# Precalculus : Graphing Functions

## Example Questions

### Example Question #6 : Amplitude, Period, Phase Shift Of A Trig Function

Which equation would produce this sine graph?

Explanation:

The graph has an amplitude of 2 but has been shifted down 1:

In terms of the equation, this puts a 2 in front of sin, and -1 at the end.

This makes it easier to see that the graph starts [is at 0] where .

The phase shift is  to the right, or

### Example Question #7 : Amplitude, Period, Phase Shift Of A Trig Function

Which of the following equations could represent a cosine function with amplitude 3, period , and a phase shift of ?

Explanation:

The form of the equation will be

First, think about all possible values of A that could give you an amplitude of 3. Either A = -3 or A = 3 could each produce amplitude = 3. Be sure to look for answer choices that satisfy either of these.

Secondly, we know that the period is . Normally we know what B is and need to find the period, but this is the other way around. We can still use the same equation and solve:

. You can cross multiply to solve and get B = 4.

Finally, we need to find a value of C that satisfies

. Cross multiply to get:

Next, plug in B= 4 to solve for C:

Putting this all together, the equation could either be:

or

### Example Question #8 : Amplitude, Period, Phase Shift Of A Trig Function

State the amplitude, period, phase shift, and vertical shift of the function

Amplitude: 7

Period: /3

Phase Shift: /6

Vertical Shift: -4

Amplitude: 7

Period: -/3

Phase Shift: -/6

Vertical Shift: -4

Amplitude: -7

Period: /3

Phase Shift: -/6

Vertical Shift: 4

Amplitude: 7

Period: /3

Phase Shift: -/6

Vertical Shift: -4

Amplitude: 7

Period: /3

Phase Shift: -/6

Vertical Shift: -4

Explanation:

A common way to make sense of all of the transformations that can happen to a trigonometric function is the following. For the equations y = A sin(Bx + C) + D,

• amplitude is |A|
• period is 2/|B|
• phase shift is -C/B
• vertical shift is D

In our equation, A=-7, B=6, C=, and D=-4. Next, apply the above numbers to find amplitude, period, phase shift, and vertical shift.

To find amplitude, look at the coefficient in front of the sine function. A=-7, so our amplitude is equal to 7.

The period is 2/B, and in this case B=6. Therefore the period of this function is equal to 2/6 or /3.

To find the phase shift, take -C/B, or -/6. Another way to find this same value is to set the inside of the parenthesis equal to 0, then solve for x.
6x+=0
6x=-
x=-/6
Either way, our phase shift is equal to -/6.

The vertical shift is equal to D, which is -4.

y=-7\sin(6x+\pi)-4

### Example Question #9 : Amplitude, Period, Phase Shift Of A Trig Function

State the amplitude, period, phase shift, and vertical shift of the function

Amplitude: 1

Period:

Phase Shift:

Vertical Shift: 3

Amplitude: 1

Period:

Phase Shift:

Vertical Shift: 3

Amplitude: 1

Period:

Phase Shift:

Vertical Shift: 3

Amplitude: 1

Period:

Phase Shift:

Vertical Shift: 0

Amplitude: 1

Period:

Phase Shift:

Vertical Shift: 3

Amplitude: 1

Period:

Phase Shift:

Vertical Shift: 3

Explanation:

A common way to make sense of all of the transformations that can happen to a trigonometric function is the following. For the equations y = A sin(Bx + C) + D,

• amplitude is |A|
• period is 2/|B|
• phase shift is -C/B
• vertical shift is D

In our equation, A=-1, B=1, C=-, and D=3. Next, apply the above numbers to find amplitude, period, phase shift, and vertical shift.

To find amplitude, look at the coefficient in front of the sine function. A=-1, so our amplitude is equal to 1.

The period is 2/B, and in this case B=1. Therefore the period of this function is equal to 2.

To find the phase shift, take -C/B, or . Another way to find this same value is to set the inside of the parenthesis equal to 0, then solve for x.
x-=0
x=
Either way, our phase shift is equal to .

The vertical shift is equal to D, which is 3.

### Example Question #2 : Amplitude, Period, Phase Shift Of A Trig Function

State the amplitude, period, phase shift, and vertical shift of the function

Amplitude: 1

Period:

Phase Shift: -3/2

Vertical Shift: 2

Amplitude: 1

Period:

Phase Shift: 3/2

Vertical Shift: 2

Amplitude: 1

Period:

Phase Shift: -3/2

Vertical Shift: -2

Amplitude: 1

Period: 3/2

Phase Shift:

Vertical Shift: 2

Amplitude: 1

Period:

Phase Shift: 3/2

Vertical Shift: 2

Explanation:

A common way to make sense of all of the transformations that can happen to a trigonometric function is the following. For the equations y = A sin(Bx + C) + D,

• amplitude is |A|
• period is 2/|B|
• phase shift is -C/B
• vertical shift is D

In our equation, A=1, B=2, C=-3, and D=2. Next, apply the above numbers to find amplitude, period, phase shift, and vertical shift.

To find amplitude, look at the coefficient in front of the sine function. A=1, so our amplitude is equal to 1.

The period is 2/B, and in this case B=2. Therefore the period of this function is equal to .

To find the phase shift, take -C/B, or 3/2. Another way to find this same value is to set the inside of the parenthesis equal to 0, then solve for x.
2x-3=0
2x=3
x=3/2
Either way, our phase shift is equal to 3/2.

The vertical shift is equal to D, which is 2.

### Example Question #1 : Trigonometric Graphs (All Six)

Which of the trigonometric functions is represented by this graph?

y = csc x

y = sec x

y = tan x

y = cot x

y = csc x

Explanation:

This graph is the graph of y = csc x. The domain of this function is all real numbers except  where n is any integer. In other words, there are vertical asymptotes at all multiples of . The range of this function is . The period of this function is .

### Example Question #2 : Trigonometric Graphs (All Six)

Which of the following functions is represented by this graph?

y = csc x

y = sec x

y = tan x

y = cot x

y = cot x

Explanation:

This graph is the graph of y = cot x. The domain of this function is all real numbers except  where n is any integer. In other words, there are vertical asymptotes at all multiples of . The range of this function is . The period of this function is .

### Example Question #3 : Trigonometric Graphs (All Six)

Which of the following functions is represented by this graph?

y = csc x

y = tan x

y = sec x

y = cot x

y = sec x

Explanation:

This graph is the graph of y = sec x. The domain of this function is all real numbers except  where n is any integer. In other words, there are vertical asymptotes at  , , and so on. The range of this function is . The period of this function is

### Example Question #4 : Trigonometric Graphs (All Six)

Which of the following functions is represented by this graph?

y = tan x

y = sec x

y = csc x

y = cot x

y = tan x

Explanation:

This graph is the graph of y = tan x. The domain of this function is all real numbers except  where n is any integer. In other words, there are vertical asymptotes at , and so on. The range of this function is . The period of this function is .

### Example Question #5 : Trigonometric Graphs (All Six)

Which of the following functions has a y-intercept of