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 rightmost point \\((2, 0)\\) is graphed as a solid dot.

part 3 of 5
the second rule defines a line with slope \\(2\\) and \\(y\\)-intercept \\((0, -4)\\).

part 4 of 5
since \\(x > 2\\), the leftmost point \\((2, 0)\\) is graphed as an open dot.

part 5 of 5
note: the solid dot from the first function \fills in\ the open dot from the second. a vertex is a point where the graph changes direction.

Explanation:

Analyze the piecewise function

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

$$ r(x) =
LATEXBLOCK0
$$

Analyze the first rule

Using the Parabola Vertex knowledge point

$$
LATEXBLOCK1
$$

Analyze the second rule

Using the Linear Function Properties knowledge point

$$
LATEXBLOCK2
$$

Determine graph behavior at boundary

Using the Piecewise Functions knowledge point

$$
LATEXBLOCK3
$$

Answer:

The piecewise function is defined by:

$$ r(x) =
LATEXBLOCK0
$$

• Part 1: The first rule defines a parabola with vertex \((0, -4)\).
• Part 2: Since \(x \le 2\), the rightmost point \((2, 0)\) is graphed as a solid dot.
• Part 3: The second rule defines a line with slope \(2\) and y-intercept \((0, -4)\).
• Part 4: Since \(x > 2\), the leftmost point \((2, 0)\) is graphed as an open dot.
• Part 5: The solid dot from the first function fills in the open dot from the second, resulting in a continuous curve that changes direction at the vertex \((0, -4)\) and the transition point \((2, 0)\).