Question

graph $y = 3|x - 3| - 2$.

Explanation:

Step1: Identify the vertex

The general form of an absolute - value function is \(y = a|x - h|+k\), where \((h,k)\) is the vertex of the V - shaped graph. For the function \(y = 3|x - 3|-2\), we have \(h = 3\) and \(k=-2\). So the vertex of the graph is at the point \((3,-2)\).

Step2: Determine the slope of the two arms

For the absolute - value function \(y = a|x - h|+k\), when \(x\geq h\), the function can be written as \(y=a(x - h)+k\) (a linear function with slope \(a\)), and when \(x\lt h\), the function can be written as \(y=-a(x - h)+k\) (a linear function with slope \(-a\)). Here, \(a = 3\).

• When \(x\geq3\), the function is \(y = 3(x - 3)-2=3x-9 - 2=3x - 11\). The slope of this line is \(m = 3\). To find a point on this line, we can use \(x = 4\). Substitute \(x = 4\) into \(y=3x - 11\), we get \(y=3\times4-11 = 12 - 11=1\). So the point \((4,1)\) is on the graph.
• When \(x\lt3\), the function is \(y=-3(x - 3)-2=-3x + 9-2=-3x + 7\). The slope of this line is \(m=-3\). To find a point on this line, we can use \(x = 2\). Substitute \(x = 2\) into \(y=-3x + 7\), we get \(y=-3\times2 + 7=-6 + 7 = 1\). So the point \((2,1)\) is on the graph.

Step3: Plot the vertex and the two points

First, plot the vertex \((3,-2)\) on the coordinate plane. Then plot the points \((4,1)\) and \((2,1)\). Then draw two lines: one line passing through \((3,-2)\) and \((4,1)\) (for \(x\geq3\)) and another line passing through \((3,-2)\) and \((2,1)\) (for \(x\lt3\)). The graph should be a V - shaped graph with the vertex at \((3,-2)\), opening upwards (since \(a = 3>0\)) and with the slopes of the two arms being \(3\) and \(-3\) respectively.

(Note: Since the task is to graph, the final answer is the graph constructed by following the above steps. But in terms of the key points, the vertex is \((3,-2)\) and the two additional points are \((2,1)\) and \((4,1)\) which help in drawing the graph.)

Answer:

The graph of \(y = 3|x - 3|-2\) has a vertex at \((3,-2)\), passes through \((2,1)\) and \((4,1)\), and is a V - shaped graph opening upwards with slopes \(3\) (for \(x\geq3\)) and \(-3\) (for \(x\lt3\)).