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# What is the equation of the midline for the function f(x) ?

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If any is willing to help me with these ,I need the help.

1) What is the equation of the midline for the function f(x) ?

f(x)= 1/ 2 cos (x) + 5

2)

What is the frequency of the function f(x)?

f(x)=1/4 cos(2x)+5

Express the answer in fraction form.

3)

The functionf(x) is multiplied by a factor of 2 and then 3 is added to the function.

f(x)=sin(x)

What effect does this have on the graph of the function?

A) The graph is vertically stretched by a factor of 2 and shifted up 3 units.

B)The graph is vertically stretched by a factor of 3 and shifted up 2 units.

C)The graph is vertically compressed by a factor of 3 and shifted up 2 units.

D) The graph is vertically compressed by a factor of 2 and shifted up 3 units.

4)

The graph of  f(x)=sin(x)  is transformed to a new function, g(x) , by stretching it vertically by a factor of 2 and shifting it 3 units up.

What is the equation of the new function  g(x) ?

g(x)=

Jan 18, 2019
edited by jjennylove  Jan 18, 2019
edited by jjennylove  Jan 18, 2019
edited by jjennylove  Jan 18, 2019

#1
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1) What is the equation of the midline for the function f(x) ?

f(x)= 1/ 2 cos (x) + 5

Notice that   the min for the cos  = -1

So....the lowest point on the graph  is (1/2)(-1) + 5  = -1/2 + 5 =  4.5

The max for the cos  = 1

So...the highest point on the graph  = (1/2)(1) + 5 = 5.5

Midline    ( Highest point + Lowest point) / 2   =    (5.5 + 4.5) / 2  =  10 / 2  =  5

So....the equation for the midline is  y = 5

Jan 18, 2019
#2
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2)

What is the frequency of the function f(x)?

f(x)=1/4 cos(2x)+5

Express the answer in fraction form.

In the form

A * trig function ( Bx + C) + D

Only  "B" affects the period

Here...B  = 2      [ and  C = 0 ]

This means that there are 2 periods in 2pi

So....to find the period  take    2pi / 2    =   pi = the period

And the frequency  =  1 / period  =   1 /pi

Jan 18, 2019
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3)

The functionf(x) is multiplied by a factor of 2 and then 3 is added to the function.

f(x)=sin(x)

What effect does this have on the graph of the function?

The "2"  vertically stretches the parent function by a factor of 2

The "3"  shifts the function up 3 units

So.... "A" is correct

Jan 18, 2019
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So you know it is vertically stretched when there is on an (x) and the "B" (period) is not included and it is compressed when there is a value for B ? such as "4x" ?

jjennylove  Jan 18, 2019
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In the form

y = A * trig function (Bx  + C ) + D

The number on "B"  only affects the horizontal compression or horizontal expansion .....

If this  number  is between 0 and 1, we have a horizontal " stretch"

If this number is > 1, then we have a horizontal "compression"

It is "A"  that affects the vertcal stretching or vertical compression

If   absolute value of A  is  > 0  but < 1,  we have a vertical  compression

If absolute value of A > 1, we have a vertical stretch

"D"  affects the shift of the graph upward   (if D is positive)  or downward ( if D is negative)

Hope that helps, Jenny   !!!!

CPhill  Jan 18, 2019
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ohhh that all makes sense now. However such as the problem I had , how did you know "x" was a vertical stretch since x does not have a value ?

jjennylove  Jan 18, 2019
#10
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Remember that we have

y =  2sin (x)  + 3

We can write this as

y = 2 sin (1x) + 3

So "B"  = 1.....this means that there is no horizontal stretch or compression

It is  "A"  [ the number in front of sin (x) ] that affects the vertical stretching or compression

Since the abs value of this  = 2...and this is > 1.......we have a "stretch"

If the abs value of "A"  is between 0 and 1, we would have a vertical compression...

CPhill  Jan 18, 2019
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4)

The graph of  f(x)=sin(x)  is transformed to a new function, g(x) , by stretching it vertically by a factor of 2 and shifting it 3 units up.

What is the equation of the new function  g(x) ?

This is just the counterpart of the last question

g(x)  =   2sin (x) + 3

Jan 18, 2019
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I want to say thank you. These are great explantaions and it has helped me a lot. I am very thankful that you helped with each one. They were more easy then it looked.

jjennylove  Jan 18, 2019
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Looking at the question I didnt realize it was simialr to the last question but now I do it was just the oppostite .

jjennylove  Jan 18, 2019