problem uses radians. If the expression includes an added constant, that introduces a vertical shift, so the range is centered on a different midline than the unshifted examples here. The cited formulas and examples establish the amplitude and period for the stated forms; they do not give a full graphing rule for every possible phase or vertical shift. For those equations, identify all transformations before plotting.
A practical checklist
Write the equation in a clear sine-or-cosine form and identify the coefficient outside the function and the multiplier of x inside it.
Calculate amplitude as |A| and period as 2π/|B|, checking that the angle is in radians.
Mark the midline and the two vertical extremes; for the unshifted examples, the midline is y = 0.
Divide one period into four equal horizontal steps and place the appropriate sine or cosine landmarks.
Check whether a negative outside coefficient reflects the curve and whether any added constant shifts the midline.
A few more useful answers
Does a negative amplitude mean the amplitude is negative?
No. Amplitude is |A|, so it is nonnegative. A negative outside coefficient reflects the graph across its midline without changing the amplitude. [S1]
What happens to the period if the inside multiplier doubles?
For the stated radian form, doubling |B| halves the period because the period is 2π/|B|. The graph completes each cycle over a shorter horizontal interval. [S1]
Do these formulas account for a vertical shift?
The cited forms and examples assume no vertical shift. If a constant is added outside the sine or cosine, account for the changed midline separately; the supplied formula does not establish a complete rule for every transformed equation.
Sources
References used for this guide.
- [S1] OpenStax Precalculus 2e: Graphs of the Sine and Cosine Functions openstax.org · Checked