Question

graph the equation.
$y = -6|x + 6|$

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. For the function \(y=-6|x + 6|\), we can rewrite it as \(y=-6|x-(-6)|+0\). So the vertex \((h,k)\) is \((-6,0)\).

Step2: Determine the direction and slope

Since \(a=-6\) and \(a<0\), the graph opens downwards. The slope of the lines forming the V - shape: For \(x\geq - 6\), \(y=-6(x + 6)=-6x-36\), the slope \(m=-6\). For \(x < - 6\), \(y=-6(-(x + 6))=6x + 36\), the slope \(m = 6\).

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

• Plot the vertex \((-6,0)\) on the coordinate plane.
• For the right side (\(x\geq - 6\)): From the vertex \((-6,0)\), when \(x=-5\) (which is \(x=-6 + 1\)), \(y=-6|-5 + 6|=-6\times1=-6\). So the point \((-5,-6)\) is on the graph.
• For the left side (\(x < - 6\)): From the vertex \((-6,0)\), when \(x=-7\) (which is \(x=-6-1\)), \(y=-6|-7 + 6|=-6\times1=-6\). So the point \((-7,-6)\) is on the graph.
• We can also find the \(y\) - intercept by setting \(x = 0\). Then \(y=-6|0 + 6|=-6\times6=-36\). But since our graph has a relatively small range (from \(x=-10\) to \(x = 10\) and \(y=-10\) to \(y = 10\)), we can focus on the points near the vertex.

To graph the function:

1. Mark the vertex \((-6,0)\) on the coordinate grid.
2. Use the slope to draw the two lines forming the V - shape. The line for \(x\geq - 6\) goes through \((-6,0)\) and \((-5,-6)\) (and other points with the same slope), and the line for \(x < - 6\) goes through \((-6,0)\) and \((-7,-6)\) (and other points with the same slope). The graph will be a downward - opening V - shaped graph with vertex at \((-6,0)\) and steeper slopes (because \(|a| = 6>1\)) compared to the parent function \(y = |x|\).

Answer:

The graph is a downward - opening V - shaped graph with vertex at \((-6,0)\), passing through points like \((-5,-6)\) and \((-7,-6)\) (and other points determined by the slope of \(\pm6\) from the vertex).