Problem 33 - Beginning and Intermediate Algebra

Question

Use the transformation techniques to graph each of the following functions. $$f(x)=|x|-5$$

Step-by-Step Solution

Answer

To graph the function \(f(x) = |x| - 5\), first identify the parent function \(f(x) = |x|\) and the vertical translation of 5 units downward, represented by "-5." The key points of the parent function are the vertex (0, 0) and the x-intercept (0, 0). Apply the transformation by moving the vertex and the x-intercept 5 units down to obtain the new vertex (0, -5). Finally, draw the V-shape absolute value graph with the new vertex at (0, -5), maintaining the same slope of the parent function's lines of 1 and -1.

Step 1: Identify the parent function and transformation

The parent function given is \(f(x) = |x|\), which is an absolute value function that typically forms a V-shape. The transformation we need to apply is a vertical translation 5 units downward, represented by "-5" in the given function \(f(x) = |x| - 5\).

Step 2: Determine the key points of the parent function

The key points for the parent function \(f(x) = |x|\) are the points that have the greatest impact on the graph's shape. The essential points for the absolute value function are the vertex (0, 0) and the points where the graph changes directions, which are the x-intercepts (0, 0).

Step 3: Apply the transformation to the key points

Now we need to apply the transformation (-5) to the key points of the parent function. The vertex of the parent function is (0, 0). The transformation will move it 5 units down, which gives us the new vertex as (0, -5). The x-intercept of the parent function is also at (0, 0). After applying the transformation, it will now fall at (0, -5), right at the new vertex.

Step 4: Draw the graph of the transformed function

We will now use the transformed points and the vertical translation information to graph the transformed function: 1. Plot the new vertex (0, -5) on the coordinate plane 2. Draw the V-shape of the absolute value function with the new vertex at (0, -5) 3. Since the slope of the parent function's line remains the same (with slopes of 1 and -1), the shape of the graph will be identical to the parent graph, only translated downwards by 5 units. By following these steps, we have successfully graphed the transformed function \(f(x) = |x| - 5\).

Key Concepts

Transformation Techniques

Graphing functions is a fascinating process because you can see how transformations affect the graph's shape and position. One important transformation technique is vertical translation. This is a straightforward method that involves moving the entire graph up or down along the y-axis.
In general, transformation techniques can include several other types such as horizontal translation, reflection, and scaling (stretching or compressing) in various directions. For our specific function, we're focusing on vertical translation, that is moving up or down, which is crucial in altering the y-values of all points on a graph.
Understanding transformation techniques helps in visualizing and predicting the changes a graph undergoes when equations are altered.

Vertical Translation

Vertical translation is particularly useful when you want to shift a graph higher or lower on the coordinate plane. This involves adding or subtracting a constant value to the function's output (y-values).

• If the constant is positive, the graph moves upwards.
• If the constant is negative, the graph moves downwards.

In the function provided, \(f(x) = |x| - 5\), the "-5" indicates a vertical translation of 5 units downward. This means every point on the parent function \(f(x) = |x|\) moves straight down by 5 units, effectively lowering the whole graph while maintaining its shape.

Parent Function

The parent function \(f(x) = |x|\) is a fundamental form of absolute value functions and serves as the reference graph. It takes a classic V-shape that opens upwards with its vertex at the origin, (0,0).
This function is particularly symmetric, making it a suitable reference for applying various transformations. Key characteristics of this parent function include:

• The vertex, which is the lowest point of the V, at (0, 0).
• Lines with slopes of 1 and -1 from the vertex, indicating symmetry across the y-axis.

Understanding this parent function helps you predict how transformations like vertical translation will affect its shape and position on the graph.

Coordinate Plane

The coordinate plane is the two-dimensional backdrop where we visualize and draw functions, including transformed ones. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
When graphing transformations, such as \(f(x) = |x| - 5\), you rely on the coordinate plane to accurately plot points and draw the graph. In our case, we started by marking the new vertex at (0, -5) on the plane.
Understanding the coordinate plane is essential because it assists in pinpointing where the function begins, how it changes, and how one transformation can result in a new graph on this grid. This helps you see and understand the overall impact of different transformations on various functions.