Quarry School

Amplitude measures height, not direction

Explain it like I am five

Picture a child on a swing that travels 2 meters forward and 2 meters back from its resting spot. That 2 meters is the swing's amplitude: the distance from the middle to one far end. The full 4 meters from end to end is twice the amplitude.

Take y = −6 sin x. The number multiplying the whole wave, A, is −6. Plain sine runs from −1 to 1, so multiply: when sin x = 1, y = −6 × 1 = −6, and when sin x = −1, y = −6 × (−1) = 6. The wave reaches 6 and −6, so the amplitude is |−6| = 6. The bars mean absolute value: a number's size without its sign.

Check with the heights: amplitude = highest−lowest2 = 6−(−6)2 = 122 = 6. The minus sign in A shrank nothing. It flipped the wave upside down, so it heads down first instead of up. A distance cannot be negative, which is why amplitude uses |A|.

In plain words

Imagine a swing moving the same distance to either side of its resting place. The distance from the resting place to one farthest position tells you how large the motion is. For a wave, that distance is the Amplitude. You measure from the middle to the top, or from the middle to the bottom. You do not measure all the way from bottom to top. The number A stretches or shrinks every height before anything is added outside. A negative A also turns the wave upside down. That flip changes which part comes first, but a distance cannot become negative.

π/2π3π/22π−4−3−2−11234amplitude 3range
The amplitude is the positive vertical distance from the middle to an extreme, half the full height.
Reminder
  • Absolute value. |−3| = 3 because −3 is 3 units from 0.
  • Signed multiplication. −3 × (−1) = 3; −3 × 1 = −3.
  • Subtracting negatives. 3 − (−3) = 3 + 3 = 6.
Why it works. The basic sine and cosine outputs reach 1 and −1. Multiplying by A puts the two extreme outputs at A and −A before a vertical addition. Those positions are each |A| from 0. Their separation is 2|A|, so half the maximum-minus-minimum is |A|. Adding the same B to both heights leaves their difference unchanged. A negative multiplier swaps the extremes, which is why we keep the sign of A separate from the amplitude.
RuleRule: amplitude = |A| = ymax−ymin2. It is the distance from the midline to an extreme, not the full height. Negative A reflects the wave vertically.
The same idea, five ways
Say it

Amplitude is the distance from the middle of the wave to a peak.

Write it

The amplitude equals the absolute value of the outside multiplier.

In math
  • Amplitude = |A|
  • |A| = ymax−ymin2
  • Graph words: half the height from trough to peak.
Like

Measure from a swing's rest position to one farthest reach.

See it
π/2π3π/22π−4−224amplitude 3
The amplitude is the positive vertical distance from the middle to an extreme, half the full height.
The same idea, other ways
As a height marker

The amplitude marker covers only half the full bottom to top span.

π/2π3π/22π−4−224amplitude 3
The amplitude is the positive vertical distance from the middle to an extreme, half the full height.
With a copier

Doubling a picture doubles its heights. Flipping the enlarged page does not change its size. Positive 3 and negative 3 both give amplitude 3.

.1Positive A

Positive A changes the distance without reversing the heights.

  • Rule: A = 2 gives amplitude 2.
π/2π3π/22π−22
Follow the horizontal input to the wave, then read its height.
The same idea, five ways
Say it

Positive A

Write it

Positive A changes the distance without reversing the heights.

In math
  • Rule: A = 2 gives amplitude 2.
Like

Enlarge a photo to double the distance from its center to its edges.

See it
π/2π3π/22π−22
Follow the horizontal input to the wave, then read its height.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: 2 sin x reaches amplitude 4 because its total height is 4.
The full span measures two equal distances from the center.
✓ Instead: The amplitude is 2, half the span.
Tips and tricks
  • Tip: Say what the formula changes before calculating.
.2Negative A

A negative multiplier changes distance and reverses which original heights point upward. This flip is a Reflection.

  • Rule: A = −3 gives amplitude 3, with a vertical reflection.
π/2π3π/22π−4−224
Follow the horizontal input to the wave, then read its height.
The same idea, five ways
Say it

Negative A

Write it

A negative multiplier changes distance and reverses which original heights point upward.

