Question

graph this function:
y = -|x|
click to plot the vertex first.

Explanation:

Step1: Find the vertex of \( y = -|x| \)

The general form of an absolute - value function is \( y=a|x - h|+k \), where \((h,k)\) is the vertex. For the function \( y=-|x| \), we can rewrite it as \( y = -|x - 0|+0 \). So, the vertex \((h,k)\) is \((0,0)\). We plot the point \((0,0)\) on the coordinate plane.

Step2: Analyze the slope of the two parts of the absolute - value function

For \( x\geq0 \), the function \( y=-|x| \) becomes \( y=-x \). The slope of the line \( y = mx + b\) (here \( b = 0\)) is \( m=- 1\). So, when \( x = 1\), \( y=-1\); when \( x = 2\), \( y=-2\), and so on. We can plot these points \((1, - 1)\), \((2,-2)\) etc. on the right - hand side of the vertex (\(x\geq0\)).
For \( x<0 \), the function \( y=-|x| \) becomes \( y = -(-x)=x\) (since \(|x|=-x\) when \(x < 0\)). The slope of the line \( y=x\) is \( m = 1\). So, when \( x=-1\), \( y=-1\); when \( x = - 2\), \( y=-2\), and so on. We can plot these points \((-1,-1)\), \((-2,-2)\) etc. on the left - hand side of the vertex (\(x < 0\)).
After plotting the vertex and a few other points on both sides, we can draw the graph of the function \( y=-|x| \), which is a V - shaped graph opening downwards with the vertex at the origin \((0,0)\).

Answer:

The vertex is at \((0,0)\), and the graph is a V - shaped graph opening downward with points like \((1, - 1)\), \((-1,-1)\), \((2,-2)\), \((-2,-2)\) etc. plotted and connected. (Note: Since the question is about graphing, the key steps are plotting the vertex \((0,0)\) and then plotting other points based on the slope of the two linear parts of the absolute - value function to form the graph of \(y=-|x|\))