Problem 92 - Precalculus Essentials

Question

Begin by graphing the absolute value function, \(f(x)=|x| .\) Then use transformations of this graph to graph the given function. $$h(x)=2|x+3|$$

Step-by-Step Solution

Answer

The graph for the function \(h(x)=2|x+3|\) starts at the point \(-3, 0\), and forms a steeper V-shape than \(f(x)=|x|\), with the slopes of -2 for \(x < -3\) and 2 for \(x > -3\).

Step 1: Graph the Absolute Value Function

Plot the function \(f(x) = |x|\). This will yield a V-shaped graph intercepting the origin, (0,0) with slopes that are -1 for \(x<0\) and 1 for \(x>0\).

Step 2: Shift the Graph

The '+3' inside the absolute value function indicates a horizontal shift. This implies that the graph will shift 3 units to the left. Now the vertex point is at \(-3, 0\).

Step 3: Apply the Vertical Stretch

The '2' outside the absolute function denotes a vertical stretch by a factor of 2. Every y-coordinate should be multiplied by 2, for the new V-shape that is twice as steep.

Step 4: Graph the Final Function

Draw the final graph using all transformations, resulting in the function \(h(x) = 2|x+3|\), which is a V-shape starting at the point \(-3, 0\), with slopes of -2 for \(x < -3\) and 2 for \(x > -3\).