In math
  • Rule: A = −3 gives amplitude 3, with a vertical reflection.
Like

Turn the enlarged photo upside down; its size stays the same.

See it
π/2π3π/22π−4−224
Follow the horizontal input to the wave, then read its height.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: y = −3 sin x has amplitude −3.
A distance from the middle cannot be negative. The minus reflects the graph.
✓ Instead: A = −3, amplitude = |−3| = 3.
Tips and tricks
  • Tip: Say what the formula changes before calculating.
.3Amplitude from maximum and minimum

Measure the total span, then take half because the center splits it into equal distances.

  • Rule: amplitude = maximum−minimum2.
π/2π3π/22π24681012amplitude 4
The amplitude is the positive vertical distance from the middle to an extreme, half the full height.
The same idea, five ways
Say it

Amplitude from maximum and minimum

Write it

Measure the total span, then take half because the center splits it into equal distances.

In math
  • Rule: amplitude = maximum−minimum2.
Like

Measure the distance from the bottom of a wheel to its top, then halve it.

See it
π/2π3π/22π24681012amplitude 4
The amplitude is the positive vertical distance from the middle to an extreme, half the full height.
Worked exampleRecover amplitude from two heights

A wave has maximum 10 and minimum 2. Find its amplitude. You want the distance from the middle to either end.

π/2π3π/22π24681012amplitude 4
The amplitude is the positive vertical distance from the middle to an extreme, half the full height.
  1. Total height = 10 − 2 = 8.Subtract the lower height from the upper height to measure the entire span.
  2. Amplitude = 82 = 4.The wave has equal height on both sides of its middle.
  3. The middle is 10+22 = 6; 10 − 6 = 4 and 6 − 2 = 4.A midpoint check independently confirms the distance.
Answer
Amplitude = 4
Check The two distances from 6 both equal 4, rather than the full span 8.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: maximum 10 and minimum 2 give amplitude 6.
6 is the average height, not half the separation.
✓ Instead: Amplitude = (10 − 2) ÷ 2 = 4.
Tips and tricks
  • Tip: Say what the formula changes before calculating.
Strategy: step by step
  1. Read A, the entire outside multiplier, including its sign.
  2. Use |A| for the amplitude.
  3. If heights are given instead, subtract minimum from maximum and divide by 2.
  4. Keep A's sign when drawing which part of the cycle rises or falls.
Strategy
Strategy: find amplitude
1
Do you have the equation?
YesRead the outside multiplier A and take |A|.
NoRead the maximum and minimum, subtract, and divide by 2.
  1. From an equation, use the absolute value of A.
  2. From a graph or data, measure the whole height and halve it.
  3. Check that your answer is nonnegative.
Worked exampleAmplitude when the outside multiplier is negative

Find A and the amplitude of y = −9 sin(π2x) + 2. The midline is y = 2, and one cycle runs from x = 0 to x = 4, with lowest point (1, −7) and highest point (3, 11). Confirm the amplitude from those two heights, then say whether the graph falls or rises first as it leaves (0, 2). The figure shows the midline with the five key points (0, 2), (1, −7), (2, 2), (3, 11) and (4, 2).

