1.Find the equation of the function with the maximum point at (pi/6, 2) and the next minimum point at (13pi/6, -4).
2. The function f(x) a sin x+d has a range of -2 less or equal that f(x) less or equal than 18. What are the values of a and d?
I would reaaaly appreciate it guys!
Thanks:)
2. The function f(x) a sin x+d has a range of -2 less or equal that f(x) less or equal than 18. What are the values of a and d?
I would reaaaly appreciate it guys!
Thanks:)
-
1.Find the equation of the function with the maximum point at (pi/6, 2) and the next minimum point at (13pi/6, -4).
Maximum = 2 and minimum = -4 puts middle at average (2+-4)/2 = -1
The distance between the max and minimum is half a period = 13π/6 minus π/6 = 12π/6 = 2π
Consequently the period of our function is 4π
Amplitude is 2 -(-1) = 3
With amplitude of 3 and period of 2π
and max at zero, minimum at π
our function would be
y = 3 cos x
To get the period to 4π
It becomes
y = 3 cos[ x/2]
that leaves maximum at zero and minimum at 2π
To get the max point over to π/6 and shift minimum to 13π/6
it becomes
y = 3 cos[½(x-(π/6))]
This has max at 3 and min at -3
To move max down to do and min down to -4 we have to subtract 1
so the function is
y = 3 cos[½(x-(π/6))] -1
2. The function f(x) a sin x+d has a range of -2 less or equal that f(x) less or equal than 18. What are the values of a and d?
Range is -2≤f(x)≤18
size of range is 18 -(-2) = 20
amplitude is half of that 20 which is 10 so a=10
Now d is the upward or downward shift, upward if positive and downward if negative.
The average of -2 and 18 is [18 +(-2) ]/2 = 16/2 = 8
That is an upward shift of 8 units so d=8
Maximum = 2 and minimum = -4 puts middle at average (2+-4)/2 = -1
The distance between the max and minimum is half a period = 13π/6 minus π/6 = 12π/6 = 2π
Consequently the period of our function is 4π
Amplitude is 2 -(-1) = 3
With amplitude of 3 and period of 2π
and max at zero, minimum at π
our function would be
y = 3 cos x
To get the period to 4π
It becomes
y = 3 cos[ x/2]
that leaves maximum at zero and minimum at 2π
To get the max point over to π/6 and shift minimum to 13π/6
it becomes
y = 3 cos[½(x-(π/6))]
This has max at 3 and min at -3
To move max down to do and min down to -4 we have to subtract 1
so the function is
y = 3 cos[½(x-(π/6))] -1
2. The function f(x) a sin x+d has a range of -2 less or equal that f(x) less or equal than 18. What are the values of a and d?
Range is -2≤f(x)≤18
size of range is 18 -(-2) = 20
amplitude is half of that 20 which is 10 so a=10
Now d is the upward or downward shift, upward if positive and downward if negative.
The average of -2 and 18 is [18 +(-2) ]/2 = 16/2 = 8
That is an upward shift of 8 units so d=8