Question

graph each equation.

1. $y = -|x| + 2$
2. $y = |x| - 4$

Explanation:

Step1: Analyze \( y = -|x| + 2 \)

The parent function is \( y = |x| \), which is a V - shaped graph with vertex at \((0,0)\). For \( y=-|x| + 2\), the negative sign reflects the graph of \( y = |x| \) over the \( x \) - axis, and the \( + 2 \) shifts the graph up by 2 units. So the vertex of \( y=-|x|+2 \) is at \((0,2)\). When \( x = 0\), \( y=2\); when \( x = 1\) or \( x=-1\), \( y=-1 + 2=1\); when \( x = 2\) or \( x = - 2\), \( y=-2 + 2 = 0\); when \( x=3\) or \( x=-3\), \( y=-3 + 2=-1\) and so on. We can plot these points \((0,2)\), \((1,1)\), \((-1,1)\), \((2,0)\), \((-2,0)\) etc. on the first grid (left - hand grid) and draw the V - shaped graph opening downwards with vertex at \((0,2)\).

Step2: Analyze \( y=|x|-4 \)

The parent function is \( y = |x| \). The \( - 4 \) shifts the graph of \( y = |x| \) down by 4 units. So the vertex of \( y = |x|-4\) is at \((0,-4)\). When \( x = 0\), \( y=-4\); when \( x = 1\) or \( x=-1\), \( y=1 - 4=-3\); when \( x = 2\) or \( x=-2\), \( y=2 - 4=-2\); when \( x = 3\) or \( x=-3\), \( y=3 - 4=-1\) and so on. We can plot these points \((0,-4)\), \((1,-3)\), \((-1,-3)\), \((2,-2)\), \((-2,-2)\) etc. on the second grid (right - hand grid) and draw the V - shaped graph opening upwards with vertex at \((0,-4)\).

(Note: Since the question says "Graph each equation", we have provided the steps to graph each of the absolute - value functions by analyzing the transformations from the parent function \( y = |x|\))

Answer:

To graph \( y=-|x| + 2\): Plot the vertex \((0,2)\), and points like \((1,1)\), \((-1,1)\), \((2,0)\), \((-2,0)\) on the left grid and draw a downward - opening V - shape. To graph \( y = |x|-4\): Plot the vertex \((0,-4)\), and points like \((1,-3)\), \((-1,-3)\), \((2,-2)\), \((-2,-2)\) on the right grid and draw an upward - opening V - shape.