Problem 91 - Precalculus Essentials

Question

Begin by graphing the absolute value function, \(f(x)=|x| .\) Then use transformations of this graph to graph the given function. $$h(x)=2|x+4|$$

Step-by-Step Solution

Answer

The graph of \(h(x)=2|x+4|\) is like the graph of the absolute value function \(f(x)=|x|\), but stretched vertically due to the factor of 2 and shifted four units to left due to the \(+4\) inside the absolute value function, placing the vertex at the point (-4,0).

Step 1: Graph the Absolute Value Function

Start by drawing the graph of \(f(x)=|x|\). This function is shaped like a 'V' with the point of the 'V' located at the point (0,0). This point is called the vertex. The line goes upwards from this point in both positive and negative direction of x-axis forming a 'V' shape.

Step 2: Apply the Scaling Factor

The next transformation to apply to the graph of \(f(x)=|x|\) is the scaling factor of 2 in the function \(h(x)=2|x+4|\). This will cause a vertical stretch of the graph of \(f(x)=|x|\). The 'V' becomes narrower or steeper.

Step 3: Apply the Horizontal Shift

The final transformation is the horizontal shift caused by \(+4\) inside the absolute value function in \(h(x)=2|x+4|\). This shifts the vertex of the 'V' four units to the left (due to positive sign inside the absolute value function, which always shifts opposite to the traditional direction). The vertex will now be at the point (-4,0).