Activity 1.9.4.
In what follows, we work to understand two different piecewise functions entirely by hand based on familiar properties of linear and quadratic functions.
(a)
Consider the function \(p\) defined by the following rule:
\begin{equation*}
p(x) =
\begin{cases}
-(x+2)^2 + 2, \amp x \lt 0 \\
\frac{1}{2}(x-2)^2 + 1, \amp x \ge 0
\end{cases}
\end{equation*}
What are the values of \(p(-4)\text{,}\) \(p(-2)\text{,}\) \(p(0)\text{,}\) \(p(2)\text{,}\) and \(p(4)\text{?}\)
(b)
What point is the vertex of the quadratic part of \(p\) that is valid for \(x \lt 0\text{?}\) What point is the vertex of the quadratic part of \(p\) that is valid for \(x \ge 0\text{?}\)
(c)
For what values of \(x\) is \(p(x) = 0\text{?}\) In addition, what is the \(y\)-intercept of \(p\text{?}\)
(d)
Sketch an accurate, labeled graph of \(y = p(x)\) on the axes provided at left in the following figure.
(e)
For the function \(f\) defined by the righthand figure in (d), determine a piecewise-defined formula for \(f\) that is expressed in bracket notation similar to the definition of \(y = p(x)\) above.
