Question

graph the function.

\
(x) = \

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

\\

part: 0 / 5

part 1 of 5

the first rule defines a select with vertex ( , ).

Explanation:

Identify the first rule of the piecewise function

Using the Piecewise Functions knowledge point
The piecewise function is defined as:

$$ r(x) =
LATEXBLOCK0
$$

The first rule is \(y = x^2 - 4\) for \(x \le 2\).

Determine the shape and vertex of the first rule

The equation \(y = x^2 - 4\) is a quadratic function in standard form \(y = ax^2 + bx + c\) where \(a = 1\), \(b = 0\), and \(c = -4\).

• Since \(a > 0\), it defines a parabola opening upwards.
• The \(x\)-coordinate of the vertex is \(x = -\frac{b}{2a} = 0\).
• The \(y\)-coordinate of the vertex is \(y = 0^2 - 4 = -4\).

Thus, the first rule defines a parabola with vertex \((0, -4)\).

Answer:

The first rule defines a <blank>parabola</blank> with vertex (<blank>0</blank>, <blank>-4</blank>).