Question

graph the function
f(x)=|x - 7| + 2
use the graphing tool to graph the function.

Explanation:

Step1: Recall the parent function

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

Step2: Analyze the transformation

For the function \( f(x)=|x - 7|+2 \), we use the transformation rules for absolute - value functions. The general form of an absolute - value function is \( y=a|x - h|+k \), where \( (h,k) \) is the vertex of the graph.

• For the horizontal shift: The term \( (x - 7) \) in \( |x - 7| \) means that the graph of \( y = |x| \) is shifted 7 units to the right. This is because if we have \( y=|x - h| \), when \( h>0 \), the graph shifts \( h \) units to the right.
• For the vertical shift: The \( + 2 \) at the end means that the graph is shifted 2 units up. This is because if we have \( y = |x - h|+k \), when \( k>0 \), the graph shifts \( k \) units up.

So, the vertex of the graph of \( f(x)=|x - 7|+2 \) is at the point \( (7,2) \).

Step3: Determine the slope of the two arms

For the absolute - value function \( y = |x| \), the slope of the right - hand arm (where \( x\geq0 \)) is 1 and the slope of the left - hand arm (where \( x<0 \)) is - 1. For the function \( f(x)=|x - 7|+2 \), since there are no vertical stretches or compressions (\( a = 1 \)) and no reflections (the coefficient of the absolute - value is positive), the slope of the right - hand arm (where \( x\geq7 \)) is 1 and the slope of the left - hand arm (where \( x<7 \)) is - 1.

To graph the function:

1. Plot the vertex at \( (7,2) \).
2. For the right - hand side ( \( x\geq7 \) ), use the slope of 1. For example, when \( x = 8 \), \( f(8)=|8 - 7|+2=1 + 2 = 3 \), so the point \( (8,3) \) is on the graph.
3. For the left - hand side ( \( x<7 \) ), use the slope of - 1. For example, when \( x = 6 \), \( f(6)=|6 - 7|+2=1 + 2 = 3 \), so the point \( (6,3) \) is on the graph.

Answer:

The graph of \( f(x)=|x - 7|+2 \) is a V - shaped graph with a vertex at \( (7,2) \), the right - hand arm has a slope of 1 and the left - hand arm has a slope of - 1, shifted 7 units right and 2 units up from the graph of \( y = |x| \). To graph it, plot the vertex \( (7,2) \), then use the slopes of 1 (for \( x\geq7 \)) and - 1 (for \( x<7 \)) to draw the two arms of the V - shape.