The exercise asks you to graph the given functions, compare their graphs to the standard absolute value function $f(x) = |x|$, and describe their domains and ranges. Let's break them down step by step.

Analyzing Each Function

1. $d(x) = |x| - 4$

   • Transformation: This graph is shifted down by 4 units compared to $f(x) = |x|$.
   • Domain: All real numbers ($\mathbb{R}$).
   • Range: $y \geq -4$.

2. $m(x) = |x + 1|$

   • Transformation: This graph is shifted left by 1 unit compared to $f(x) = |x|$.
   • Domain: All real numbers ($\mathbb{R}$).
   • Range: $y \geq 0$.

3. $r(x) = |x| + 5$

   • Transformation: This graph is shifted up by 5 units compared to $f(x) = |x|$.
   • Domain: All real numbers ($\mathbb{R}$).
   • Range: $y \geq 5$.

4. $v(x) = |x - 3|$

   • Transformation: This graph is shifted right by 3 units compared to $f(x) = |x|$.
   • Domain: All real numbers ($\mathbb{R}$).
   • Range: $y \geq 0$.

5. $p(x) = \frac{1}{3}|x|$

   • Transformation: This graph is vertically compressed by a factor of $\frac{1}{3}$.
   • Domain: All real numbers ($\mathbb{R}$).
   • Range: $y \geq 0$.

6. $j(x) = 3|x|$

   • Transformation: This graph is vertically stretched by a factor of 3.
   • Domain: All real numbers ($\mathbb{R}$).
   • Range: $y \geq 0$.

7. $a(x) = -5|x|$

   • Transformation: This graph is vertically reflected and stretched by a factor of 5.
   • Domain: All real numbers ($\mathbb{R}$).
   • Range: $y \leq 0$ (since it’s reflected down).

8. $q(x) = -\frac{3}{2}|x|$

   • Transformation: This graph is vertically reflected and stretched by a factor of $\frac{3}{2}$.
   • Domain: All real numbers ($\mathbb{R}$).
   • Range: $y \leq 0$.

Summary of Transformations

• Vertical shifts: $d(x), r(x)$.
• Horizontal shifts: $m(x), v(x)$.
• Vertical stretch/compression: $p(x), j(x)$.
• Reflections: $a(x), q(x)$.

Would you like a detailed graph or further exploration of these transformations?

Related Questions

1. What are the key differences between vertical and horizontal shifts for absolute value functions?
2. How does vertical stretching/compression affect the steepness of the graph?
3. What is the significance of reflecting the graph vertically?
4. Can you describe the domain and range of a general absolute value function?
5. How would combining transformations (e.g., shift + stretch) affect the graph?

Tip

Always start by identifying the type of transformation (shift, stretch, compression, or reflection) before analyzing the graph.