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.