precalculus

11 questions

1

.

How do you find the

period

of a cosine function of the form

 

y 

=

cos bx?

The period of y = cos bx is 

2 π
	b

.

The period of y = cos bx is 2πb.

    

The period of y = cos bx is 

b

π

.

The period of y = cos bx is πb.

2. What is the

amplitude

of 

y

 = 

6 

sin 

x?

3

. Use the graph of the function to answer the following.

f(x)

= cos x

 on the interval [−2π, 2π]

(a) Find the x-intercepts of the graph of 

y = f(x)

.

 (Enter your answers as a comma-separated list.)
x =  
(b) Find the y-intercepts of the graph of 

y = f(x).

(Enter your answers as a comma-separated list.)
y =  
(c) Find the intervals on which the graph of 

y = f(x)

 is increasing and the intervals on which the graph of 

y = f(x)

 is decreasing. (Enter your answers using interval notation.)

increasing

decreasing

(d) Find the relative extrema of the graph of 

y = f(x).

 (Enter your answers as a comma-separated list of ordered pairs.)

relative maxima

relative minima

4

. Find the period and the amplitude.

y = 8 cos(3x)

=

period		=
amplitude

5. Find the period and amplitude.

y = −4 sin 

	x
		3

 

	period
	amplitude

6. Find the period and amplitude.

y = 

3

	2

 cos 

πx
	4

 

period    

amplitude    

7. Find the period and amplitude.

y = 

3

2

 cos 

x

4

 

(period)

(amplitude)

8. Describe the relationship between the graphs of f and g. Consider amplitude, period, and shifts.

 = 

		f(x)						=	sin 2x
		g(x)	sin(−2x)

The graph of g is a vertical shift of the graph of f 2 units downward.The graph of g is a horizontal shift of the graph of f 2 units to the right.    The graph of g is a vertical shift of the graph of f 2 units upward.The graph of g is a reflection of the graph of f in the y-axis.The graph of g is a horizontal shift of the graph of f 2 units to the left.

9. Describe the relationship between the graphs of f and g. Consider amplitude, period, and shifts.

f(x)

 = 

g(x)

 = 

cos 4x

−3 + cos 4x

The graph of g is a vertical shift of the graph of f 3 units downward.The graph of g is a reflection of the graph of f in the y-axis.    The period of g is 3 that of f.The graph of g is a horizontal shift of the graph of f 3 units to the right.The graph of g has an amplitude 3 times that of the graph of f.

10 Sketch the graphs of f and g in the same coordinate plane. (Include two full periods.)

f(x)

 = 

g(x)

 = 

sin x

sin  x 4

11. Sketch the graph of the function by hand. Use a graphing utility to verify your sketch. (Include two full periods.)

f(x) = 3 cos (x + π/7)