Trigonometric Functions and Their Graphs

Test your understanding of sine, cosine, and tangent functions, their key properties, and how they appear on a graph. This quiz covers amplitude, period, phase shift, and vertical shift, along with graph identification and basic transformations.

1 / 20

What is the period of the function

$f(x) = sin(4x)$

pi

2pi

frac{pi}{2}

4pi

The period of

$sin(Bx)$

$frac{2pi}{B}$

. Here,

$B = 4$

, so the period is

$frac{2pi}{4} = frac{pi}{2}$

2 / 20

The amplitude of

$f(x) = -5 cos(x)$

is:

= 5.

5

-5

1

0

Amplitude is the absolute value of the coefficient, so

$∣ - 5 ∣ = 5$

3 / 20

The range of

$f(x) = 3 + 2 sin(x)$

is:

[-1, 1]

[1, 5]

[3, 5]

(-infty, infty)

The range of

$sin(x)$

$[-1, 1]$

, so

$3 + 2(-1) = 1$

and

$3 + 2(1) = 5$

4 / 20

The phase shift of

$f(x) = 2 sinleft(3x – frac{pi}{4}right)$

is:

frac{pi}{4}

to the left

frac{pi}{3}

to the right

No phase shift

frac{pi}{12}

to the right

Rewrite as

$2 sinleft(3left(x – frac{pi}{12}right)right)$

. The phase shift is

$frac{pi}{12}$

units right.

5 / 20

The function

$f(x) = tan(x)$

has vertical asymptotes at:

x = npi

x = frac{pi}{2} + npi

x = frac{pi}{4} + npi

x = 0

$tan(x) = frac{sin(x)}{cos(x)}$

, which is undefined where

$cos(x) = 0$

6 / 20

Which identity is correct?

sin^2 x + cos^2 x = 1

sin(2x) = sin x + cos x

tan x = frac{cos x}{sin x}

sec x = frac{1}{sin x}

7 / 20

The general solution to

$cos x = frac{1}{2}$

is:

x = frac{pi}{6} + npi

x = npi

x = frac{pi}{3} + 2npi

x = frac{5pi}{3} + 2npi

x = frac{pi}{2} + 2npi

Cosine equals

$frac{1}{2}$

$frac{pi}{3}$

and

$frac{5pi}{3}$

plus any full period

$2npi$

8 / 20

The exact value of

$sinleft(frac{7pi}{6}right)$

is:

frac{sqrt{3}}{2}

frac{1}{2}

-1

-frac{1}{2}

$frac{7pi}{6}$

is in the third quadrant where sine is negative, and

$sinleft(pi + frac{pi}{6}right) = -sinleft(frac{pi}{6}right) = -frac{1}{2}$

9 / 20

The expression

$sin(x) cos(x)$

can be rewritten as:

cos^2 x – sin^2 x

frac{1}{2} sin(2x)

tan x

sec x

This is the double-angle identity:

$sin(2x) = 2 sin x cos x$

10 / 20

The simplified form of

$frac{1 – cos^2 x}{sin x}$

is:

sin x

cos x

tan x

cot x

$1 – cos^2 x = sin^2 x$

, so

$frac{sin^2 x}{sin x} = sin x$

11 / 20

The graph of

$f(x) = cos(x)$

crosses the x-axis at:

x = npi

x = frac{pi}{4} + npi

x = frac{pi}{2} + npi

x = 0

$cos(x) = 0$

when

$x = frac{pi}{2} + npi$

12 / 20

The maximum value of

$f(x) = 2 – 3 sin(x)$

is:

5

2

-1

3

The maximum of

$sin(x)$

$1$

, so

$2 – 3(-1) = 5$

13 / 20

The graph of

$f(x) = tan(x)$

has a period of:

2pi

frac{pi}{2}

4pi

pi

The tangent function repeats every

$pi$

units.

14 / 20

The function

$f(x) = sin(x) + 1$

has a midline at:

y = 0

y = 1

y = -1

y = 2

15 / 20

The graph of

$f(x) = -2 cos(3x)$

has an amplitude of:

= 2.

-2

3

2

frac{2}{3}

Amplitude is always non-negative, so

$∣ - 2 ∣ = 2$

16 / 20

The graph of

$f(x) = sec(x)$

has vertical asymptotes where:

cos(x) = 0

sin(x) = 0

tan(x) = 0

cot(x) = 0

$sec(x) = frac{1}{cos(x)}$

, so it is undefined where

$cos(x) = 0$

17 / 20

The range of

$f(x) = csc(x)$

is:

[-1, 1]

(0, infty)

(-infty, infty)

(-infty, -1] cup [1, infty)

Since

$csc(x) = frac{1}{sin(x)}$

, and

$sin(x) in [-1, 1]$

$csc(x)$

is outside

$[-1, 1]$

18 / 20

The graph of

$f(x) = cot(x)$

has a period of:

2pi

pi

frac{pi}{2}

4pi

The cotangent function repeats every

$pi$

units.

19 / 20

The function

$f(x) = 4 sinleft(2x – frac{pi}{3}right)$

has a phase shift of:

frac{pi}{6}

to the right

frac{pi}{3}

to the left

frac{pi}{2}

to the right

No phase shift

Rewrite as

$4 sinleft(2left(x – frac{pi}{6}right)right)$

20 / 20

The graph of

$f(x) = tan(x)$

is symmetric about:

The origin

The y-axis

The x-axis

The line

y = x

$tan(-x) = -tan(x)$

, so it is an odd function.