Graphing the Sine Function

The sine function produces the most recognizable shape in mathematics after the straight line: a smooth wave that rises to a crest, falls through a trough, and repeats forever in both directions. Every sine wave you will ever be asked about is the parent wave y = sin(x) stretched, squeezed, or slid, and exactly four numbers control those changes. Once you can name what each number does, a question that looks like a wall of symbols becomes a matter of reading off four values.

The two tests treat this topic differently. The digital SAT keeps trigonometry light: it asks you to convert between radians and degrees, to use the unit circle to find a sine or cosine value, and occasionally to recognize a translated trig graph, but it rarely asks you to compute a period. The ACT asks directly. Expect an ACT question that hands you an equation like y = 3 sin(2x) - 1 and asks for the amplitude or the period, or one that shows a wave and asks which equation produced it. That makes this lesson a guaranteed point or two on the ACT and a reliable one on the SAT.

The common failure here is treating the four transformation numbers as four unrelated facts to memorize. They are not. Each one answers a physical question about the wave, namely how tall it is, how long one cycle takes, where the cycle starts, and where its center line sits. Ask those four questions of any equation and the graph draws itself.

Start with the parent function y = sin(x). Its outputs are never larger than 1 or smaller than -1, and it repeats every 2π units, so one full cycle runs from x = 0 to x = 2π. Five points define that cycle and you should know them cold: the wave starts on the axis at (0, 0), rises to a crest at (π/2, 1), returns to the axis at (π, 0), falls to a trough at (3π/2, -1), and comes back to the axis at (2π, 0). Plot those five points, join them with a smooth curve, and copy the shape left and right forever. In decimal terms, a full cycle is about 6.28 units long, so the crest sits near x = 1.57 and the trough near x = 4.71.

Why those values? A sine is a coordinate on the unit circle, the circle of radius 1 centered at the origin. Rotate counterclockwise from the positive x-axis through an angle θ, and the point you reach has coordinates (cos θ, sin θ). The sine is the height of that point. At θ = 0 the point is on the x-axis, so the height is 0; at θ = π/2 you are at the top of the circle, height 1; at θ = π you are back on the axis, height 0; at θ = 3π/2 you are at the bottom, height -1. Since a full turn is 2π radians, the heights repeat every 2π, which is exactly why the period of sin(x) is 2π. The same picture explains the SAT's radian and degree questions: 2π radians is one full turn, so 2π radians equals 360 degrees and π radians equals 180 degrees, and the sine of an angle is the same number whether you name the angle in degrees or radians.

The general sine function is y = A sin(B(x - C)) + D, and each letter answers one question about the wave. The amplitude is |A|, the distance from the center line up to a crest or down to a trough. It is a distance, so it is never negative; a negative A flips the wave upside down, so it starts by falling instead of rising, but the height of the wave is unchanged. The full vertical span of the wave, from trough to crest, is 2|A|. So y = 2 sin(x) rises to 2 and falls to -2, twice as tall as the parent wave, while y = (1/2) sin(x) stays between -1/2 and 1/2.

The period is the horizontal length of one full cycle, and it is controlled by B through the formula period = 2π / B. The reason is that the parent wave completes a cycle when its argument runs from 0 to 2π, so B(x - C) must grow by 2π, which takes a change in x of 2π / B. A larger B squeezes more cycles into the same space and shortens the period. So y = sin(2x) has period 2π / 2 = π and completes two full cycles between 0 and 2π, while y = sin(x / 2) has period 2π / (1/2) = 4π and needs twice as much room as the parent. Reading in the other direction, if you know the period from a graph, then B = 2π / period.

The phase shift is C, and it slides the whole wave horizontally: C units to the right if the equation reads (x - C), and C units to the left if it reads (x + C). The sign feels backward, and the way to trust it is to ask where the cycle begins. The parent wave starts a cycle where its argument equals zero, and B(x - C) equals zero when x = C, so the cycle now begins at x = C instead of at x = 0. When an equation is written as y = A sin(Bx + E) + D with the coefficient not factored, the shift is -E / B, which is the same thing as factoring B out of the parentheses first. On both tests a phase shift usually appears only in the multiple-choice matching questions, where you must tell sin(x - π/2) from sin(x + π/2), so it is worth being precise about the sign.

The vertical shift is D, and it moves the center line of the wave, called the midline, from y = 0 to y = D. Amplitude and midline together give the maximum and minimum of any sine wave with no graphing at all: the maximum value is D + |A| and the minimum value is D - |A|. The reason is that sin of anything lies between -1 and 1, so A sin(anything) lies between -|A| and |A|, and adding D slides that whole interval up or down. Running the formulas backward is how you read a graph: the midline is halfway between the maximum and minimum, $D = \frac{\text{max} + \text{min}}{2}$, and the amplitude is half the distance between them, $|A| = \frac{\text{max} - \text{min}}{2}$.

In the interactive graph below, drag the amplitude slider and watch the crests and troughs move away from the x-axis while the crossing points stay fixed. Drag the period slider and watch the crossings spread apart or bunch together while the height never changes, which is the visual proof that amplitude and period are independent. Drag the phase shift and notice that the entire wave slides sideways as a rigid shape, and drag the vertical shift and watch the midline rise and fall carrying the wave with it. The app writes the sine as A sin(2π x / period + phase) + shift, so the period slider is the actual length of one cycle in x-units, and the parent wave corresponds to a period of about 6.28.

Using the coefficient of x as the period. In y = sin(4x) the period is 2π / 4 = π/2, not 4. The coefficient tells you how many cycles fit into 2π, so a bigger number means a shorter period. Every time you write down a period, make sure you have divided into 2π, and sanity check that a coefficient bigger than 1 gave a period smaller than 2π.

Taking the full height of the wave as the amplitude. A wave that runs from -1 to 5 has amplitude 3, not 6, because amplitude is measured from the midline to a crest, not from trough to crest. Compute (max - min) / 2 every time, and pair it with (max + min) / 2 for the midline; the two numbers together should rebuild the original maximum and minimum.

Reading the phase shift without factoring. In y = sin(2x - π/2) the shift is π/4, not π/2, because 2x - π/2 = 2(x - π/4). Factor the coefficient of x out of the parentheses before you read C. Then remember the sign is backward: (x - π/4) is a shift to the right, and (x + π/4) is a shift to the left.

Letting a negative amplitude change the maximum and minimum. In y = -4 sin(x) + 1 the amplitude is 4, the midline is 1, and the wave still reaches 1 + 4 = 5 and 1 - 4 = -3. The negative sign only reflects the wave across the midline so it falls first instead of rising. Use |A| in the max and min formulas and treat the sign as a separate fact about direction.

Mixing amplitude and vertical shift. Amplitude controls how tall the wave is and vertical shift controls where its center line sits; a wave can be tall and low, or short and high, and the two numbers are independent. When you read a graph, find the midline first, because a shifted wave no longer crosses the x-axis at its key points and measuring amplitude from the x-axis instead of from the midline will give the wrong answer.

Forgetting that radians and degrees describe the same angle. The period of sin(x) is 2π when x is in radians and 360 when x is in degrees, and sin(π/2) and sin(90 degrees) are both 1. On the SAT, convert using π radians = 180 degrees before you compare an angle to the key points, and make sure your calculator is in the mode the problem uses.