Q. Which graph represents the function \(f(x) = 4|x|\)?

Step-by-step solution: Which graph represents f(x) = 4|x| ?

1. Rewrite the absolute-value expression as a piecewise linear function.By the definition of absolute value,

   \( |x| = \begin{cases} x &\text{if } x \ge 0, \\ -x &\text{if } x < 0, \end{cases} \)
   so

   \( f(x) = 4|x| = \begin{cases} 4x &\text{if } x \ge 0, \\ -4x &\text{if } x < 0. \end{cases} \)

2. Find the vertex (corner) of the V-shaped graph.The absolute-value function has its minimum where the inside is zero. Evaluate at \(x=0\):

   \( f(0) = 4|0| = 0. \)

   The vertex is at the point \( (0,0) \).

3. Describe each branch (slope and direction).For \(x \ge 0\), the function is \(f(x)=4x\). This is a straight line through the origin with slope 4 (rises 4 units for each 1 unit to the right). Example points: \( (1,4), (2,8) \).

   For \(x < 0\), the function is \(f(x)=-4x\). This is a straight line through the origin with slope \(-4\) as a linear rule, but because \(x\) is negative on this side the graph also rises as you move left from the origin. Example points: \( (-1,4), (-2,8) \).

4. Domain, range, and symmetry.Domain: all real numbers, \( (-\infty,\infty) \).

   Range: all real numbers greater than or equal to 0, \( [0,\infty) \).

   Symmetry: the function is even because \(f(-x)=f(x)\); the graph is symmetric about the y-axis.

5. Visual description to identify the correct graph.The correct graph is a V-shaped graph with its corner at the origin \( (0,0) \). Both arms open upward. The arms are straight lines with slopes of magnitude 4: the right arm follows the line \(y=4x\) for \(x\ge 0\), and the left arm follows the line \(y=-4x\) for \(x\le 0\). Compared to \(y=|x|\), this graph is vertically stretched by a factor of 4, so it is steeper.
6. Plotting quick checklist (if you are matching to multiple-choice graphs):
   • Look for a V with vertex at the origin.
   • Check that y-values at x=1 and x=-1 are both 4.
   • Check that the graph is steeper than \(y=|x|\) (points at x=2 should give y=8).
   • Range must be nonnegative; there must be no part of the graph below the x-axis.

Conclusion: The graph that represents \( f(x)=4|x| \) is the V-shaped graph with vertex at (0,0), symmetric about the y-axis, with the right branch following \(y=4x\) for \(x\ge 0\) and the left branch following \(y=-4x\) for \(x\le 0\). This graph is a vertical stretch by factor 4 of the graph of \(y=|x|\).