Problem 85 - Precalculus : Building Concepts and Connections

Question

When using transformations with both vertical scaling and vertical shifts, the order in which you perform the transformations matters. Let \(f(x)=|x|\). (a) Find the function \(g(x)\) whose graph is obtained by first vertically stretching \(f(x)\) by a factor of 2 and then shifting the result upward by 3 units. A table of values and/or a sketch of the graph will be helpful. (b) Find the function \(g(x)\) whose graph is obtained by first shifting \(f(x)\) upward by 3 units and then multiplying the result by a factor of 2 A table of values and/or a sketch of the graph will be helpful. (c) Compare your answers to parts (a) and (b). Explain why they are different.

Step-by-Step Solution

Answer

In both cases when the transformations are applied to \(f(x) = |x|\), vertical stretch and shift upwards, we get different new functions depending on the order of the transformations. For (a) the resulting function is \(2|x| + 3\), for (b) the result is \(2(|x| + 3)\). The differences highlight the importance of the order of transformations in function manipulations.

Step 1: Apply Transformations for Part (a)

For part (a), we are told to begin by vertically stretching the function \(f(x) = |x|\), by a factor of 2, which gives us the function \(2f(x) = 2|x|\). Afterwards, the function is shifted upwards by 3 units, which gives us \(g(x) = 2|x| + 3\). Thus, \(g(x)\) is the result of the vertical stretching and shifting for part (a).

Step 2: Apply Transformations for Part (b)

For part (b), we start from \(f(x) = |x|\) and shift the function upward by 3 units to get \(f(x) + 3 = |x| + 3\). Next, we multiply this result by a factor of 2 to vertically stretch it: \(2(|x| + 3)\). Thus, \(g(x) = 2(|x| + 3)\) is the result for part (b).

Step 3: Compare the Result of Part (a) and Part (b)

The result from part (a) is \(g(x) = 2|x| + 3\) and the result from part (b) is \(g(x) = 2(|x| + 3)\). These equations look a bit different, and when graphed they will also look different. This shows that the order in which we apply the transformations does alter the final results.

Key Concepts

Absolute Value Transformations

Understanding absolute value transformations is crucial when addressing complex function modifications. With the absolute value function, such as the example we have featuring the function f(x) = |x|, specific transformations like vertical stretching and shifting can drastically change its graph.

Transforming the absolute value function involves visual shifts that may be vertical or horizontal, reflections across the x-axis or y-axis, and stretching or compressing the graph. Students often find these transformations initially confusing, but realizing that the absolute value graph is V-shaped can simplify the process. For instance, a vertical stretch will make the V-shape narrower, while a shift will move the entire graph up, down, or sideways without altering its shape.

Vertical Stretching

Vertical stretching of a function occurs when we multiply the function by a scale factor greater than one. In our exercise, the absolute value function f(x) = |x| has been stretched by a factor of 2, resulting in 2f(x) = 2|x|. This action effectively doubles the distance of all points on the graph from the x-axis, creating a steeper, narrower graph.

Think of vertical stretching as pulling the graph away from the x-axis. Students should visualize this to better understand how each point on the graph is affected individually, which will aid in grasping the concept more thoroughly.

Vertical Shifting

Vertical shifting is a more straightforward transformation than stretching. It entails adding (or subtracting) a constant to the entire function, which moves the graph up (or down) without altering the shape of the graph.

In our textbook problem, after we apply a vertical stretch to f(x), we add 3 units, resulting in a vertical shift upwards yielding the function g(x) = 2|x| + 3. It's as if you picked up the graph and moved it up along the y-axis by the distance of the constant value added. Students should be attentive to the fact that the V-shape remains intact, only the position on the graph changes.

Transformation Sequence

The order in which transformations are applied to a function, known as the transformation sequence, is vital. It must be understood that performing a vertical shift first and then stretching will yield a different result compared to stretching first and then shifting.

As highlighted in the comparison between parts (a) and (b) of our exercise, the resulting functions g(x) = 2|x| + 3 and g(x) = 2(|x| + 3) are notably different, which emphasizes the importance of transformation sequence. This understanding will allow students to approach function transformations with a clear strategy, knowing that the sequence of their actions will have direct and varying impacts on the function's graph.

Function Transformation Properties

Every transformation applied to a function has specific properties that dictate how the graph will be altered. The properties governing the absolute value function transformations incorporate a combination of shifts, stretches, reflections, and sometimes compressions.

It is essential for students to learn that a vertical stretch multiplies the output value of the function without changing the x-coordinates, a vertical shift adds a constant value to the output affecting the y-coordinates, and the sequence in which these are done will affect the shape and position of the final graph. Recognizing these properties provides a systematic approach to predicting and understanding the impact of any given transformation on the graph of a function.