Question

question
use technology to find points and then graph the function $y = -2|x| - 1$, following the
instructions below.

Explanation:

Step1: Analyze the function type

The function \( y = -2|x| - 1 \) is an absolute - value function. 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. In our function, \( h = 0\) and \( k=-1\), and \( a=-2\).

Step2: Find key points

• When \( x = 0\):

Substitute \( x = 0\) into the function \( y=-2|0|-1\). Since \(|0| = 0\), we have \( y=-2\times0 - 1=-1\). So the point \((0,-1)\) is on the graph.

• When \( x = 1\):

Substitute \( x = 1\) into the function \( y=-2|1|-1\). Since \(|1| = 1\), we get \( y=-2\times1-1=-3\). So the point \((1, - 3)\) is on the graph.

• When \( x=-1\):

Substitute \( x=-1\) into the function \( y=-2|-1|-1\). Since \(|-1| = 1\), we have \( y=-2\times1 - 1=-3\). So the point \((-1,-3)\) is on the graph.

• When \( x = 2\):

Substitute \( x = 2\) into the function \( y=-2|2|-1\). Since \(|2| = 2\), we get \( y=-2\times2-1=-5\). So the point \((2,-5)\) is on the graph.

• When \( x=-2\):

Substitute \( x=-2\) into the function \( y=-2|-2|-1\). Since \(|-2| = 2\), we have \( y=-2\times2-1=-5\). So the point \((-2,-5)\) is on the graph.

Step3: Graph the function

The graph of the function \( y = - 2|x|-1\) is a V - shaped graph (because it is an absolute - value function) that opens downwards (because \( a=-2<0\)) with the vertex at \((0,-1)\). We can plot the points we found \((0,-1)\), \((1, - 3)\), \((-1,-3)\), \((2,-5)\), \((-2,-5)\) and then draw a V - shaped graph passing through these points. The left side of the graph (for \(x<0\)) has a slope of \(2\) (since the slope \(m\) of the line \(y=-2|x|-1\) for \(x < 0\) is \(m = 2\) because \(y=-2(-x)-1 = 2x-1\) when \(x<0\)) and the right side (for \(x>0\)) has a slope of \(- 2\) (since \(y=-2x - 1\) when \(x>0\)).

Answer:

To graph \(y=-2|x|-1\):

1. Identify the vertex at \((0, - 1)\) (from the form \(y=a|x - h|+k\) with \(h = 0,k=-1,a=-2\)).
2. Find additional points:

• For \(x = 1\), \(y=-3\) (point \((1,-3)\)).
• For \(x=-1\), \(y=-3\) (point \((-1,-3)\)).
• For \(x = 2\), \(y=-5\) (point \((2,-5)\)).
• For \(x=-2\), \(y=-5\) (point \((-2,-5)\)).

1. Plot the vertex and the additional points. Then draw a V - shaped graph opening downward (since \(a=-2<0\)) passing through these points. The left - hand branch (for \(x<0\)) has a slope of \(2\) and the right - hand branch (for \(x > 0\)) has a slope of \(-2\).