Question

graph the equation shown below by transforming the given graph of the parent function.
\\( y = |x| + 5 \\)

Explanation:

Step1: Identify the parent function

The parent function here is \( y = |x| \), whose graph is a V - shaped graph with the vertex at the origin \((0,0)\) (as shown in the given graph with the vertex at \((0,0)\) and the two arms going through points like \((1,1)\), \(( - 1,1)\) etc.).

Step2: Analyze the transformation

The given function is \( y=|x| + 5 \). For a function of the form \( y = f(x)+k \), when \( k>0 \), the graph of \( y = f(x) \) is shifted vertically upwards by \( k \) units. Here, \( f(x)=|x| \) and \( k = 5 \), so we need to shift the graph of \( y = |x| \) (the parent function) upwards by 5 units.

Step3: Determine the new vertex and key points

• The vertex of the parent function \( y = |x| \) is at \((0,0)\). After shifting upwards by 5 units, the new vertex will be at \((0,0 + 5)=(0,5)\).
• For the right - hand arm (where \( x\geq0 \)) of the parent function \( y = |x| \), when \( x = 1 \), \( y=1 \); when \( x = 2 \), \( y = 2 \); when \( x = 3 \), \( y=3 \); when \( x = 4 \), \( y = 4 \); when \( x=5 \), \( y = 5 \). After shifting upwards by 5 units, the new \( y \) - values for these \( x \) - values will be: when \( x = 1 \), \( y=1 + 5=6 \); when \( x = 2 \), \( y=2 + 5 = 7 \); when \( x=3 \), \( y=3 + 5=8 \); when \( x = 4 \), \( y=4 + 5 = 9 \); when \( x=5 \), \( y=5 + 5 = 10 \).
• For the left - hand arm (where \( x<0 \)) of the parent function \( y = |x| \), when \( x=-1 \), \( y = 1 \); when \( x=-2 \), \( y = 2 \); when \( x=-3 \), \( y = 3 \); when \( x=-4 \), \( y = 4 \); when \( x=-5 \), \( y = 5 \). After shifting upwards by 5 units, the new \( y \) - values for these \( x \) - values will be: when \( x=-1 \), \( y=1 + 5=6 \); when \( x=-2 \), \( y=2 + 5 = 7 \); when \( x=-3 \), \( y=3 + 5=8 \); when \( x=-4 \), \( y=4 + 5 = 9 \); when \( x=-5 \), \( y=5 + 5 = 10 \).

To graph \( y = |x|+5 \), we plot the new vertex \((0,5)\) and the new key points (e.g., \((1,6)\), \(( - 1,6)\), \((2,7)\), \(( - 2,7)\) etc.) and draw the V - shaped graph with the vertex at \((0,5)\) and the two arms having the same slope (slope of 1 for the right - hand arm and slope of - 1 for the left - hand arm) as the parent function.

Answer:

To graph \( y = |x|+5 \), shift the graph of the parent function \( y = |x| \) (with vertex at \((0,0)\)) vertically upward by 5 units. The new graph has a vertex at \((0,5)\), and for \( x\geq0 \), the points on the right - hand arm follow \( y=x + 5 \), and for \( x<0 \), the points on the left - hand arm follow \( y=-x + 5 \).