To graph the equation $y = 3|x - 3|$, follow these steps:

Step 1: Understand the equation

The equation is in the form $y = a|x - h| + k$, where:

• $a = 3$ (which affects the slope and stretching of the graph)
• $h = 3$ (which shifts the graph horizontally)
• $k = 0$ (which shifts the graph vertically, but in this case, it remains at 0)

Step 2: Find the vertex

The vertex of the graph is located at $(h, k) = (3, 0)$. This is the point where the graph changes direction.

Step 3: Analyze the graph's shape

• The absolute value function $|x - 3|$ creates a "V" shape.
• The coefficient $3$ outside the absolute value means that the graph is steeper than the standard $y = |x|$ by a factor of 3.

Step 4: Plot points

1. Vertex: The vertex is at $(3, 0)$.

2. To the right of the vertex: For $x > 3$, the graph is a straight line with a slope of 3.

   • For $x = 4$, $y = 3(4 - 3) = 3 \times 1 = 3$.
   • For $x = 5$, $y = 3(5 - 3) = 3 \times 2 = 6$.

3. To the left of the vertex: For $x < 3$, the graph is also a straight line with a slope of -3.

   • For $x = 2$, $y = 3(3 - 2) = 3 \times 1 = 3$.
   • For $x = 1$, $y = 3(3 - 1) = 3 \times 2 = 6$.

Step 5: Draw the graph

• The graph will have a "V" shape with its vertex at $(3, 0)$.
• The arms of the "V" will rise on both sides, but with different directions:
  • The right arm (for $x > 3$) rises to the right.
  • The left arm (for $x < 3$) rises to the left.

The final graph will look like this:

• A "V" shape centered at $(3, 0)$, with both sides of the "V" rising sharply due to the multiplier 3.

Would you like more details or have any questions?

Practice Questions:

1. Graph the equation $y = 2|x + 1|$.
2. What is the vertex of the graph $y = 4|x - 2| + 1$?
3. How does the graph of $y = -|x - 5|$ differ from $y = |x - 5|$?
4. Graph $y = 3|x + 2| - 4$.
5. Determine the vertex and graph $y = 0.5|x - 7|$.

Tip:

When dealing with absolute value functions, remember that the graph is symmetric about the vertical line through the vertex. This symmetry can help you quickly sketch both sides of the graph.