Amplitude is half the difference between a sinusoidal function's maximum and minimum values. For f(θ) = a sin(b(θ - c)) + d, the amplitude is |a| (always positive), and the parent functions sin θ and cos θ each have an amplitude of 1.
Amplitude measures how far a sinusoidal function swings above and below its midline. The CED defines it as half the difference between the maximum and minimum values of the function. If a sine curve peaks at 7 and bottoms out at 1, the amplitude is (7 - 1)/2 = 3. The midline sits at y = 4, and the wave reaches 3 units above it and 3 units below it.
In equation form, when you write a sinusoidal function as f(θ) = a sin(b(θ - c)) + d or g(θ) = a cos(b(θ - c)) + d, the amplitude is |a|. The absolute value matters. A negative value of a flips the graph over its midline (a reflection), but it doesn't change how tall the wave is. So g(θ) = -5cos(3θ) + 2 has an amplitude of 5, not -5. Graphically, amplitude controls the vertical stretch or compression of the parent sine or cosine curve, which both start with amplitude 1.
Amplitude lives in Topic 3.5 (Sinusoidal Functions) in Unit 3 and directly supports learning objective 3.5.A, identifying key characteristics of the sine and cosine functions. It's one of the four parameters (amplitude, period, phase shift, midline) you need to read off a graph, pull out of an equation, or build into a model. Sinusoidal functions are the workhorse of Unit 3 because they model anything that cycles, like tides, temperatures, and Ferris wheels. Amplitude answers the question "how big are the swings?" In a modeling context, that's the difference between high tide and the average water level, or between the hottest day and the yearly average temperature. If you can't extract amplitude quickly, almost every sinusoidal question in Unit 3 slows down.
Keep studying AP Precalculus Unit 3
Period (Unit 3)
Period and amplitude are the two sizes of a wave, but they measure different directions. Amplitude is the vertical reach from the midline, while period is the horizontal length of one full cycle. In f(θ) = a sin(bθ), |a| controls amplitude and b controls period, and changing one has zero effect on the other.
Frequency (Unit 3)
Frequency is the reciprocal of period, so it tells you how many cycles fit in one unit, not how tall those cycles are. A wave can oscillate fast with a tiny amplitude or slowly with a huge one. Keeping amplitude (height) and frequency (speed of repetition) separate in your head prevents a lot of wrong answers.
Phase Shift (Unit 3)
Phase shift slides the wave left or right without touching its height. It's why cos θ = sin(θ + π/2) works. Sine and cosine are the same shape with the same amplitude of 1, just shifted horizontally. When you decode f(θ) = 3cos(2(θ - π/8)) + 1, the 3 is amplitude and the π/8 is phase shift, and they never interfere with each other.
Amplitude shows up in multiple-choice questions in two main directions. Going equation-to-value, you'll see stems like "What is the amplitude of g(θ) = -5cos(3θ) + 2?" where the answer is 5 because amplitude is |a|. The -5 is the classic trap; picking -5 means you forgot the absolute value. Going description-to-equation, you'll be asked to build a function, like writing a cosine curve with amplitude 1 and midline y = -2.5. Questions also bundle amplitude with period and phase shift, asking you to identify all three from one function like f(θ) = 3cos(2θ - π/4) + 1, which forces you to factor out the 2 before reading the phase shift but lets you grab the amplitude of 3 immediately. One more trap worth knowing is that tangent functions have a vertical dilation factor A, but tangent has no maximum or minimum, so it has no amplitude. Don't call the 2 in 2tan(3x + π) - 5 an amplitude.
Amplitude and midline both come from the max and min, but they answer different questions. The midline is the average of the max and min, the horizontal center line the wave oscillates around, and it comes from the d in a sin(b(θ - c)) + d. Amplitude is half the difference between max and min, the distance from that center line up to a peak, and it comes from |a|. For g(θ) = -5cos(3θ) + 2, the midline is y = 2 and the amplitude is 5. Mixing up "average of max and min" with "half the difference of max and min" is one of the most common Topic 3.5 errors.
Amplitude is half the difference between a sinusoidal function's maximum and minimum values, so a wave with max 7 and min 1 has amplitude 3.
In the form f(θ) = a sin(b(θ - c)) + d, the amplitude is |a|; a negative a reflects the graph over the midline but the amplitude stays positive.
The parent functions sin θ and cos θ both have an amplitude of 1, and multiplying by a stretches or compresses the graph vertically.
Amplitude measures vertical distance from the midline to a peak, while period measures the horizontal length of one cycle; they are completely independent.
Tangent functions have no amplitude because they have no maximum or minimum, even though they can have a vertical dilation factor.
In modeling problems, amplitude represents the size of the swing away from average, like how far high tide rises above the mean water level.
It's half the difference between the function's maximum and minimum values, which equals |a| in f(θ) = a sin(b(θ - c)) + d. The basic sine and cosine functions both have an amplitude of 1.
No. Amplitude is a distance, so it's always positive. If the coefficient a is negative, like in -5cos(3θ) + 2, the graph is reflected over its midline but the amplitude is still |−5| = 5.
The midline is the average of the max and min (the center line, given by d), while the amplitude is half the difference between them (the distance from that line to a peak, given by |a|). For -5cos(3θ) + 2, the midline is y = 2 and the amplitude is 5.
No. Tangent has no maximum or minimum value, so amplitude isn't defined for it. In f(x) = 2tan(3x + π) - 5, the 2 is a vertical dilation factor, not an amplitude.
Read off the maximum and minimum y-values, subtract them, and divide by 2. Equivalently, find the midline first, then measure the vertical distance from the midline up to a peak.