Problem 22 - Beginning and Intermediate Algebra

Question

Sketch the graph of \(f(x) .\) Then, graph \(g(x)\) on the same axes using the transformation techniques. $$\begin{aligned}&f(x)=|x|\\\&g(x)=|x|-4\end{aligned}$$

Step-by-Step Solution

Answer

To sketch the graphs of \(f(x) = |x|\) and \(g(x) = |x| - 4\), start by plotting the key points for the absolute value function: (-2, 2), (-1, 1), (0, 0), (1, 1), and (2, 2). The graph of \(f(x) = |x|\) is a V-shape. For \(g(x) = |x| - 4\), apply a vertical shift 4 units downward to get the new key points (-2, -2), (-1, -3), (0, -4), (1, -3), and (2, -2). The graph of \(g(x)\) will also be a V-shape, shifted 4 units down from the graph of \(f(x)\). Combine both graphs on the same axes to observe their relationship.

Step 1: Sketch the graph of f(x) = |x|

To sketch the graph for f(x) = |x|, we can start by plotting some key points. Some key points for the absolute value function are (-2, 2), (-1, 1), (0, 0), (1, 1), and (2, 2). Connect these points with two diagonal lines that form a V-shape, and we have our graph for f(x) = |x|.

Step 2: Transformation for g(x) = |x| - 4

To relate g(x) = |x| - 4 to f(x) = |x|, we need to identify what kind of transformation is happening. We see that g(x) is just f(x) with a subtraction of 4. The subtraction of a constant number from a function does not affect the x-values, but it shifts the graph vertically in the coordinate plane. Specifically, subtracting 4 means that the graph will shift 4 units downward from its origin in f(x).

Step 3: Sketch the graph of g(x) = |x| - 4

To sketch the graph for g(x) = |x| - 4, we first apply the transformation mentioned in Step 2. Start by revisiting the key points that we plotted for f(x): (-2, 2), (-1, 1), (0, 0), (1, 1), and (2, 2). We will keep the x-values same, but decrease the y-values by 4. The new points to plot would then be (-2, -2), (-1, -3), (0, -4), (1, -3), and (2, -2). Connect these points with two diagonal lines that form a V-shape, and we have our graph for g(x) = |x| - 4.

Step 4: Combine the graphs on the same axes

Finally, we have both f(x) = |x| and g(x) = |x| - 4 plotted separately. Draw them on the same axes and observe their relationship. The graph of g(x) should be a downward-shifted version of f(x), moving 4 units down in the coordinate plane.

Key Concepts

Absolute Value Function

The absolute value function is represented as \(f(x) = |x|\). This function is fascinating because of its unique graph. It forms a V-shape when plotted. The reason for this V-shape comes from how the absolute value works. It always converts negative inputs of \(x\) into positive outputs. Therefore, the graph features two symmetrical straight-line segments, one for positive \(x\) values and one mirroring it for negative values.
Here are some important properties of the absolute value function:

• The vertex of the graph is at the origin, which is the point (0,0).
• The graph is symmetric about the y-axis, meaning the left side is a mirror reflection of the right side.
• The graph consists of two lines: one sloping upwards from left to right on the right side and the other upwards from right to left on the left side.

These properties make it straightforward to identify and sketch the absolute value function graph with just a few points.

Vertical Shifts

A vertical shift in graph transformations refers to moving the entire graph of a function up or down along the y-axis.
In our example, we see this with \(g(x) = |x| - 4\). This transformation involves subtracting 4 from the function \(f(x) = |x|\).
The effect of this operation is a downward shift.

• The whole graph of \(f(x) = |x|\) moves down 4 units.
• This involves keeping the x-values the same and subtracting 4 from all y-values.
• The vertex of this transformed graph \(g(x)\) will be at (0, -4).

Remember that vertical shifts do not affect the shape of the graph, only its position on the coordinate plane.
Thus, the original V-shape of the absolute value function remains intact, but is simply relocated.

Plotting Graphs

Plotting graphs of transformed functions demands understanding both the basic graph and the transformation applied.
Here’s a step-by-step method to plot the graph of \(g(x) = |x| - 4\) based on \(f(x) = |x|\):

• First, sketch the graph of \(f(x) = |x|\) by identifying the key points, such as (-2, 2), (-1, 1), (0, 0), (1, 1), and (2, 2).
• Draw lines connecting these points to emphasize the V-shape.
• For \(g(x) = |x| - 4\), take each of the corresponding y-values from \(f(x)\) and subtract 4, resulting in new points: (-2, -2), (-1, -3), (0, -4), (1, -3), and (2, -2).
• Plot these new points and connect them to form the transformed V-shape on the same axes as \(f(x)\).
• Observe the shift and how the overall shape and symmetry remain, noticing that all y-coordinates are merely adjusted downward.

By systematically following these steps, any transformation of graph functions can be sketched accurately.