To graph the equation \( y = -5|x+3| \), follow these steps:

### 1. **Understand the Basic Shape**:
   The function \( y = |x+3| \) is an absolute value function, which creates a "V" shape. The absolute value means that the graph reflects any negative values back to positive ones, making the graph symmetric around its vertex.

### 2. **Transform the Graph**:
   - **Horizontal Shift**: The expression \( |x + 3| \) shifts the basic \( |x| \) graph **3 units to the left**. This changes the vertex of the graph from \( (0, 0) \) to \( (-3, 0) \).
   - **Vertical Stretch and Reflection**: The coefficient of \(-5\) outside the absolute value causes two transformations:
     - **Vertical Stretch**: The factor \( 5 \) makes the graph steeper by multiplying the y-values by 5.
     - **Reflection**: The negative sign flips the graph upside down, meaning the "V" shape points downwards instead of upwards.

### 3. **Plot the Vertex**:
   The vertex is at \( (-3, 0) \). Start by plotting this point.

### 4. **Find Other Points**:
   To get a sense of how steep the graph is, choose a few points to the left and right of the vertex and calculate their y-values:
   
   - For \( x = -4 \), 
     \[
     y = -5|-4+3| = -5|1| = -5
     \]
     So, the point is \( (-4, -5) \).
     
   - For \( x = -2 \),
     \[
     y = -5|-2+3| = -5|1| = -5
     \]
     So, the point is \( (-2, -5) \).
     
   - For \( x = -6 \),
     \[
     y = -5|-6+3| = -5|3| = -15
     \]
     So, the point is \( (-6, -15) \).

   - For \( x = 0 \),
     \[
     y = -5|0+3| = -5|3| = -15
     \]
     So, the point is \( (0, -15) \).

### 5. **Draw the Graph**:
   - Plot the points you calculated: \( (-3, 0) \), \( (-4, -5) \), \( (-2, -5) \), \( (-6, -15) \), and \( (0, -15) \).
   - Connect the points with straight lines to form the inverted "V" shape.
   - Remember, the vertex is the highest point, and the graph extends downward on both sides.

### 6. **Label the Axes**:
   - Ensure your axes are labeled properly, especially the y-values which will have negative numbers since the graph points downward.

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This completes the graph of \( y = -5|x+3| \). Would you like further details on any of the transformations or specific points?

### 5 Related Questions:
1. How would the graph change if the coefficient of \( x \) inside the absolute value was modified?
2. What happens to the graph of \( y = -5|x+3| \) if we remove the negative sign?
3. How do vertical and horizontal stretches/compressions affect absolute value functions in general?
4. How would you graph \( y = 3|x - 2| \)? What are the main differences?
5. How does changing the vertex impact the symmetry of the graph?

### Tip:
When graphing absolute value functions, always start by identifying the vertex and how the coefficients transform the graph. This simplifies plotting points.

Learn how to graph the absolute value function y = −5|x + 3| step by step. This guide covers vertical stretches, reflections, and horizontal shifts, ideal for students in Grades 9-11.

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