Question

graph the equation shown below by transforming the given graph of the parent function.
$y = |x| + 3$

Explanation:

Step 1: Identify the parent function

The parent function here is \( y = |x| \), which has a vertex at \( (0, 0) \) and a V - shape with slopes \( 1 \) (for \( x\geq0 \)) and \( - 1 \) (for \( x < 0 \)).

Step 2: Analyze the transformation

The given function is \( y=|x| + 3 \). For a function of the form \( y = f(x)+k \), when \( k>0 \), the graph of \( y = f(x) \) is shifted up by \( k \) units. Here, \( f(x)=|x| \) and \( k = 3 \), so we need to shift the graph of \( y = |x| \) up by 3 units.

Step 3: Determine the new vertex and key points

• The vertex of \( y = |x| \) is \( (0,0) \). After shifting up 3 units, the new vertex is \( (0,0 + 3)=(0,3) \).
• For the right - hand side of the V - shape (where \( x\geq0 \), \( y=x\)):
• When \( x = 0 \), \( y=|0|+3 = 3 \) (vertex).
• When \( x = 1 \), \( y=|1|+3=4 \), so the point is \( (1,4) \).
• When \( x = 2 \), \( y=|2|+3 = 5 \), so the point is \( (2,5) \).
• For the left - hand side of the V - shape (where \( x<0 \), \( y=-x\)):
• When \( x=- 1 \), \( y=|-1|+3 = 4 \), so the point is \( (-1,4) \).
• When \( x=-2 \), \( y=|-2|+3 = 5 \), so the point is \( (-2,5) \).

To graph \( y = |x|+3 \), we take the graph of \( y = |x| \) and move every point on it 3 units up. The vertex moves from \( (0,0) \) to \( (0,3) \), and the other points follow the same vertical shift. For example, the point \( (1,1) \) on \( y = |x| \) moves to \( (1,1 + 3)=(1,4) \), the point \( (-1,1) \) moves to \( (-1,1 + 3)=(-1,4) \), etc.

Answer:

To graph \( y = |x|+3 \), shift the graph of \( y = |x| \) (with vertex at \( (0,0) \), points like \( (1,1),(-1,1),(2,2),(-2,2) \) etc.) 3 units upward. The new vertex is \( (0,3) \), and key points are \( (0,3),(1,4),(-1,4),(2,5),(-2,5) \) etc. (The graph is a V - shaped graph with vertex at \( (0,3) \), opening upwards, with slope 1 for \( x\geq0 \) and slope - 1 for \( x < 0 \)).