Q. which graph represents the function \( f(x) = 2|x| \)?

Which graph represents the function \( f(x) = 2|x| \)?

1. Recall the definition of absolute value.

   The absolute value function is
   \[
   |x| = \begin{cases}
   x & \text{if } x \ge 0,\\[6pt]
   -x & \text{if } x < 0. \end{cases} \]

   This means for nonnegative x we keep x, and for negative x we take its negative so the result is nonnegative.

2. Apply the factor 2 to the absolute value.

   Multiply the definition by 2 to get a piecewise formula for f:
   \[
   f(x) = 2|x| = \begin{cases}
   2x & \text{if } x \ge 0,\\[6pt]
   -2x & \text{if } x < 0. \end{cases} \]

   This gives two linear expressions, one for the right side (x ≥ 0) and one for the left side (x < 0).

3. Find the vertex (the corner point).

   Evaluate at x = 0:
   \[
   f(0) = 2|0| = 0.
   \]

   The graph has a corner (vertex) at the point (0, 0).

4. Determine slopes of the two pieces.

   For x ≥ 0 the graph is the line y = 2x, which has slope 2.
   For x < 0 the graph is the line y = −2x, which has slope −2.

   So the right arm rises to the right with slope 2 and the left arm rises to the left with slope 2 in absolute value (equivalently the left arm falls to the right with slope −2).

5. Check symmetry, domain, and range.

   Symmetry: f is even because
   \[
   f(-x) = 2|-x| = 2|x| = f(x).
   \]
   The graph is symmetric about the y-axis.

   Domain: all real numbers, written as (−∞, ∞).

   Range: f(x) ≥ 0 for all x, so the range is [0, ∞).

6. Plot a few key points to see the shape.

   Compute f at simple x-values:

   • f(0) = 0 → point (0, 0)
   • f(1) = 2|1| = 2 → point (1, 2)
   • f(2) = 4 → point (2, 4)
   • f(−1) = 2 → point (−1, 2)
   • f(−2) = 4 → point (−2, 4)

   These points lie on two straight lines that meet at the origin.

7. Describe the graph.

   The graph is a V-shaped curve with vertex at (0, 0). The right branch (x ≥ 0) is the line y = 2x. The left branch (x ≤ 0) is the line y = −2x. The V opens upward, is symmetric about the y-axis, and is steeper than y = |x| because of the vertical stretch factor 2.

8. Final conclusion — which graph represents f(x) = 2|x|?

   The correct graph is the V-shaped graph with vertex at the origin, arms given by the lines y = 2x for x ≥ 0 and y = −2x for x ≤ 0, domain (−∞, ∞), and range [0, ∞). Equivalently, the graph obtained by taking y = |x| and stretching it vertically by a factor of 2.