Question

g is a trigonometric function of the form g(x) = a sin(bx + c) + d. Below is the graph g(x). The function intersects its midline at 3/4 *pi, 3) and has a maximum point at (pi,7) Find a formula for g(x). Give an exact expression.​

Answer 1

The trigonometric function with the given properties is determined to be g(x) = 4 sin(x - 3/4 π) + 3, reflecting the amplitude, period, phase shift and midline value.

Step-by-step explanation:

The given function g(x) = a sin(bx + c) + d intersects its midline at (3/4 π, 3) and has a maximum point at (π, 7). From these points, we can deduce certain properties of the trigonometric function. The midline value, d, corresponds to the y-coordinate of the point where the function intersects the midline, which is 3. The amplitude, a, can be found by taking the difference between the maximum y-value and the midline, which is 7 - 3 = 4. Since the period of a sine function is 2π and the function completes one cycle between 3/4 π and 2π + 3/4 π, we find that the period is 2π, indicating that b = 1. The phase shift can be found from the point where the function intersects the midline, which indicates that c = -3/4 π. Thus, the formula for g(x) is g(x) = 4 sin(x - 3/4 π) + 3.

Answer 2

`g(x)=4sin(2x+(\pi)/(2))+3`

Step-by-step explanation:

Because the function intersects its midline at
`((3\pi)/(4),3)`, then the midline is
`d=3`.

Additionally, the amplitude is just the positive distance between the maximum/minimum and the midline, so the amplitude is
`a=7-3=4`.

Also, given that period is
`(2\pi)/(b)` and the fact that the period is
`\pi-0=\pi` from our given maximum, we have the equation
`(2\pi)/(b)=\pi` where
`b=2`.

Lastly, to find
`c`, we know that the phase shift,
`-(c)/(b)`, is
`-(\pi)/(4)` (or
`(\pi)/(4)` to the left) since
`\pi-(3\pi)/(4)=-(\pi)/(4)`. Therefore, we have the equation
`-(c)/(2)=-(\pi)/(4)` where
`c=(\pi)/(2)`.

Putting it all together, our final equation is
`g(x)=4sin(2x+(\pi)/(2))+3`