Question

find the graph of the function
( g(x) = |x + 1| - 7 )

use the graphing tool to graph the function.
click to enlarge graph

Explanation:

Step1: Recall the parent function

The parent function of absolute value is \( y = |x| \), which has a vertex at \((0, 0)\), opens upwards, and has a V - shape.

Step2: Analyze the transformations

For the function \( g(x)=|x + 1|-7 \), we use the transformation rules for absolute - value functions.

• The form of a transformed absolute - value function is \( y=a|x - h|+k \), where \((h,k)\) is the vertex of the graph.
• For the function \( g(x)=|x+1|-7=|x-(- 1)|+(-7) \), by comparing with \( y = a|x - h|+k \) (here \( a = 1\), \( h=-1\), \( k=-7\)):
• The horizontal shift: The \( h=-1\) means the graph of \( y = |x|\) is shifted 1 unit to the left. Because for the function \( y = |x - h|\), when \( h\lt0\), the graph is shifted \(|h|\) units to the left.
• The vertical shift: The \( k = - 7\) means the graph is shifted 7 units down. Because for the function \( y=|x - h|+k\), when \( k\lt0\), the graph is shifted \(|k|\) units down.

Step3: Determine the vertex

From the transformation, the vertex of the graph of \( g(x)=|x + 1|-7 \) is at \((h,k)=(-1,-7)\).

Step4: Find two more points to sketch the graph

• When \( x = 0\):

\(g(0)=|0 + 1|-7=1 - 7=-6\). So the point \((0,-6)\) is on the graph.

• When \( x = 1\):

\(g(1)=|1 + 1|-7=2 - 7=-5\). So the point \((1,-5)\) is on the graph.

• When \( x=-2\):

\(g(-2)=|-2 + 1|-7=|-1|-7 = 1-7=-6\). So the point \((-2,-6)\) is on the graph.

To graph the function:

1. Plot the vertex \((-1,-7)\).
2. Plot the points \((0,-6)\), \((1,-5)\), \((-2,-6)\) (and other points if needed).
3. Draw a V - shaped graph (since \( a = 1\gt0\), the graph opens upwards) passing through these points with the vertex at \((-1,-7)\).

Answer:

The graph of \( g(x)=|x + 1|-7 \) is a V - shaped graph with vertex at \((-1,-7)\), passing through points like \((0,-6)\), \((1,-5)\), \((-2,-6)\) (and symmetric points) and is obtained by shifting the graph of \( y = |x|\) 1 unit left and 7 units down.