Answer :
Step 1: Identify the translation.
The parent absolute value function is translated three units up.
Step 2: Understand the basic shape of the parent function.
The parent function, \( f(x) = |x| \), is a V-shaped graph with its vertex at the origin \((0, 0)\). It opens upwards and is symmetric about the y-axis.
Step 3: Apply the vertical translation.
Since the function \( f(x) = |x| + 3 \) translates the parent function three units up, the new vertex of the translated function is at \((0, 3)\).
Step 4: Sketch the graph.
- Start by plotting the new vertex at \((0, 3)\).
- From this vertex, since the shape remains the same, continue to draw the V shape. For every unit you move to the right or left on the x-axis, move upwards by one unit on the y-axis.
The result is that the entire V-shaped graph which originally had its vertex at \((0, 0)\) is now shifted up to have its vertex at \((0, 3)\), but otherwise retains its shape.
The graph of \( f(x) = |x| + 3 \) will look like this:
1. The vertex is at \((0, 3)\).
2. The lines extend upwards from the vertex forming a V shape, just like the parent function \( f(x) = |x| \), but starting three units higher on the y-axis.
This completes the translation of the absolute value function.
The parent absolute value function is translated three units up.
Step 2: Understand the basic shape of the parent function.
The parent function, \( f(x) = |x| \), is a V-shaped graph with its vertex at the origin \((0, 0)\). It opens upwards and is symmetric about the y-axis.
Step 3: Apply the vertical translation.
Since the function \( f(x) = |x| + 3 \) translates the parent function three units up, the new vertex of the translated function is at \((0, 3)\).
Step 4: Sketch the graph.
- Start by plotting the new vertex at \((0, 3)\).
- From this vertex, since the shape remains the same, continue to draw the V shape. For every unit you move to the right or left on the x-axis, move upwards by one unit on the y-axis.
The result is that the entire V-shaped graph which originally had its vertex at \((0, 0)\) is now shifted up to have its vertex at \((0, 3)\), but otherwise retains its shape.
The graph of \( f(x) = |x| + 3 \) will look like this:
1. The vertex is at \((0, 3)\).
2. The lines extend upwards from the vertex forming a V shape, just like the parent function \( f(x) = |x| \), but starting three units higher on the y-axis.
This completes the translation of the absolute value function.
