Question

What is the equation of the following graph?

	A.	f(x) = cos x
	B.	f(x) = sin x
	C.	f(x) = tan x
	D.	f(x) = cot x

What is the equation of the following graph?

	A.	f(x) = cos x
	B.	f(x) = sin x
	C.	f(x) = tan x
	D.	f(x) = cot x

What is the equation of the following graph?

	A.	f(x) = sin(x - 45°)
	B.	f(x) = sin(x + 45°)
	C.	f(x) = sin x + 2
	D.	f(x) = 2sin x

 

Answer 1

1)

Observe the graph, this is tangent function graph and its contains all properties of the Tangent Function.

• The domain is the set of all real numbers,except odd multiples of π/2.
• The range is the set of all real numbers.
• The tangent function is an odd function, as the symmetry of the graph
  with respect to the origin indicates.
• The tangent function is periodic,with period π.

(Period is length of smallest domain interval which corresponds to a complete cycle of values of the function.)

In this case a full period occur between x =  - π/2 and x = π/2.

• The x-intercepts are . . . , - 2π, - π, 0, π, 2π, . . . ; the y-intercept is 0.
• Vertical asymptotes occur at x = . . . , - 3π/2, - π/2, π/2, 3π/2, . . . . .

Amplitude of the graph (maximum y value )

In this case there is no amplitude, since there is no maximum value of y.

Therfore, the function is f(x) = tan(x)

Option C is correct.

Answer 2

2) Observe the graph,this is sine function graph and its contains all properties of the sine function.

• The domain is all real numbers .
• Range is [-1, 1]

All real numbers between including between -1 and 1 including -1 and 1.

• The sine function is an odd function, as the symmetry of the graph
  with respect to the origin indicates.
• The sine function is periodic,with period 2π.

(Period is length of smallest domain interval which corresponds to a complete cycle of values of the function.)

• The x-intercepts are . . . , - 2π, - π, 0, π, 2π, . . . ; the y-intercept is 0.

• Amplitude of the graph (maximum y value )

  In this case amplitude is 1.

Therfore, the function is f(x) = sin(x)

Option B is correct.

 

Answer 3

3)Observe the graph,

• Amplitude of the graph (maximum y value )

  In this case amplitude is 1.

Period is length of smallest domain interval which corresponds to a complete cycle of values of the function.

Period of the graph is 2π.

The x-intercepts of the y = sinx are . . . , - 2π, - π, 0, π, 2π, . . . ; the y-intercept is 0.

From the graph, the x-intercepts are . . . , - 9π/4, - 5π/4, -π/4, 3π/4,7π/4. . . .

From the sin function properties,

The absolute maximum is 1 and occurs at x = . . . . . , - 3π/2, π/2, 5π/2, 9π/2, . . . . ;

The absolute minimum is - 1 and occurs at x = . . . . , - π/2, 3π/2, 7π/2, 11π/2, . . . .

In this case

The absolute maximum is 1 and occurs at x = . . . . , - 7π/4, π/4,9π/4 . . . . ;

The absolute minimum is - 1 and occurs at x = . . . , - 3π/4, 5π/4, . . . .

From the above conclusion, The graph shifts to the left by π/4 units.

 

Therfore, the function is f(x) = sin(x + π/4)

f(x) = sin(x + 45°)

Option B is correct choice.