Problem 82 - Precalculus Essentials

Question

Begin by graphing the absolute value function, \(f(x)=|x| .\) Then use transformations of this graph to graph the given function. $$g(x)=|x|+3$$

Step-by-Step Solution

Answer

The graph of \( g(x) = |x|+3 \) is a 'V' shaped graph, same as the parent function \( f(x) = |x| \), but shifted upward by 3 units. The vertex is at (0, 3)

Step 1: Graph the parent function

Start by sketching the parent function, \( f(x) = |x| \). This graph is a straight line shaped like a 'V', with the vertex at the origin (0, 0). The line increases at a 45 degree angle in both the positive and negative direction from the vertex.

Step 2: Identify the transformation

Now, consider the function \( g(x) = |x|+3 \). This is a transformation of the parent function where every \(y\)-value in \(f(x) = |x|\) is increased by 3. This means the graph of \(g(x)\) will be the same shape as the graph of \(f(x)\), just shifted upward by 3 units.

Step 3: Graph the transformed function

Now, draw the new graph, taking into consideration the vertical shift. The vertex, originally at (0, 0), has now moved up to (0, 3). The shape of the graph is still 'V', but now it is located 3 units higher than the original graph.