Question

graph the following absolute value expression:
$y = -|x| - 2$
show your work here
draw

Explanation:

Step1: Analyze the parent function

The parent absolute value function is \( y = |x| \), which has a V - shape with vertex at \((0,0)\), opening upwards, with slopes \(1\) (for \(x\geq0\)) and \(- 1\) (for \(x < 0\)).

Step2: Analyze transformations

For the function \( y=-\vert x\vert - 2\), we have two transformations:

• The negative sign in front of \(|x|\) reflects the graph of \(y = |x|\) over the \(x\) - axis. So the graph will open downwards.
• The \(-2\) at the end shifts the graph of \(y=-\vert x\vert\) down by 2 units.

Step3: Find the vertex

The vertex of \(y = |x|\) is \((0,0)\). After reflection and vertical shift, the vertex of \(y=-\vert x\vert - 2\) is \((0,-2)\).

Step4: Find the slopes and equations for each piece

• For \(x\geq0\), \(y =-\vert x\vert - 2=-x - 2\). The slope \(m=- 1\). Using the point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(0,-2)\), we get \(y=-x - 2\) (for \(x\geq0\)).
• For \(x < 0\), \(y =-\vert x\vert - 2=-(-x)-2=x - 2\). The slope \(m = 1\). Using the point - slope form with \((x_1,y_1)=(0,-2)\), we get \(y=x - 2\) (for \(x < 0\)).

Step5: Plot points

• For \(x = 0\), \(y=-|0|-2=-2\), so the point is \((0,-2)\).
• For \(x = 1\) (since \(x\geq0\)), \(y=-1 - 2=-3\), so the point is \((1,-3)\).
• For \(x=-1\) (since \(x < 0\)), \(y=-1 - 2=-3\), so the point is \((-1,-3)\).
• For \(x = 2\) (since \(x\geq0\)), \(y=-2 - 2=-4\), so the point is \((2,-4)\).
• For \(x=-2\) (since \(x < 0\)), \(y=-2 - 2=-4\), so the point is \((-2,-4)\).

We can then draw a V - shaped graph opening downwards with vertex at \((0,-2)\), passing through the points we calculated.

Answer:

The graph of \(y =-\vert x\vert - 2\) is a V - shaped graph opening downwards with vertex at \((0,-2)\), with the left - hand side (for \(x < 0\)) having a slope of \(1\) (equation \(y=x - 2\)) and the right - hand side (for \(x\geq0\)) having a slope of \(-1\) (equation \(y=-x - 2\)).