Question

Graphing \(f\) from \(f^{\prime}\) Sketch the graph of a continuous function \(f\) with \(f(0)=-1\) and \(f^{\prime}(x)=\left\\{\begin{array}{ll}{1,} & {x<-1} \\\ {-2,} & {x>-1}\end{array}\right.\)

Short answer

The graph starts from some point at \(x<-1\), increases until \(x=-1\), where it takes a sharp turn, passes through the point (0, -1), and then keeps decreasing as \(x\) moves further in the positive direction.

Step-by-step solution

Understand the derivative and constant values

Start by understanding the derivative \(f^\prime\). The derivative represents the slope of the function \(f\). A positive slope indicates an increasing function, while a negative slope indicates a decreasing function. In this case, for \(x<-1\), the derivative equals 1, which means the function is increasing for this subset of x-values. For \(x>-1\), the derivative equals -2, so the function is decreasing for this subset of x-values. We are also given that the function passes through the point \( (0,-1)\).

Sketch the increasing and decreasing intervals

On the graph, sketch an increasing line for \(x<-1\), due to the positive derivative. Initiate this line at some arbitrary point but make sure it's increasing. Then, sketch a decreasing line for \(x>-1\) due to the negative derivative. Make sure there's a sharp turn in the graph at \(x=-1\) to represent the change in increase-decrease behavior of the graph. This sharp turn is because the derivative, and thus the slope, changes abruptly at this point.

Position the graph according to given point

We have a known point on this graph, i.e., (0, -1). Locate this point on your axis. Now, adjust the graph sketched in the previous step so that it goes through this point, maintaining the increasing-decreasing behavior around \(x=-1\).