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A short class with Sarah and her students. The other ways are under More.
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Compare y = 3 cos(2x) with y = cos x. What does the 3 do?
It multiplies every y-value by 3, so the wave goes up to 3 and down to −3. The amplitude is 3.
And the 2 inside the cosine?
The angle doubles, so the wave repeats twice as fast. The period is 360° divided by 2, which is 180°.
So is that a horizontal stretch by 2?
No, by a scale factor of ½. The graph gets squeezed towards the y-axis.
One cycle is 180°. Split it into quarters. What x-values do we use?
0°, 45°, 90°, 135° and 180°.
Find y at each.
3 cos 0° is 3, 3 cos 90° is 0, 3 cos 180° is −3, 3 cos 270° is 0 and 3 cos 360° is 3.
So between 0° and 360°, how many waves are there?
Two complete waves, with minimums at 90° and 270°.
What changes in y = 2 sin x + 1?
The amplitude is 2 and the whole graph moves up 1, so it goes from −1 to 3.
And in y = −2 sin(3x)?
Amplitude 2, period 120°, and the minus sign flips it, so it goes down first.
A cosine graph has max 5, min −1 and period 90°. What is its equation?
Amplitude is half of 6, which is 3. The middle is 2, and b is 360 over 90, so 4. It's y = 3 cos(4x) + 2.
in y = a cos(bx), the amplitude is |a|, so y = 3 cos(2x) oscillates between 3 and −3.
the period is 360° ÷ b, so y = 3 cos(2x) repeats every 180°.
replacing x by bx stretches the graph horizontally by scale factor 1/b.
quarter-periods give the maxima, zeros and minima needed for an accurate sketch.
y = a cos(b(x − c)) + d adds a horizontal shift c and a vertical shift d with principal axis y = d.
amplitude = (max − min) ÷ 2, principal axis = (max + min) ÷ 2, b = 360° ÷ period.
y = 3 cos(2x): amplitude 3, period 180°, key points (0°, 3), (45°, 0), (90°, −3), (135°, 0) and (180°, 3).
y = 2 sin x + 1: amplitude 2, period 360°, principal axis y = 1, range −1 ≤ y ≤ 3.
y = −2 sin(3x): amplitude 2, period 120°, reflected, so y = −2 at x = 30° and y = 2 at x = 90°.
A cosine graph with maximum 5 at x = 0°, minimum −1 and period 90° has equation y = 3 cos(4x) + 2.
Outside number, height to show; inside number, fast or slow; three-sixty divided by b, that's the period you will see.
A wave machine has two dials. Turn the outside dial to 3 and the waves stand three times as tall; turn the inside dial to 2 and twice as many waves pass by each lap of 360°.
"Out is Up, In is Across" - the number outside the cosine stretches up and down, the number inside stretches across, by its reciprocal.
A.P.S. - Amplitude from a, Period from 360 over b, Shifts from c and d.
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What is the amplitude of y = 3 cos(2x)?
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What is the period of y = 3 cos(2x)?
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What transformation maps y = cos x to y = cos(2x)?
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What is the period of y = a sin(bx)?
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How do you find the principal axis from a graph?
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Half the distance between the maximum and minimum values of a periodic graph; for y = a cos(bx) it is |a|.
The horizontal length of one complete cycle; for y = a cos(bx) it is 360° ÷ b.
The horizontal line midway between the maximum and minimum; for y = a cos(bx) + d it is y = d.
A transformation that multiplies every y-value by a scale factor, such as 3 in y = 3 cos x.
A transformation that multiplies every x-value by a scale factor; replacing x by 2x gives scale factor ½.
A slide of the whole graph without changing its shape, either sideways or up and down.
The set of output values a function takes; for y = 3 cos(2x) it is −3 ≤ y ≤ 3.
Example 2 sketches y = 3 cos(2x) by transforming y = cos x: the 3 stretches the graph vertically to amplitude 3, and the 2 squeezes it horizontally so the period becomes 360° ÷ 2 = 180°. One full wave runs from 0° to 180°, between heights 3 and −3.
Graphs of the form y = a cos(bx) are transformations of y = cos x. In y = 3 cos(2x), the factor 3 is a vertical stretch that makes the amplitude 3 and the range −3 ≤ y ≤ 3, and the factor 2 is a horizontal stretch with scale factor ½ that halves the period from 360° to 180°. This lesson finds the key points of one cycle (a maximum at 0°, zeros at 45° and 135°, a minimum at 90° and the next maximum at 180°), sketches two full cycles between 0° and 360°, and then extends the method to vertical shifts, reflections and phase shifts in the general form y = a cos(b(x − c)) + d, including reading an equation back from a graph.
Starting point: y = cos x. The basic cosine graph has amplitude 1 and period 360°. It starts at a maximum of 1 when x = 0°, crosses the x-axis at 90°, reaches its minimum of −1 at 180°, crosses again at 270° and returns to 1 at 360°.
Example 2: sketch y = 3 cos(2x) for 0° ≤ x ≤ 360°.
The general pattern. For y = a cos(b(x − c)) + d, and likewise for sine:
More examples.
Reading an equation from a graph. Suppose a cosine-shaped graph has a maximum of 5 at x = 0°, a minimum of −1, and repeats every 90°. Then the amplitude is (5 − (−1)) ÷ 2 = 3, the principal axis is y = (5 + (−1)) ÷ 2 = 2, and b = 360° ÷ 90° = 4. The equation is y = 3 cos(4x) + 2.
Common mistakes. Multiplying 360° by b instead of dividing, taking the amplitude to be the full height from minimum to maximum instead of half of it, and stretching horizontally by factor b instead of 1/b.