Problem 40 - 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 \(f\). \(g (x) = -\frac{1}{2}(x + 1)^3 \)

Step-by-Step Solution

Answer

The parent function is \(f(x) = x^3\). Transformations include a horizontal shift 1 unit to the left, a reflection over the x-axis, and a vertical compression by a factor of \(\frac{1}{2}\). \(g(x)\) in terms of \(f\) would be written as \(g(x) = -\frac{1}{2}f(x+1)\).

Step 1: Identify the Parent Function

Here, the parent function is \(f(x) = x^3\) as the given function is a cubic function.

Step 2: Describe the Sequence of Transformations

The function \(g(x)\) is derived from function \(f(x)\) by following transformations: \n1. Horizontal shift to the left by 1 unit as indicated by \(+1\). A positive value inside brackets causes a shift to the left for cubic function. \n2. A vertical reflection due to the negative sign. A negative sign in front of the function reflects the graph across the x-axis. \n3. Vertical compression by a factor of \(\frac{1}{2}\). The presence of factor \(\frac{1}{2}\) in front of the function decreases the height of the cubic function by \(\frac{1}{2}\).

Step 3: Sketch the Graph

To sketch the graph, start by drawing the parent function's graph \(f(x)=x^3\), then apply the transformations as described in step 2 sequentially, meaning first shift the graph 1 unit to left, then reflect it across x axis and finally compress the graph vertically by a factor of \(\frac{1}{2}\).

Step 4: Write g in terms of f

The function \(g\) can be written in terms of \(f\) as \(g(x) = -\frac{1}{2}f(x+1)\), taking into account all the transformations applied to the parent function \(f(x) = x^3\).

Key Concepts

Cubic Function

A cubic function is a type of polynomial function where the highest degree of any variable is three. In its simplest form, the parent cubic function is given by the equation \( f(x) = x^3 \). These functions are known for their distinctive 'S-shaped' curves that have an inflection point at the origin (0,0), where the function changes concavity.

The cubic function graph is symmetric with respect to the origin, illustrating that it is an odd function. This means that for every point on the graph at \((x, y)\), there will be a corresponding point at \((-x, -y)\). Understanding the basic shape of a cubic graph is essential for predicting and sketching graphs of transformed cubic functions.

Graph Transformations

Graph transformations refer to the changes made to the basic parent function's graph in order to obtain a new function. There are several types of transformations:

• Vertical Shifts: Moving the graph up or down without changing its shape.
• Horizontal Shifts: Moving the graph left or right.
• Reflexions: Flipping the graph over the x-axis or y-axis.
• Vertical Compressions and Stretches: Altering the 'steepness' or 'flatness' of the graph by multiplying the function by a factor greater or smaller than one.
• Horizontal Compressions and Stretches: Making the function 'wider' or 'narrower' by multiplying the x-variable by a factor.

Understanding how these transformations affect the graph helps you predict the appearance of the graph of a transformed function.

Function Notation

Function notation is a way to denote functions for clarity and convenience in mathematical expressions. Using the notation \( f(x) \), 'f' represents the name of the function, and 'x' represents the input value.

When dealing with transformations, function notation becomes particularly useful. For instance, if a function \(g\) is derived from another function \(f\) through certain transformations, we can represent \(g\) in terms of \(f\) by accounting for these changes directly within the notation. For example, if function \(g(x)\) is a shifted and scaled version of function \(f(x)\), we can write \(g(x) = a \times f(b(x - c)) + d\) where:\(a\) is a vertical stretch or compression and reflection,\(b\) is a horizontal stretch or compression,\(c\) is a horizontal shift, and\(d\) is a vertical shift. This notation helps visualize the relationship between functions and their transformations.

Sketching Graphs

The skill of sketching graphs involves drawing a quick visual representation of a function's behavior without necessarily computing every value. The process typically begins with identifying the parent function and its key features, such as intercepts, asymptotes, inflections, and intervals of increase or decrease.

When sketching transformed graphs, you sequentially apply each transformation to the parent function one by one, starting with horizontal moves (shifts and stretches/compressions), followed by reflections, and ending with vertical moves. This method helps maintain accuracy and provides a systematic approach to graph sketching. It's also useful to mark key points, like turns or intercepts, after each transformation to understand how they're affected.