CPhill

f(x) = sin x

g(x)  =    4sin (x - 3)

See the graphs here : https://www.desmos.com/calculator/js4vt924c0

C- the graph is shifted 4 units right

See the graphs here :   https://www.desmos.com/calculator/xaomug9is8

What's the question ???

f(x)=x^2-2x How would you find f(f(f(f(f(f(-1))))))  ?

We (seemingly) have six evaluations to make

(1)  f ( - 1) =  (-1)^2 -2(-1)  = 1 + 2  = 3

(2)   f ( f  (-1) ) =  f (3) =    (3)^2 - 2(3) =  9 - 6 =  3

(3)  f (  f ( f  (-1) ) ) =  f(3)  = 3

Note that this pattern continues....so

f(f(f(f(f(f(-1))))))  =  3

B- the graph shifts 5 units left and 3 units up 

1.

 ax-b=c, for x         add b to both sides

ax = b + c               divide both sides by 

x = [ b + c ] / a

2.

15x+1=y, for x          subtract 1 from both sides

15x = y - 1                divide both sides by 15

x = [ y - 1 ] / 15

3.

(x+f)+2=j, for x       subtract 2 from both sides

x + f  = j - 2             subtract f  from both sides

 x = j - 2 - f

Let the ordering of the integers from lowest to highest be a, b, c, d and e

And we know that  [a + b + c + d + e ] / 5    =  6

a + b + c + d + e  = 30

And the median is 6, so....c = 6

And   ...    e - a =  6  →  e = 6 + a

So....

a + b + c +  d + e =  30

a + b + 6 + d +( 6 + a)  = 30

2a + b + d =   18

b + d =  18 - 2a

[ b + d ] / 2  =   9 - a

"a" must be odd.....so...."e" must be odd

And b, d must be of the same parity....both even or both odd

And since the mode is 6, either b or d  (or both ) are  6

Let a = 1 and b = 6... then d would have to be 10  

But e = 7....which means that d > e......not possible

And if d = 6, then b = 10....but b > c   ....not possible

So a cannot be 1

Next....let a = 3

[ b + d  ] / 2 = 9 - 3

b + d = 12        

So....if either b, d  = 6....then the other must also = 6

So....

3 + 6 + 6 +  6 + 9 = 30        is true

The mean, mode, range and median = 6

So....the numbers are just as EP found  !!!!

No real solutions.....we could solve this algebraically....but this  shows that the graphs do not intersect....

https://www.desmos.com/calculator/g69psgs4hq

(1)  This is just the basic sine graph shifted down 2 units

Here is the graph : https://www.desmos.com/calculator/6ppigmmzbe

(2) This is the basic cosine graph with a period of pi shifted up 1 unit

Here is the graph : https://www.desmos.com/calculator/okxkxbmj5d

(3) A scientist recorded the movement of a pendulum for 10 s. The scientist began recording when the pendulum was at its resting position. The pendulum then moved right (positive displacement) and left (negative displacement) several times. The pendulum took 4 s to swing to the right and the left and then return to its resting position. The pendulum’s furthest distance to either side was 6 in. Graph the function that represents the pendulum’s displacement as a function of time.

The form is

y = Asin (Bx)

A = the amplitude =  6

The period is 4 seconds

B =  2pi / period  =  2pi / 4  =   pi/2

The function is

y = 6sin ( pi * t / 2 )

Here is the graph :  https://www.desmos.com/calculator/u7pe5zrrft

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