−2246−8−6−4−224681012(0, 2)(1, −7)(2, 2)(3, 11)(4, 2)
The horizontal line is the midline y = 2. The five marked points are the key points of one cycle from x = 0 to x = 4: (0, 2), (1, −7), (2, 2), (3, 11) and (4, 2).
  1. Read the outside multiplier of y = −9 sin(π2x) + 2: A = −9.A is the entire number multiplying the sine, sign included. The π2 inside the sine and the + 2 added at the end are not part of it.
  2. Take the absolute value: amplitude = |A| = |−9| = 9.Amplitude is a distance from the midline to an extreme, and a distance cannot be negative, so the sign is dropped when measuring the size of the wave.
  3. Confirm with the heights. The highest point (3, 11) gives ymax = 11 and the lowest point (1, −7) gives ymin = −7, so amplitude = ymax−ymin2 = 11−(−7)2 = 182 = 9.Subtracting the minimum from the maximum gives the full height, 18, which is two amplitudes: one below the midline and one above. Dividing by 2 leaves one amplitude, and it matches |A|.
  4. Measure from the midline y = 2: up to the top is 11 − 2 = 9, and down to the bottom is 2 − (−7) = 9.The amplitude runs from the midline to an extreme, not from bottom to top, so each of these distances must be 9. The full height 18 is not the amplitude.
  5. Bring the sign back for the shape. At x = 1, y = −9 sin(π2) + 2 = −9 × 1 + 2 = −7, so the graph leaves (0, 2) going down to (1, −7), comes back to (2, 2), climbs to (3, 11) and returns to (4, 2).With A = 9 the graph would rise first, to (1, 11). The negative A reflects the wave vertically across the midline, so the fall comes first and the rise second. Only the direction changes; the size stays 9.
Answer
A = −9, and the amplitude is |−9| = 9 (not −9, and not 18). The heights agree: 11−(−7)2 = 182 = 9, so the full height 18 is twice the amplitude. Because A is negative, the graph falls first, from (0, 2) to its lowest point (1, −7), and rises to its highest point (3, 11) afterwards.
Check Since sin(π2x) stays between −1 and 1, −9 sin(π2x) stays between −9 and 9, so y stays between 2 − 9 = −7 and 2 + 9 = 11. Substituting confirms both ends are reached: at x = 1, sin(π2) = 1 gives y = −9 × 1 + 2 = −7, and at x = 3, sin(3π2) = −1 gives y = −9 × (−1) + 2 = 11. The formula gives the same lowest and highest heights as the problem, and 11 − (−7) = 18 = 2 × 9, so the amplitude is 9.

Work to write

  1. A = −9
  2. amplitude = |A| = |−9| = 9
  3. ymax−ymin2 = 11−(−7)2 = 182 = 9
  4. 11 − 2 = 9 and 2 − (−7) = 9
  5. A is negative, so the graph falls first from (0, 2) to (1, −7), then rises to (3, 11)

A = −9, and the amplitude is |−9| = 9 (not −9, and not 18). The heights agree: 11−(−7)2 = 182 = 9, so the full height 18 is twice the amplitude. Because A is negative, the graph falls first, from (0, 2) to its lowest point (1, −7), and rises to its highest point (3, 11) afterwards.

Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: y = −3 sin x has amplitude −3.
A distance from the middle cannot be negative. The minus reflects the graph.
✓ Instead: A = −3, amplitude = |−3| = 3.
Tips and tricks
  • Tip: Memory device: distance drops the sign.
  • Tip: If the total height is 8, the amplitude is 4.
Trap. Do not report maximum-minus-minimum as the amplitude. It contains two amplitude distances.
Keep in mind
  • Amplitude is half the top-to-bottom distance: a wave from 2 to 8 has amplitude 8−22 = 3, not 6.
  • Amplitude is never negative: y = −6 sin x has amplitude 6, and its minus sign only flips the wave.
  • Adding a number outside moves the wave up or down but keeps its amplitude: y = 2 sin x + 7 still has amplitude 2, swinging from 5 to 9.
  • Only the number multiplying the whole wave sets the amplitude: y = sin(8x) has amplitude 1, since the 8 sits inside.
Memory hookAmplitude is a swing's reach: middle to one end, never end to end. Take half of top minus bottom, and let the absolute value bars drop the minus sign.
Flash cards: say the answer out loud, then flip
What is the amplitude of a wave?
The distance from the midline to a peak, which equals the distance from the midline to a trough.
What does |A| mean?
The absolute value of A, its size without the sign: |−6| = 6.
What is the amplitude of y = −58 cos x?
58
A wave's highest value is 13 and its lowest is 5. What is its amplitude?
13−52 = 4
For y = −7 sin x, what is y at x = π2?
−7 × sin π2 = −7 × 1 = −7, a low point, because A is negative.
A wave runs from −4 up to 4. Is its amplitude 8?
No. 8 is top to bottom. The amplitude is half of that, 4.