Question

use technology to find points and then graph the function $y = -2|x - 4| - 2$, following the instructions below.
plot at least five points that fit on the axes below. click a point to delete it.

Explanation:

Step1: Choose x-values

We can choose x - values around the vertex of the 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 function \(y=-2|x - 4|-2\), the vertex is at \((4,-2)\). Let's choose \(x = 2\), \(x = 3\), \(x = 4\), \(x = 5\), \(x = 6\).

Step2: Calculate y for x = 2

Substitute \(x = 2\) into the function \(y=-2|x - 4|-2\). First, calculate the absolute - value part: \(|2 - 4|=|-2| = 2\). Then, \(y=-2\times2-2=-4 - 2=-6\). So the point is \((2,-6)\).

Step3: Calculate y for x = 3

Substitute \(x = 3\) into the function. \(|3 - 4|=|-1| = 1\). Then \(y=-2\times1-2=-2 - 2=-4\). The point is \((3,-4)\).

Step4: Calculate y for x = 4

Substitute \(x = 4\) into the function. \(|4 - 4|=|0| = 0\). Then \(y=-2\times0-2=0 - 2=-2\). The point is \((4,-2)\).

Step5: Calculate y for x = 5

Substitute \(x = 5\) into the function. \(|5 - 4|=|1| = 1\). Then \(y=-2\times1-2=-2 - 2=-4\). The point is \((5,-4)\).

Step6: Calculate y for x = 6

Substitute \(x = 6\) into the function. \(|6 - 4|=|2| = 2\). Then \(y=-2\times2-2=-4 - 2=-6\). The point is \((6,-6)\).

To graph the function, we can plot the points \((2,-6)\), \((3,-4)\), \((4,-2)\), \((5,-4)\), \((6,-6)\) and then draw a V - shaped graph (since it's an absolute - value function) opening downwards (because \(a=-2<0\)) with the vertex at \((4,-2)\).

Answer:

The five points are \((2,-6)\), \((3,-4)\), \((4,-2)\), \((5,-4)\), \((6,-6)\) (and the graph is a downward - opening V - shape with vertex at \((4,-2)\)).