Question

graph the function.

\
(x) = \

$$\begin{cases} x^2 - 4 & \\text{for } x \\le 2 \\\\ 2x - 4 & \\text{for } x > 2 \\end{cases}$$

\\

part 1 of 5
the first rule defines a parabola with vertex \\((0, -4)\\).

part 2 of 5
since \\(x \le 2\\), the select point \\((\quad, \quad)\\) is graphed as a solid dot.

Explanation:

Evaluate the first rule at the boundary

Using the Piecewise Functions knowledge point
The first rule of the piecewise function is \(r(x) = x^2 - 4\) for \(x \le 2\).
To find the boundary point (or endpoint) of this piece, we evaluate the function at \(x = 2\):

$$r(2) = 2^2 - 4 = 4 - 4 = 0$$

Determine the endpoint type and coordinates

Using the Piecewise Graphing knowledge point
Since the inequality is \(x \le 2\) (which includes equality), the boundary point is an endpoint of this section of the graph.
The coordinates of this endpoint are \((2, 0)\).
Because the inequality includes the "or equal to" sign (\(\le\)), this boundary point is plotted as a solid dot.
Thus, the dropdown selection is "endpoint" and the coordinates are \((2, 0)\).

Answer:

Since \(x \le 2\), the <blank>endpoint</blank> point <blank>\((2, 0)\)</blank> is graphed as a solid dot.