Question

y = -4|x - 3| + 2

Explanation:

Step1: Identify vertex form of absolute value function

The general form of an absolute value function is \( y = a|x - h| + k \), where \((h, k)\) is the vertex. For \( y = -4|x - 3| + 2 \), the vertex is \((3, 2)\).

Step2: Determine direction and vertical stretch

The coefficient \( a = -4 \) is negative, so the graph opens downward. The absolute value of \( a \) (4) means it's vertically stretched by a factor of 4.

Step3: Plot key points

• Vertex: \((3, 2)\)
• For \( x = 3 + 1 = 4 \), \( y = -4|4 - 3| + 2 = -4(1) + 2 = -2 \), so point \((4, -2)\)
• For \( x = 3 - 1 = 2 \), \( y = -4|2 - 3| + 2 = -4(1) + 2 = -2 \), so point \((2, -2)\)
• For \( x = 3 + 2 = 5 \), \( y = -4|5 - 3| + 2 = -4(2) + 2 = -6 \), so point \((5, -6)\)
• For \( x = 3 - 2 = 1 \), \( y = -4|1 - 3| + 2 = -4(2) + 2 = -6 \), so point \((1, -6)\)

Step4: Draw the graph

Connect the points with two straight lines (since it's an absolute value graph) opening downward from the vertex \((3, 2)\), passing through the calculated points.

Answer:

The graph of \( y = -4|x - 3| + 2 \) has a vertex at \((3, 2)\), opens downward, is vertically stretched by a factor of 4, and passes through points like \((2, -2)\), \((4, -2)\), \((1, -6)\), \((5, -6)\) (graph drawn as described).