Question

graph the piecewise function given below.
$f(x) = \

$$\begin{cases} -4 & \\text{for } x \\leq 4 \\\\ (x - 5)^2 & \\text{for } 4 < x \\leq 7 \\end{cases}$$

$
step 1: select a function then move with blue dot (if necessary) done

Explanation:

Step 1: Graph \( f(x) = -4 \) for \( x \leq 4 \)

This is a horizontal line. The \( y \)-value is always -4, and \( x \) can be any real number less than or equal to 4. So we draw a horizontal line at \( y = -4 \), starting from the left (as \( x \) approaches negative infinity) and ending at \( x = 4 \) (with a closed dot at \( (4, -4) \) because \( x = 4 \) is included in this interval).

Step 2: Graph \( f(x) = (x - 5)^2 \) for \( 4 < x \leq 7 \)

This is a parabola in vertex form \( y = a(x - h)^2 + k \), where the vertex is at \( (h, k) = (5, 0) \). Since \( a = 1 > 0 \), the parabola opens upwards. We need to graph this parabola for \( x \) values between 4 (not including 4, so open dot at \( x = 4 \)) and 7 (including 7, so closed dot at \( x = 7 \)).

First, find some key points:

• At \( x = 5 \), \( f(5) = (5 - 5)^2 = 0 \) (vertex).
• At \( x = 4 \), \( f(4) = (4 - 5)^2 = 1 \) (open dot at \( (4, 1) \) because \( x = 4 \) is not included in this interval).
• At \( x = 7 \), \( f(7) = (7 - 5)^2 = 4 \) (closed dot at \( (7, 4) \) because \( x = 7 \) is included in this interval).

Plot the vertex \( (5, 0) \), the open dot at \( (4, 1) \), and the closed dot at \( (7, 4) \), then draw the upward - opening parabola segment connecting these points for \( 4 < x \leq 7 \).

(Note: Since this is a graphing problem, the final answer would be the visual graph as described above. If we were to describe the graph in text, it has a horizontal line at \( y=-4 \) for \( x\leq4 \) and an upward - opening parabolic segment from \( x = 4 \) (open dot) to \( x = 7 \) (closed dot) with vertex at \( (5,0) \).)

Answer:

The graph consists of a horizontal line \( y = - 4 \) for \( x\leq4 \) (with a closed dot at \( (4,-4) \)) and an upward - opening parabolic segment \( y=(x - 5)^2 \) for \( 4