What Function Makes a Narrow, Bell-Shaped Peak?

If by “peaks quickly” you mean a narrow, symmetric peak, a Gaussian function is a convenient starting point. Its center sets where the maximum occurs, and its width parameter controls how quickly the curve falls away on either side.

A bell curve is only one possible model: real data may have a sloping baseline, unequal sides, or a peak that does not look Gaussian. The sections below show how to adjust the basic model and when to consider an alternative.

The Gaussian peak

Use

$$f(x)=C+A\exp\left(-\frac{(x-a)^2}{2\sigma^2}\right),$$

where $a$ is the peak’s horizontal location, $A>0$ is its height above the baseline, $C$ is the baseline, and $\sigma>0$ controls its width. The maximum occurs at $x=a$, where $f(a)=C+A$.

The parameter $\sigma$ is a scale, not the full width of the peak. At one width from the center,

$$f(a\pm\sigma)=C+A e^{-1/2}\approx C+0.607A.$$

So reducing $\sigma$ makes the peak narrower; at corresponding scaled positions, such as one $\sigma$ from the center, the sides are steeper in absolute $x$-units. The curve is smooth, and its slope at the exact maximum is zero; “peaks quickly” usually means it is narrow, not that the slope at the maximum is large.

Choose the width from a desired half-height

A useful measure is the full width at half maximum (FWHM): the distance between the two points where the curve is halfway from the baseline to its maximum. Setting $f(x)=C+A/2$ gives

$$\text{FWHM}=2\sigma\sqrt{2\ln 2}\approx 2.355\sigma.$$

If you know the desired FWHM, solve this relation for $\sigma$:

$$\sigma=\frac{\text{FWHM}}{2\sqrt{2\ln 2}}.$$

This is often easier than guessing $\sigma$ from a graph. The half-height is measured relative to $C$, not relative to zero, when the curve has a nonzero baseline.

Worked example

Suppose a peak should be centered at $x=2$, reach a maximum of $4$, and sit on a baseline of $1$. Its height above baseline is $A=3$. If you want a relatively narrow peak with $\sigma=0.4$, use

$$f(x)=1+3\exp\left(-\frac{(x-2)^2}{2(0.4)^2}\right).$$

At the center, $f(2)=4$. One width from the center, at $x=1.6$ or $x=2.4$,

$$f(x)=1+3e^{-1/2}\approx 2.82.$$

The half-height above baseline is $1+3/2=2.5$. The full width at that level is approximately

$$2.355(0.4)=0.942.$$

For a sharper peak with the same center and height, reduce $\sigma$; for a broader peak, increase it. Changing $A$ adjusts the height without changing the width, while changing $C$ shifts the whole curve vertically.

If one side is steeper

A symmetric Gaussian has the same shape on both sides of its center. To make the curve rise sharply and then trail off more gradually, give the two sides different widths:

$$f(x)=C+A\begin{cases} \exp\left(-\dfrac{(x-a)^2}{2\sigma_L^2}\right),&x<a,\\ \exp\left(-\dfrac{(x-a)^2}{2\sigma_R^2}\right),&x\ge a. \end{cases}$$

Choose a smaller $\sigma_L$ for a narrower left side and a larger $\sigma_R$ for a broader right side. The two pieces meet at the same peak, and their first derivatives are both zero there. However, their second derivatives need not match, so the curve may not have a perfectly smooth change in curvature at the center. If that matters, use a smooth asymmetric model, such as a skew-normal curve, or fit a model chosen for the process that generated the data.

Common mistakes

• Using $\sigma$ as the full width. The Gaussian FWHM is about $2.355\sigma$, not $\sigma$.
• Forgetting the baseline. If the curve sits above zero, measure the peak height and half-height relative to $C$.
• Making the peak arbitrarily narrow. A narrow curve may match a few points but miss the data between them. Fit the width to measurements or a stated requirement.
• Assuming every bell-like curve is Gaussian. A Gaussian is a useful symmetric model, not a universal law. Skewed, flat-topped, or heavy-tailed data may need another function.

If you only need a graph with a quick, symmetric peak, start with the Gaussian and tune $a$, $A$, $C$, and $\sigma$. If the left and right sides behave differently, use an asymmetric model rather than trying to fix the mismatch by changing the height or center.