Here are the equations for each problem:

5. Graph function $h$:

   • The graph $f(x) = 2|x|$ is translated down 3 units.
   • The translation shifts the graph vertically. Subtract 3 from $f(x)$.
   • Equation:
     $h(x) = 2|x| - 3$

6. Graph function $d$:

   • The graph $g(x) = |x|$ is translated left 2 units.
   • The translation shifts the graph horizontally. Replace $x$ with $(x + 2)$.
   • Equation:
     $d(x) = |x + 2|$

7. Graph function $w$:

   • The graph $b(x) = 3|x|$ is translated right 4 units.
   • The translation shifts the graph horizontally. Replace $x$ with $(x - 4)$.
   • Equation:
     $w(x) = 3|x - 4|$

Let me know if you’d like further clarification or have additional questions.

5 Follow-Up Questions:

1. What happens to the graph of $|x|$ when you multiply it by a constant, like in $2|x|$?
2. How do vertical and horizontal translations affect the vertex of an absolute value function?
3. Could you describe the vertex form of an absolute value function and how to identify transformations?
4. How would you combine multiple transformations (e.g., translation and reflection) in one equation?
5. What practical situations could these transformations represent, such as scaling or translating absolute value graphs?

Tip:

When working with absolute value functions, always identify changes to the vertex caused by horizontal and vertical shifts to better understand the transformation.