Question

Can you please help me with 64Use the graph to write an equation for the function.

Answer 1

We must write an equation for the function in the graph.

From the graph, we see that the function f(x):

0. has a vertical asymptote at x = -2 → f must be proportional to 1/(x + 2),

,

1. has a vertical asymptote at x = 4 → f must be proportional to 1/(x - 4),

,

2. is zero at x = 2 → f must be proportional to (x - 2).

Now, we also see that the function:

• is negative for x < -2,

,

• looks like a parabola in the interval -2 < x < 4 → f is proportional 1/(x + 2)²,

,

• is positive in the interval -2 < x < 4 → f must be proportional 1/(x - 4)² to have a positive sign for values x < 4.

Taking into consideration all the points above, we write the following function:

`f(x)=k\cdot((x-2)^2)/((x+2)\cdot(x-4)^2),k>0.`

Where k is a positive constant.

Selecting the value k = 1 and plotting the function f(x), we get the following graph:

Answer

An equation for the function in the graph is:

`f(x)=((x-2)^2)/((x+2)\cdot(x-4)^2)\text{.}`