Question

1. graph: $y = 2|x - 3| - 5$

Explanation:

Step1: Identify the 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 the given function \( y = 2|x - 3| - 5 \), we can identify \( a = 2 \), \( h = 3 \), and \( k = -5 \). So the vertex of the graph is at \((3, -5)\).

Step2: Determine the direction and slope

Since \( a = 2 > 0 \), the graph opens upwards. The slope of the right side (when \( x \geq h \)) is \( a = 2 \), and the slope of the left side (when \( x < h \)) is \( -2 \).

Step3: Plot the vertex and use the slopes to find other points

• Plot the vertex \((3, -5)\) on the coordinate plane.
• For \( x > 3 \) (right side), use the slope \( 2 \). For example, when \( x = 4 \), \( y = 2|4 - 3| - 5 = 2(1) - 5 = -3 \), so the point is \((4, -3)\). When \( x = 5 \), \( y = 2|5 - 3| - 5 = 2(2) - 5 = -1 \), so the point is \((5, -1)\).
• For \( x < 3 \) (left side), use the slope \( -2 \). For example, when \( x = 2 \), \( y = 2|2 - 3| - 5 = 2(1) - 5 = -3 \), so the point is \((2, -3)\). When \( x = 1 \), \( y = 2|1 - 3| - 5 = 2(2) - 5 = -1 \), so the point is \((1, -1)\).

Step4: Draw the graph

Connect the points with two straight lines, forming a V - shape with the vertex at \((3, -5)\), opening upwards, passing through the points calculated above.

Answer:

The graph is a V - shaped graph (absolute value graph) with vertex at \((3, -5)\), opening upwards, passing through points like \((2, -3)\), \((4, -3)\), \((1, -1)\), \((5, -1)\) etc. (The actual drawing should be done on the given coordinate grid with the vertex at \((3, -5)\) and the two arms having slopes \(2\) (right) and \(-2\) (left)).