Graphing Trig Functions: Amplitude, Period & Midline

Sine and cosine don't just give single values — graphed over all angles, they trace smooth, repeating waves. The beauty is that a handful of numbers in the equation completely control the wave's shape: one stretches it taller, one squeezes the cycles closer together, and one slides it up or down. Learn what amplitude, period, and midline mean and where to read them, and you can sketch or recognize any sine or cosine graph.

This guide decodes the trig graph, building on the unit circle, with worked and practice problems matched to real ACT difficulty at Northside Tutoring.

Why Trig Graphs Matter

Trig graphs appear on the ACT and are central to Pre-Calculus, where they model anything periodic — sound, tides, seasons. Reading amplitude and period off an equation (or a graph) is a quick, learnable skill. (The SAT touches trig graphs only lightly; this is more an ACT/Pre-Calc topic.)

The Basic Wave

The general form is y = a·sin(bx) + d (cosine works the same way). Each constant controls one feature of the wave.

Amplitude (a)

The amplitude is |a| — the distance from the midline up to a peak (or down to a trough). It sets the wave's height. In y = 3sin(x), the amplitude is 3, so the wave rises 3 above and falls 3 below its midline. A negative a flips the wave upside down but doesn't change the amplitude.

Period (from b)

The period is the length of one full cycle, controlled by b:

A larger b squeezes more cycles into the same space (shorter period). In y = sin(2x), the period is 2π/2 = π — twice as fast as the basic sine wave.

Midline (d)

The midline is the horizontal line the wave oscillates around, set by d (a vertical shift). In y = sin(x) + 4, the whole wave slides up so it centers on y = 4. The max is then midline + amplitude, and the min is midline − amplitude.

Sine vs. Cosine

Sine and cosine have the same shape, just shifted. Sine starts at the midline and rises; cosine starts at a maximum. That's the main difference to recognize when matching a graph to an equation — check where the curve begins.

Where You'll See This — Test by Test

No reference sheet covers trig graphs. The ACT tests reading amplitude and period; the SAT tests trig graphs only lightly. It's core Pre-Calculus, building on the unit circle. Beyond the SSAT.

Watch the Lesson

Sometimes a diagram needs a voice. In the short video below, one of our Northside tutors walks through the core idea and works through test-style problems in real time.

Worked Example Problems

These problems are calibrated to the difficulty you'll actually see on test day. Try each one before opening the solution.

Common Mistakes to Avoid

Practice Problems — You Try

Three problems below. Work each before checking the solution.

The Northside Method — How We Teach This 1-on-1

Reading a blog is a great starting point. But there's a meaningful gap between understanding a concept and reflexively applying it under timed conditions. That gap is exactly what our tutors close.

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