Question

graph the following absolute value expression:
$y = |x - 5| - 6$
show your work here

Explanation:

Step1: Identify the vertex

The general form of an absolute - value function is \(y = |x - h|-k\), where the vertex is \((h,k)\). For the function \(y=|x - 5|-6\), \(h = 5\) and \(k=6\), so the vertex of the graph is \((5,-6)\).

Step2: Find two points to the left of the vertex

Let's choose \(x = 0\) (a value less than \(5\)). Substitute \(x = 0\) into the function:
\(y=|0 - 5|-6=| - 5|-6 = 5-6=-1\). So we have the point \((0,-1)\).
Let's choose \(x = 1\) (a value less than \(5\)). Substitute \(x = 1\) into the function:
\(y=|1 - 5|-6=| - 4|-6 = 4-6=-2\). So we have the point \((1,-2)\).

Step3: Find two points to the right of the vertex

Let's choose \(x = 10\) (a value greater than \(5\)). Substitute \(x = 10\) into the function:
\(y=|10 - 5|-6=|5|-6 = 5 - 6=-1\). So we have the point \((10,-1)\).
Let's choose \(x = 6\) (a value greater than \(5\)). Substitute \(x = 6\) into the function:
\(y=|6 - 5|-6=|1|-6 = 1-6=-5\). So we have the point \((6,-5)\).

Step4: Plot the vertex and the points

Plot the vertex \((5,-6)\), the points \((0,-1)\), \((1,-2)\), \((10,-1)\), \((6,-5)\) on the coordinate plane. Then, draw two rays: one ray passes through the vertex and the points to the left of the vertex (with a slope of \(- 1\) for \(x<5\)), and the other ray passes through the vertex and the points to the right of the vertex (with a slope of \(1\) for \(x > 5\)).

Answer:

The graph of \(y = |x - 5|-6\) is a V - shaped graph with vertex at \((5,-6)\), passing through points like \((0,-1)\), \((1,-2)\), \((6,-5)\), \((10,-1)\) and having a slope of \(-1\) for \(x<5\) and \(1\) for \(x > 5\).