Question

graph this function:
y = -|x| - 6
click to plot the vertex first.
(graph with x-axis from -10 to 10 and y-axis from -10 to 10, grid lines, and a coordinate plane)

Explanation:

Step1: Identify the vertex form

The function \( y = -|x| - 6 \) is a transformation of the parent absolute - value function \( y = |x| \). The general form of an absolute - value function is \( y=a|x - h|+k \), where \((h,k)\) is the vertex. For the function \( y=-|x|-6 \), we can rewrite it as \( y = -|x - 0|-6 \). So, \( h = 0 \) and \( k=-6 \).

Step2: Plot the vertex

Using the values of \( h \) and \( k \) from Step 1, the vertex of the function \( y=-|x|-6 \) is at the point \((0,-6)\). To plot the vertex, we find the point on the coordinate plane where \( x = 0 \) and \( y=-6 \).

Step3: Find other points

We can find two more points to graph the function.

• When \( x = 1 \):

Substitute \( x = 1 \) into the function \( y=-|x|-6 \). We know that \( |1| = 1 \), so \( y=-1 - 6=-7 \). So, the point \((1,-7)\) is on the graph.

• When \( x=-1 \):

Substitute \( x = - 1 \) into the function \( y=-|x|-6 \). We know that \( |-1| = 1 \), so \( y=-1 - 6=-7 \). So, the point \((-1,-7)\) is on the graph.
We can also find more points for better accuracy. For example, when \( x = 2 \), \( y=-|2|-6=-2 - 6=-8 \), so the point \((2,-8)\) is on the graph. When \( x=-2 \), \( y=-|-2|-6=-2 - 6=-8 \), so the point \((-2,-8)\) is on the graph.
After plotting the vertex \((0,-6)\) and other points like \((1,-7)\), \((-1,-7)\), \((2,-8)\), \((-2,-8)\) etc., we can draw the graph of the function \( y=-|x|-6 \). The graph is a "V - shaped" graph (since it is an absolute - value function) opening downwards (because \( a=-1<0 \)) with the vertex at \((0,-6)\).

Answer:

The vertex of the function \( y = -|x|-6 \) is at \((0,-6)\), and the graph can be drawn by plotting the vertex \((0,-6)\) and other points such as \((1,-7)\), \((-1,-7)\), \((2,-8)\), \((-2,-8)\) and then connecting them to form a downward - opening V - shaped graph.