The given function is:

$y = -|x - 3| - 5$

Step 1: Understand the form of the function

This is a transformation of the absolute value function $y = |x|$. Here's how the transformations apply:

• $|x - 3|$: The graph shifts 3 units to the right.
• The negative sign in front of the absolute value means a reflection across the x-axis.
• The -5 at the end means the graph shifts 5 units down.

Step 2: Find key points on the graph

1. Vertex: The vertex occurs where the expression inside the absolute value equals 0, i.e., when $x = 3$. At $x = 3$, we calculate:

   $y = -|3 - 3| - 5 = -0 - 5 = -5$ So, the vertex is $(3, -5)$.

2. Other points: We can plug in other values of $x$ around 3 to get more points.

   • For $x = 2$: $y = -|2 - 3| - 5 = -| -1 | - 5 = -1 - 5 = -6$ So, $(2, -6)$ is a point.

   • For $x = 4$: $y = -|4 - 3| - 5 = -| 1 | - 5 = -1 - 5 = -6$ So, $(4, -6)$ is a point.

   • For $x = 0$: $y = -|0 - 3| - 5 = -| -3 | - 5 = -3 - 5 = -8$ So, $(0, -8)$ is a point.

   • For $x = 6$: $y = -|6 - 3| - 5 = -| 3 | - 5 = -3 - 5 = -8$ So, $(6, -8)$ is a point.

Step 3: Graph the function

• The vertex is at $(3, -5)$, and the graph opens downward (because of the negative sign).
• The points $(2, -6)$ and $(4, -6)$ show the symmetry of the graph around the vertex.
• The points $(0, -8)$ and $(6, -8)$ are further out, confirming the V-shape.

Would you like to visualize the graph or need more details?

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Follow-up Questions:

1. Can you find the vertex of $y = |x - 4| + 2$?
2. What effect does adding a constant outside the absolute value have on the graph?
3. How would the graph change if the equation was $y = |x - 3| + 5$?
4. What happens to the graph if the reflection is removed, i.e., if we consider $y = |x - 3| - 5$?
5. Can you determine the domain and range of the function?

Tip: For any absolute value function, the vertex is the point where the expression inside the absolute value equals zero.