Problem 62 - Algebra and Trigonometry Real Mathematics, Real People

Question

Describing Transformations \(g\) is related to one of the six parent functions on page 122 (a) Identify the parent function \(f .\) (b) Describe the sequence of transformations from \(f\) to \(g .\) (c) Sketch the graph of \(g\) by hand. (d) Use function notation to write \(g\) in terms of the parent function \(f\).$$g(x)=\frac{1}{2}|x-2|-3$$.

Step-by-Step Solution

Answer

The parent function is \(f(x) = |x|\). The sequence of transformations are: a vertical shrink by factor of 1/2, a horizontal shift to the right by 2 units, and a vertical shift down by 3 units. The transformed function \(g(x)\) in terms of parent function \(f(x)\) can be written as \(g(x) = \frac{1}{2}f(x-2) - 3\).

Step 1: Identify the Parent Function

The parent function of \(g(x)\) is the absolute value function given by \(f(x) = |x|\). This is because \(g(x)\) includes an absolute value expression.

Step 2: Sequence of Transformations

To find the transformations involved in going from the parent function \(f(x)\) to the given function \(g(x)\), we observe a series of transformations occurring in three steps. The parent function \(f(x) = |x|\) got multiplied by 1/2 - this is a vertical shrink by the factor of 1/2. Then, \(x\) is replaced by \(x - 2\) - this is a horizontal shift to the right by 2 units. Lastly, 3 is subtracted - this is a vertical shift downward by 3 units.

Step 3: Graph Sketching

To sketch the graph of \(g(x)\), first draw the graph of the parent function \(f(x) = |x|\), which is a V-Shape line intercepting the origin. Then apply the transformations identified in step 2: first vertically shrink the graph by half times. Then, shift it to the right by 2 units and finally shift it down by 3 units.

Step 4: Function Notation

Now write the function \(g(x)\) in terms of the parent function \(f(x)\). It can be written as \(g(x) = \frac{1}{2}f(x-2) - 3\). This clearly shows the transformations of the parent function \(f(x)\).

Key Concepts

Absolute Value Function

The absolute value function is a foundational piece of mathematical understanding, and its graph is distinctive with its sharp V-shape. This function, defined as
\( f(x) = |x| \),
takes any real number as input and outputs its absolute value, which means it converts negative numbers to positive numbers, while positive numbers and zero remain unchanged. Visually, the graph consists of two rays that intersect at the origin and extend outward, with one ray going up and to the right and the other up and to the left.

Recognizing the absolute value function is crucial, as it serves as a ‘parent function’ for more complex functions that involve absolute value. These are often modified with transformations to create graphs that share the basic shape but may be adjusted in size, direction, or position.

Sequence of Transformations

Understanding the sequence of transformations is akin to following a recipe for modifying a graph. Each step changes the graph's appearance systematically. In the context of the absolute value function, transformations typically include vertical and horizontal shifts, reflections over the x or y-axis, and vertical stretches or shrinks.

In the case of our function \( g(x) \), which is a transformed version of the absolute value parent function \( f(x) \), we can identify a three-step transformation sequence. Firstly, there's a vertical shrink by a factor of 1/2, which compresses the graph towards the x-axis. Then, a horizontal shift moves the graph 2 units to the right, changing its position but not its shape. Lastly, a vertical shift moves the graph 3 units down. It is crucial to perform these steps in the correct order to achieve the intended outcome.

Graph Sketching

Graph sketching transforms abstract function expressions into visual representations, providing a tangible way to comprehend functions. With sketching, we literally ‘see’ the function’s behavior, which helps in understanding the underlying mathematics. When sketching a transformed absolute value function like \( g(x) \), you start with the parent function and apply the transformations one by one, in sequence.

Though technology can assist in plotting these graphs, there’s educational value in learning to draw them by hand. Doing so deepens a student’s grasp of how each transformation affects the graph’s appearance. In classrooms, graph sketching is not just an exercise in drawing, but also in visualization and analytical thinking.

Function Notation

Function notation is the way we write and communicate functions, and it’s a vital part of understanding and working with mathematical concepts. In function notation, the symbol \( f(x) \) tells us that \( f \) is a function of \( x \). When dealing with transformations, function notation neatly summarizes the changes made to the parent function.

For instance, in the function \( g(x) \), the notation captures the sequence of transformations: \( g(x) = \frac{1}{2}f(x-2) - 3 \). This indicates the original function \( f \) is first shifted horizontally, then vertically shrunk, and finally shifted downward. Mastering function notation is an essential skill for students as it is a standard convention in mathematics to express functions clearly and concisely.