Question

graph the function
$f(x)=|x - 5|+1$
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 \((0, 0)\) and opens upwards (V - shaped graph).

Step2: Analyze the transformation

For the function \( f(x)=|x - 5|+1 \), we use the transformation rules for absolute - value functions. The general form of a transformed absolute - value function is \( y=a|x - h|+k \), where \((h,k)\) is the vertex of the graph.

• For the horizontal shift: The term \( x-5 \) in \( |x - 5| \) means that the graph of \( y = |x| \) is shifted 5 units to the right. This is because when we have \( y=|x - h| \), the graph of \( y = |x| \) is shifted \( h \) units to the right (if \( h>0 \)).
• For the vertical shift: The \( + 1 \) at the end means that the graph is shifted 1 unit up. This is because when we have \( y=|x - h|+k \), the graph is shifted \( k \) units up (if \( k > 0 \)).

Step3: Determine the vertex

From the transformation, the vertex of the graph of \( f(x)=|x - 5|+1 \) is at the point \((h,k)=(5,1)\).

Step4: Find some additional points

• When \( x = 4 \):

\( f(4)=|4 - 5|+1=|-1|+1 = 1 + 1=2 \)

• When \( x = 6 \):

\( f(6)=|6 - 5|+1=|1|+1 = 1+1 = 2 \)

• When \( x=0 \):

\( f(0)=|0 - 5|+1=|-5|+1=5 + 1 = 6 \)

• When \( x = 10 \):

\( f(10)=|10 - 5|+1=|5|+1=5 + 1=6 \)

To graph the function:

1. Plot the vertex \((5,1)\).
2. Plot the points \((4,2)\), \((6,2)\), \((0,6)\), \((10,6)\) (and other points if needed).
3. Draw a V - shaped graph with the vertex at \((5,1)\), passing through the plotted points. The left side of the V (for \( x<5 \)) has a slope of - 1 (since for \( x<5 \), \( f(x)=-(x - 5)+1=-x + 5 + 1=-x+6 \), and the slope of \( y=-x + 6 \) is - 1) and the right side of the V (for \( x\geq5 \)) has a slope of 1 (since for \( x\geq5 \), \( f(x)=(x - 5)+1=x-5 + 1=x - 4 \), and the slope of \( y=x - 4 \) is 1).

Answer:

The graph of \( f(x)=|x - 5|+1 \) is a V - shaped graph with vertex at \((5,1)\), shifted 5 units right and 1 unit up from the graph of \( y = |x| \), passing through points like \((4,2)\), \((6,2)\), \((0,6)\), \((10,6)\) etc. (The actual graph can be drawn using the above - mentioned steps on the given grid.)