Q. Which graph represents the function \(f(x) = -\lvert x + 3 \rvert\)?

Problem

Which graph represents the function

\[ f(x) = -\lvert x + 3 \rvert \]

Step-by-step explanation

1. Start with the basic absolute value graph

   The graph of

   \[ y = \lvert x \rvert \]

   is a V-shaped curve with vertex at (0,0). For this basic graph the left branch has slope -1 and the right branch has slope 1. The domain is all real numbers and the range is \[ y \ge 0 \].

2. Apply the horizontal shift

   Replacing x by x + 3 shifts the graph left by 3 units. So consider

   \[ y = \lvert x + 3 \rvert. \]

   This graph is the same V-shape translated left, so its vertex is at

   \[ (-3,\,0). \]

   For this graph the piecewise linear form is

   \[ \lvert x + 3 \rvert = \begin{cases}
   -(x+3), & x < -3,\\[6pt] x+3, & x \ge -3. \end{cases} \]

3. Apply the vertical reflection (multiply by -1)

   Now multiply the shifted absolute value by -1 to get the given function:

   \[ f(x) = -\lvert x + 3 \rvert. \]

   Multiplying by -1 reflects the graph across the x-axis. The vertex remains at (-3,0) because 0 times -1 is still 0, but the V now opens downward instead of upward. The range becomes

   \[ y \le 0. \]

4. Write the piecewise linear form of f(x)

   Use the piecewise expression for \lvert x + 3 \rvert and multiply by -1:

   \[ f(x) = -\lvert x + 3 \rvert = \begin{cases}
   x+3, & x < -3,\\[6pt] -x-3, & x \ge -3. \end{cases} \]

   So the left arm (for x < -3) is the line y = x + 3 with slope 1, and the right arm (for x ≥ -3) is the line y = -x – 3 with slope -1. Both meet at the vertex (-3,0).

5. Find key points and intercepts
   • Vertex: (-3, 0).
   • X-intercept(s): solve \[ -\lvert x + 3 \rvert = 0 \] so \[ \lvert x + 3 \rvert = 0 \] which gives \[ x = -3. \] Thus the only x-intercept is (-3,0).
   • Y-intercept: evaluate \[ f(0) = -\lvert 0 + 3 \rvert = -3. \] So the graph passes through (0, -3).
   • End behavior: as x goes to positive infinity, f(x) goes to negative infinity along the line y = -x – 3; as x goes to negative infinity, f(x) goes to negative infinity along the line y = x + 3.
6. Description that identifies the correct graph

   The correct graph is a V-shaped graph pointing downward with its apex (vertex) at (-3,0). It has a right branch that slopes down at 45 degrees and a left branch that slopes up at 45 degrees (when looking leftwards), it passes through (0,-3), and its entire graph lies at or below the x-axis. The domain is all real numbers and the range is y ≤ 0.

Final answer

The graph representing \[ f(x) = -\lvert x + 3 \rvert \] is the downward-opening V with vertex at (-3,0), passing through (0,-3), with left-arm line y = x + 3 for x < -3 and right-arm line y = -x – 3 for x ≥ -3, and range \[ y \le 0. \]