Find a and d for the function f(x) = a cos(x) + d such that the graph off matches the figure.

We want to find the coefficients of a given function such that the graph of the function matches the given graph.

We will get:

a = -0.5

d = -3

We start with the general function:

f(x) = a*cos(x) + d.

Let's analyze the graph

In the figure, we can see that the graph passes through the point (0, -3.5)

This means that our function evaluated in x = 0 is equal to -3.5, then we can write:

f(0) = a*cos(0) + d = -3.5

        a + d = -3.5

We also can see that the distance between a peak and a through is 1 unit, and a, the amplitude, must be (in absolute value) half of that. Also note that when x = 0 we have a local minimum, while for the normal cosine we have a local maximum at x = 0, then a must be a negative number.

Then we have that:

a = - (1/2) = -0.5

Now that we know the value of a we can replace it in the other equation to get:

a + d = -3.5

-0.5 + d = -3.5

d = -3.5 + 0.5 = -3

Then the values of a and d are:

a = -0.5

d = -3

The function is:

f(x) = -0.5*cos(x) - 3

And the graph of this can be seen at the end of the answer, where it does not match exactly because the x-scale in the image is in units of pi, while in my graph is in integers.

If you want to learn more, you can read:

link

a = amplitude

d = shift

Just from looking at the graph, I can tell that d = -3.5

The amplitude = 1/2