Question

Help asap no wrong answers----------------------

Answer 1

`y=-2(sin(2x))-7`

Explanation:

1. Approach

Given information:

• The graph intersects the midline at (0, -7)
• The graph has a minimum point at (
  `(\pi)/(4)`, 9).

What conclusions can be made about this function:

• The graph is a sine function, as its y-intercept intersects the midline
• This graph has a negative coefficient, this is because after intersecting the midlines at the y-intercept, the function has a minimum.
• This graph does not appear to have undergone any horizontal shift, as it intercepts the midlines with its y-intercept

Therefore, one has the following information figured out:

`y=-n(sin(ax))+b`

Now one has to find the following information:

• amplitude
• midline
• period

2. Midline

The midlines can simply be defined as a line that goes through a sinusoidal function, cutting the function in half. This is represented by the constant (b). One is given that point (0, -7) is where the graph intersects the midline. The (y-coordinate) of this point is the midline. Therefore, the midline is the following:

y = -7

2. Amplitude

The amplitude is represented by the coefficient (n). It can simply be defined by the distance from the midline to point of maximum (the highest part of a sinusoidal function) or point of minimum (lowest point on the function). Since the function reaches a point of minimum after intercepting the (y-axis) at its midlines, the amplitude is a negative coefficient. One can find the absolute value of the amplitude by finding the difference of the (y-coordinate) of the point of minimum (or maximum) and the absolute value of the midline.

point of minimum:
`((\pi)/(4),9)`

midline:
`y=-7`

Amplitude: 9 - |-7| = 9 - 7 = 2

3. Period

The period of a sinusoidal function is the amount of time it takes to reach the same point on the wave. In essence, if one were to select any point on the sinusoidal function, and draw a line going to the right, how long would it take for that line to reach a point on the function that is identical to the point at which it started. This can be found by taking the difference of the (x- coordinate) of the intersection point of the midline, and the (x-coordinate) of the point of minimum, and multiplying it by (4).

point of minimum:
`((\pi)/(4),9)`

midline intersection:
`(0, -7)`

Period:
`4((\pi)/(4)-0)=4((\pi)/(4))=\pi`

However, in order to input this into the function in place of the variable (a), one has to divide this number by (
`2\pi`).

`a=(2\pi)/(\pi)=2`

4. Assemble the function

One now has the following solutions to the variables:

`n =-16\\a=2\\b=-7\\`

Substitute these values into the function:

`y=-2(sin(2x))-7`