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Determining trigonometric functions given their graphs

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Determining Trig Functions Given Their Graphs

This lesson shows how to work backward from a trig graph to its equation. You will learn to identify amplitude, period, midline, phase shift, and reflection, then combine them into a sine or cosine formula, with fully worked examples.

What it means to determine a trig function from its graph

Up to now you have probably practiced going from an equation to a graph: given something like \(y = 2\sin(x)\), you sketch the curve. This lesson works in the opposite direction. You are handed a picture of a wave, and your job is to figure out which equation produced it. This skill shows up constantly in physics, engineering, and any situation where periodic data (tides, sound waves, seasonal temperatures) is modeled with a trig function.

Every graph you will meet in this lesson is a transformed sine or cosine curve, so it helps to already be comfortable with the basic shapes from the sine graph lesson and with how shifting, stretching, and reflecting a curve changes its equation in the transformations of trigonometric functions lesson.

Any sine or cosine graph can be written in the general form \(y = A\sin(B(x - C)) + D\) or \(y = A\cos(B(x - C)) + D\). Instead of guessing, you read each of the four constants directly from the picture:

Feature What it looks like Amplitude, |A| Distance from the midline up to a peak Midline, D The horizontal line the wave oscillates around Period, 2π/B Horizontal length of one full wave cycle Phase shift, C How far the wave is shifted left or right

There is also a fifth thing to watch for that is not a separate number: whether the curve is reflected over its midline. If a curve that "should" start by rising instead starts by falling at the same point, the leading coefficient \(A\) is negative.

  1. Find the midline \(D\): average the maximum and minimum \(y\)-values, \(D = \dfrac{\max + \min}{2}\).
  2. Find the amplitude: \(A = \dfrac{\max - \min}{2}\). Keep it positive for now.
  3. Find the period: measure the horizontal distance for one complete cycle, then solve \(\)period\( = \dfrac{2\pi}{B}\) for \(B\).
  4. Decide between sine and cosine as your starting shape, based on where the curve crosses the midline or hits an extreme value.
  5. Find the phase shift \(C\) by comparing where your chosen parent function normally starts to where the actual graph starts.
  6. Check whether the curve is flipped upside down; if so, make \(A\) negative.

Suppose a graph crosses its midline at \(y = 0\) while rising, reaches a maximum of \(2\), a minimum of \(-2\), and completes one full cycle over a length of \(2\pi\).

Graph of y = 2 sin(x), a sine curve with amplitude 2 and period 2 pi Plot of y = 2*sin(x) for x in [-6.2832, 6.2832] -6 -4 -2 0 2 4 6 -2 -1 0 1 2 x (radians) y midline crossing, rising maximum minimum
A sine-shaped curve rising through the origin, reaching a maximum near \(x = \tfrac{\pi}{2}\).
  • Midline: \(D = \dfrac{2 + (-2)}{2} = 0\)
  • Amplitude: \(A = \dfrac{2 - (-2)}{2} = 2\)
  • Period is \(2\pi\), so \(B = \dfrac{2\pi}{2\pi} = 1\)
  • The curve rises through the midline at \(x = 0\), exactly where an ordinary sine curve does, so no shift is needed and there is no reflection.

Putting the pieces together gives \(y = 2\sin(x)\).

Now suppose a graph has a maximum of \(3\), a minimum of \(-1\), it reaches its lowest point at \(x = 0\), and one full cycle again spans \(2\pi\).

Graph of y = -2 cos(x) + 1, a reflected cosine curve shifted up 1 unit Plot of y = -2*cos(x) + 1 for x in [-6.2832, 6.2832] -6 -4 -2 0 2 4 6 -1 0 1 2 3 x (radians) y minimum (reflected) maximum
A cosine-shaped curve that starts at a minimum instead of a maximum, showing a reflection.
  • Midline: \(D = \dfrac{3 + (-1)}{2} = 1\)
  • Amplitude: \(A = \dfrac{3 - (-1)}{2} = 2\)
  • Period is \(2\pi\), so \(B = 1\)
  • An ordinary cosine curve starts at a maximum at \(x = 0\); this graph starts at a minimum instead, which means it is reflected, so \(A\) is negative rather than positive.

Combining these gives \(y = -2\cos(x) + 1\). Notice that the same wave could also be described using a shifted sine equation; a phase shift can turn a cosine curve into an equivalent sine curve and vice versa, so there is often more than one correct answer.

A few mistakes come up again and again when students determine a trig equation from a graph:

  • Forgetting to subtract the midline before reading the amplitude, which produces a value that is too large.
  • Measuring the period from a peak to the next peak but reading the x-axis in degrees when the intended answer is in radians (or vice versa).
  • Assuming every graph must be written with sine; cosine, and for other shapes, tangent, secant, cosecant, or cotangent, are equally valid starting functions.
  • Missing a reflection because only the amplitude was checked and the direction of the curve at the midline crossing was ignored.

If the curve you are given has vertical asymptotes rather than smooth peaks and valleys, it is not a sine or cosine graph at all. In that case check whether it matches a tangent graph, a secant, cosecant, or cotangent shape instead, since those functions are read differently and are covered in their own lessons.

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