Question

use technology to find points and then graph the function $y = |x - 5| + 2$, following the instructions below.

Explanation:

Step1: Analyze the function type

The function \( y = |x - 5|+2 \) 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 this function, \( a = 1\), \( h = 5\), and \( k=2 \), so the vertex of the graph is at the point \((5,2)\).

Step2: Find additional points

• When \( x=4 \):

Substitute \( x = 4\) into the function \( y=|x - 5|+2\). We have \( y=|4 - 5|+2=|-1|+2 = 1 + 2=3\). So the point \((4,3)\) is on the graph.

• When \( x = 6\):

Substitute \( x=6\) into the function \( y=|x - 5|+2\). We get \( y=|6 - 5|+2=|1|+2=1 + 2 = 3\). So the point \((6,3)\) is on the graph.

• When \( x=3\):

Substitute \( x = 3\) into the function \( y=|x - 5|+2\). Then \( y=|3 - 5|+2=|-2|+2=2 + 2 = 4\). So the point \((3,4)\) is on the graph.

• When \( x=7\):

Substitute \( x = 7\) into the function \( y=|x - 5|+2\). We have \( y=|7 - 5|+2=|2|+2=2 + 2 = 4\). So the point \((7,4)\) is on the graph.

To graph the function:

1. Plot the vertex \((5,2)\).
2. Plot the points we found: \((4,3)\), \((6,3)\), \((3,4)\), \((7,4)\) and so on.
3. Draw a V - shaped graph passing through these points. The graph opens upwards because \( a = 1>0\).

Answer:

The vertex of the graph of \( y=|x - 5|+2 \) is \((5,2)\). Some other points on the graph are \((4,3)\), \((6,3)\), \((3,4)\), \((7,4)\) etc. The graph is a V - shaped graph opening upwards with vertex at \((5,2)\) and passing through the calculated points.