find points and then graph the function y, equals, minus, vertical bar, x, minus, 3, vertical bar, minus, 5, comma*y*=−∣*x*−3∣−5,

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.

find points and then graph the function y = −|x − 3| − 5

Explore how to graph the absolute value function y = −|x − 3| − 5 with detailed steps. Learn about key concepts such as graph transformations, vertex calculation, and symmetry. Suitable for students in Grades 8-10.

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