# How do you determine whether the function #f(x)= (x-1) / (x+52)# is concave up or concave down and its intervals?

##### 1 Answer

Use the sign of the second derivative (or knowledge of transformations of the reciprocal function).

#### Explanation:

**Calculus**

Using calculus, the general method of determining concavity is to investigate the sign of the second derivative.

For this function, the sign of

So the graph of

Because

(The definition of inflection point that I am accustomed to is: a point **on the graph** at which the concavity changes.)

**Reciprocal Function**

From the graph of

graph{y=1/x [-20.28, 20.27, -10.14, 10.14]}

we obtain the graph of

translating

graph{y=(x-1)/(x+52) [-123.7, 42.94, -35.4, 48]}

Because of the reflection the graph is concave up on the left and concave down on the right. The horizontal translation moves the change in concavity from