Problem 87 - Precalculus Essentials

Question

Begin by graphing the absolute value function, \(f(x)=|x| .\) Then use transformations of this graph to graph the given function. $$h(x)=-|x+4|$$

Step-by-Step Solution

Answer

The solution graph of the function \(h(x)= -|x+4|\) is a v-shaped curve that opens downwards (due to the negative absolute value) and is shifted 4 units left from the origin.

Step 1: Graph of Absolute Value Function

To begin, a basic understanding of the absolute value function is needed. To graph the absolute function \(f(x)=|x|\), note that the absolute value of \(x\) makes the output nonnegative. The resulting graph is a V shape which intersects the origin (0,0) and opens upwards because all values of \(x\) are positive or zero.

Step 2: Translate and Reflect function

The function \(h(x)=-|x+4|\) is a transformation of the absolute value function. This function can be obtained by shifting the function \(|x|\) to the left by 4 units and then reflecting it about the x-axis by multiplying it by -1. So, move all points on the \(|x|\) graph to the left by 4 units and then flip it over the x-axis. Ensure to change only the x-coordinates of the graph by subtracting 4 from them, then change all y-coordinates to their opposite value (if positive, make them negative), thus obtaining the graph for the function \(h(x)=-|x+4|\).

Step 3: Validate the Graph

Ensure your final graph has been moved to the left by 4 units relative to the original graph and is opening downwards. You can validate the graph accuracy by checking a few key points. For example, input x = 0 into \(h(x)=-|x+4|\), gets h(0) = -4, meaning the point (0, -4) should be on your graph. This way you can verify that the transformations were done correctly.