Answer:
Formula for g(x) is [tex]g(x) = 3 sin(\frac{\pi }{5}x + \frac{\pi }{5} ) + 6[/tex]
Step-by-step explanation:
Given - g is a trigonometric function of the form g(x)=a sin (bx+c)+d. The function intersects its midline at (-1 , 6) and has a minimum point at (-3.5 , 3)
To find - Find a formula for g (x). Give an exact expression.
Proof -
Given that,
g(x)=a sin (bx+c)+d
We know that, Midline is present in between maximum and minimum
Here given that, minimum is present is 3 and midline is present at 6
So, Maximum occurs at 9.
Now,
We know that,
Standard form of sine function is - g(x) = Asin(B(x-C)) + D
Where
A = Amplitude
and Amplitude = (Maximum - minimum) / 2
= (9 - 3)/ 2
= 6/2 = 3
⇒A = 3
Now,
Period = [tex]\frac{2\pi }{B}[/tex]
⇒B = (2π) / Period
= (2π) / 10
= π/5
⇒B = π/5
Now,
Phase Shift : C = -1 ( i.e. 1 to the left)
Vertical Shift : D = 6
So,
We get
g(x) = Asin(B(x-C)) + D
= [tex]3 sin(\frac{\pi }{5}(x -(- 1))) + 6[/tex]
⇒[tex]g(x) = 3 sin(\frac{\pi }{5}x + \frac{\pi }{5} ) + 6[/tex]
∴ we get
Formula for g(x) is [tex]g(x) = 3 sin(\frac{\pi }{5}x + \frac{\pi }{5} ) + 6[/tex]
