08/10 Trigonometric Functions and Graphs Free Online FlashCards

What domain and range do sine and cosine share?

Sine and cosine are defined for every real number, have range [−1, 1], and repeat every 2π.

What are the periods of tangent, cotangent, secant, and cosecant?

Tangent and cotangent have period π. Secant and cosecant have period 2π.

How do A, B, h, and D transform sine or cosine?

For y = A sin(B(x − h)) + D or y = A cos(B(x − h)) + D, the amplitude is |A|, the period is 2π/|B|, the phase shift is h, and the midline is y = D.

What is the range of y = A sin(B(x − h)) + D?

The range is [D − |A|, D + |A|]. The maximum is D + |A|, and the minimum is D − |A|.

What does a negative A do to a sinusoidal graph?

If A < 0, the sine or cosine graph is reflected across its midline y = D.

What are the key features of f(x) = 3 cos((π/4)(x − 2)) + 5?

For f(x) = 3 cos((π/4)(x − 2)) + 5, the amplitude is 3, period is 8, phase shift is right 2, midline is y = 5, and range is [2, 8].

Where are the vertical asymptotes of transformed tangent?

For y = A tan(B(x − h)) + D, vertical asymptotes satisfy B(x − h) = π/2 + kπ, so x = h + (π/2 + kπ)/B.

Why does tangent have no amplitude?

Tangent has no amplitude because its range is unbounded. In y = A tan(B(x − h)) + D, A controls vertical stretch or reflection, while the period is π/|B|.

How are csc, sec, and cot defined?

Cosecant is the reciprocal of sine, secant is the reciprocal of cosine, and cotangent is the reciprocal of tangent: csc x = 1/sin x, sec x = 1/cos x, and cot x = cos x/sin x.

Where do secant and cosecant have vertical asymptotes?

Secant has asymptotes where cos x = 0, at x = π/2 + kπ. Cosecant has asymptotes where sin x = 0, at x = kπ.

What are the period and asymptotes of transformed cotangent?

For y = A cot(B(x − h)) + D, the period is π/|B|, asymptotes satisfy B(x − h) = kπ, and the range is all real numbers.

What is a reliable procedure for graphing a transformed trigonometric function?

Factor the input to show B(x − h); then identify the parent, calculate the period, locate the phase shift, apply vertical changes, mark asymptotes if needed, and plot reference points.