Problem 52 - Precalculus with Limits

Question

In Exercises 25-54, \(g\) is related to one of the parent functions described in Section 1.6. (a) Identify the parent function \(f\). (b) Describe the sequence of transformations from \(f\) to \(g\). (c) Sketch the graph of \(g\). (d) Use function notation to write \(g\) in terms of \(f\). \(g (x) = -\frac{1}{2}\sqrt{x + 3} - 1\)

Step-by-Step Solution

Answer

The parent function for the given function \(g (x) = -\frac{1}{2}\sqrt{x + 3} - 1\) is \(f(x)=\sqrt{x}\). The sequence of transformations are: shift three units to left, vertical compression by a factor of 1/2 with a reflection across the x-axis and shifting down by 1. The function \(g\) can be written in terms of \(f\) as \(g(x) = -\frac{1}{2}f(x + 3) - 1\).

Step 1: Identify the Parent Function

The parent function can be determined by replacing every parameter in the function with its standard form. So, the parent function \(f(x)\) in this case is \(\sqrt{x}\).

Step 2: Describe the Transformation Sequence

The sequence of transformations from \(f\) to \(g\) can be achieved as follows: \n1) Shift \(f(x)\) three units to the left, creating \(f(x + 3)\). \n2) Compress \(f(x + 3)\) vertically by a factor of 1/2 and reflect it across the x-axis, creating \(-\frac{1}{2}f(x + 3)\).\n3) Finally, shift \(-\frac{1}{2}f(x + 3)\) one unit down to get \(g(x)= -\frac{1}{2}\sqrt{x + 3} - 1\).

Step 3: Sketch the Graph of \(g(x)\)

To sketch the graph of \(g(x)\), start with the parent function \(\sqrt{x}\), which looks like the right half of a U. First, horizontally shift the graph 3 units left, then vertically compress it by a factor of 1/2 and reflect it vertically. Finally, shift it down by 1. The resulting graph represents \(g(x)= -\frac{1}{2}\sqrt{x + 3} - 1\).

Step 4: Use Function Notation to Write \(g\) in Terms of \(f\)

Now, write the function \(g\) in terms of the parent function \(f\). This is achieved by applying all the transformation operations on \(f\). Therefore, \(g(x)\) can be written as \(g(x) = -\frac{1}{2}f(x + 3) - 1\).

Key Concepts

Parent Functions

Understanding the idea of a parent function is crucial when studying transformations. A parent function is the simplest form of a particular type of function. It's kind of like the "blueprint" for other functions of its type. For example, the parent function for square root functions is \(f(x) = \sqrt{x}\). This serves as the baseline model which all variations or transformations of that function will be compared.

Parent functions are essential in understanding how graphs behave, because every transformation starts from these foundational functions. With them, you can predict the general shape and direction of the graph. Remember to key in on the most basic version first, before thinking about shifts and changes!

Function Transformations

Function transformations involve modifying a parent function to obtain a new function, often denoted as \(g(x)\) when transforming from \(f(x)\). There are several transformation techniques to understand:

• Translation: This involves shifting the graph vertically or horizontally. For example, shifting a graph three units to the left can be done by replacing \(x\) with \(x + 3\). A vertical shift, like moving a graph down 1 unit, adjusts the entire function with a subtraction or addition to the output, e.g., subtracting 1 at the end of the function.
• Reflection: This "flips" the graph over a specific axis. For instance, a negative sign in front of the function \(-f(x)\) reflects the graph across the x-axis.
• Compression and Stretching: These affect the graph's width or height. A vertical compression by a factor of \(\frac{1}{2}\), for example, makes the graph "squish" down, reducing its height, typically depicted as \(\frac{1}{2}f(x)\).

Combined, these transformations allow for a dynamic modification of the graph, customizing it to fit specific needs or equations.

Graph Transformations

Graph transformations are a powerful visual tool that help students and mathematicians understand how functions change. After an understanding of parent functions and the types of transformations, it's time to apply these concepts to graphs.

When you plot a function, start by drawing its parent form. From there, apply the transformations sequentially:

• Horizontal shifts visualize the translation of the graph left or right.
• Vertical compressions or stretches show how "tall" or "short" the graph becomes.
• Reflections, like flipping over the x-axis, dramatically change the appearance.

In exercises, sketching each step ensures accuracy, allowing you to verify each transformation stage. It’s a systematic way of checking your work and gaining deeper insights into function behavior. The transformation creates a dynamic understanding of how a simple, basic form evolves into its complex versions.