Question

sketch a graph of the piecewise defined function.
f(x)=\begin{cases}4 & \text{if }x < -2\\x^{2} & \text{if }-2leq xleq2\\-x + 6 & \text{if }x > 2end{cases}

Explanation:

Step1: Graph \(y = 4\) for \(x<-2\)

This is a horizontal line. Since \(x < - 2\), we draw an open - circle at \(x=-2\) (because the point \(x = - 2\) is not included in this part of the function) and draw the line \(y = 4\) to the left of \(x=-2\).

Step2: Graph \(y=x^{2}\) for \(-2\leq x\leq2\)

We can find some points. When \(x=-2\), \(y=(-2)^{2}=4\); when \(x = 0\), \(y = 0\); when \(x=2\), \(y=2^{2}=4\). This is a parabola \(y=x^{2}\) (a U - shaped curve) between \(x=-2\) and \(x = 2\) (including the endpoints \(x=-2\) and \(x = 2\)).

Step3: Graph \(y=-x + 6\) for \(x>2\)

First, find the value of the function at \(x = 2\). When \(x = 2\), \(y=-2 + 6=4\). Since \(x>2\), we draw an open - circle at \(x = 2\). The slope of the line \(y=-x + 6\) is \(-1\) and the \(y\) - intercept is \(6\). We use the slope - intercept form \(y=mx + b\) (where \(m=-1\) and \(b = 6\)) to draw the line to the right of \(x = 2\).

Answer:

The graph consists of a horizontal line \(y = 4\) for \(x<-2\) (open - circle at \(x=-2\)), a parabola \(y=x^{2}\) for \(-2\leq x\leq2\) (closed - circles at \(x=-2\) and \(x = 2\)), and a line \(y=-x + 6\) for \(x>2\) (open - circle at \(x = 2\)).