Question

G is a trigonometric function of the form g(x)=a sin (bx+c)+d.

Answer 1

Answer:

Formula for g(x) is $g(x) = 3 sin(\frac{\pi }{5}x + \frac{\pi }{5} ) + 6$

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 = $\frac{2\pi }{B}$

⇒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

      = $3 sin(\frac{\pi }{5}(x -(- 1))) + 6$

⇒$g(x) = 3 sin(\frac{\pi }{5}x + \frac{\pi }{5} ) + 6$

∴ we get

Formula for g(x) is $g(x) = 3 sin(\frac{\pi }{5}x + \frac{\pi }{5} ) + 6$