Buy Trigonometric Graphs and Transformations

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Important concepts to recall (continued):

6. Graphs of the three main trig functions: As shown in the following graphs,

the sine, cosine, and tangent functions are periodic functions. Recall that y

= sin x and y = cos x have period 2π, amplitude 1, maximum value 1, and

minimum value –1; and y = tan x has period π, no amplitude, and no

maximum or minimum.

7. The functions y = A sin(Bx C) K and y = A cos(Bx C) K are

transformations of the basic sine and cosine functions. The constant A

induces a vertical compression or stretch; and if A < 0, a reflection about

the x-axis. The constant B induces a horizontal compression or stretch, the

constant C induces a horizontal shift (usually called a phase shift), and the

constant K induces a vertical shift. Therefore, the functions y = A sin(Bx

C) K and y = A cos(Bx C) K have amplitude |A|, period , horizontal

shift — to the right if 0, and vertical shift |K|

—up if K > 0 and down if K < 0. The maximum height of the graph is |A|

K and the minimum height of the graph is –|A| K. Note: To more easily

determine a horizontal shift, rewrite (Bx C) as B because the

horizontal shift is actually .

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8. The function y = A tan(Bx C) K is a transformation of the basic tangent

function. The constant A induces a vertical compression or stretch; and if A

< 0, a reflection about the x-axis. The constant B induces a horizontal

compression or stretch, the constant C induces a phase shift, and the

constant K induces a vertical shift. Therefore, the function y = A tan(Bx C)

K has period , horizontal shift —to the right if 0, and vertical shift |K|—up if K > 0 and down if K < 0. However,

unlike the sine and cosine functions, the tangent function has neither a

maximum nor a minimum value.

For questions 408 to 412, find the period of the function.

408. y = 3sin x

409. y = 2 cos3x 5

410. y = sin(–5x 2)

411. y = 4cos ( x – π)

412. y = –3sin(x 2π) – 1

For questions 413 to 417, determine the (a) phase shift and (b) vertical shift.

413. y = sin(x 2π) 4

414. y = 3cos (2x –

415. y = 2 sin (3x – 1

416. y = –2 cos( x – π)

417. y = 4sin(2x – )

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For questions 418 to 422, find the amplitude A of the function.

418. y = 3sin x

419. y = –4 cos x – 3

420. y = sin(5x – 2π)

421. y = π cos(πx 3π) 3

422. y = –12 cos(22x –17) 25

For questions 423 to 427, determine the (a) period, (b) phase shift, and (c)

vertical shift.

423. y = 3tan (2x)

424. y = tan(–3 2π) –5

425. y = 3tan(πx – 2) 10

426. y = 7 tan ( x 5)

427. y 2 = cos(3x – 6)

428. Using the following figure, solve sin x = 0 when –2π ≤ x ≤ 2π.

429. Using the following figure, solve cos x = 0 when –2π ≤ x ≤ 2π.

For questions 430 to 432, sketch the graph of the function.

430. y = 3sin(2x – π)

431. y = –2 cos(–3x – 6π)

432. y = tan(3x – 6)

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