Find the equation of a cosine curve that is obtained by shifting the graph of y=cos(x) to the left 5 units and upward 9 units and vertically compressed by a factor of 4 and vertically flipped

We want to find the equation of a cosine function after we apply some given transformations to it.

The equation of the resulting cosine curve is:

g(x) = -(1/4)*cos(x + 5) - 9/4.

Let's see how we got that equation:

First, we need to define all the transformations that we will be using:

Vertical shift:

For a given function f(x) a vertical shift of N units is written as:

g(x) = f(x) + N.

Horizontal shift:

For a given function f(x) a horizontal shift is written as:

g(x) = f(x + N).

Vertical compression.

For a function f(x) a vertical compression by a factor k is written as:

g(x) = (1/k)*f(x).

Vertical flip (or vertical reflection).

For a general function f(x) a vertical reflection is written as:

g(x) = -f(x).

So we start with:

f(x) = cos(x).

First we shift it to the left 5 units, so we get:

g(x) = f(x + 5)

Then we shift it up 9 units, then we get:

g(x) = f(x + 5) + 9

Then we compress it vertically by a factor of 4:

g(x) = (1/4)*(f(x + 5) + 9)

Then we flip it vertically:

g(x) = -(1/4)*(f(x + 5) + 9)

Replacing f by the cosine function we get:

g(x) = -(1/4)*( cos(x + 5) + 9)

g(x) = -(1/4)*cos(x + 5) - 9/4.

This is the equation of the resulting cosine curve.

If you want to learn more, you can read:

link

3; 9 is 8 to -4

step-by-step explanation: