Problem 38 - 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 \(f\). \(g (x) = (x + 1)^3 - 10\)

Step-by-Step Solution

Answer

The parent function is \(f(x) = x^3\). The sequence of transformations involves a horizontal shift to the left by 1 unit and a vertical shift down by 10 units. The given function can be represented in terms of the parent function as \(g(x) = f(x+1) - 10\).

Step 1: Identify the Parent Function

For the given function \(g (x) = (x + 1)^3 - 10\), the parent function \(f\) is \(f(x) = x^3\). This is because the structure of the function \(g\), without the transformations, follows the cubic function pattern.

Step 2: Describe the Transformations

The given function \(g (x) = (x + 1)^3 - 10\) can be understood from the parent function \(f(x) = x^3\) as being horizontally shifted to the left by 1 unit (due to \(x + 1\)) and vertically shifted down by 10 units (due to \(- 10\)).

Step 3: Sketch the Graph

To sketch the graph of \(g\), first draw the parent function \(f(x) = x^3\) which is a cube curve. Then apply the transformations which involve moving every point on \(f(x)\) 1 unit to the left and 10 units downward. This will provide the graph of \(g\).

Step 4: Write \(g\) in Terms of \(f\)

We know that the function \(g(x)\) is the result of transforming \(f(x)\) by a horizontal shift to the left by 1 unit and a vertical shift down by 10 units. Therefore, \(g\) can be written in terms of \(f\) as follows: \(g(x) = f(x+1) - 10\). This notation indicates that \(x\) in \(f(x)\) is replaced with \(x + 1\) and then 10 is subtracted from the outcome.

Key Concepts

Parent Functions

Understanding parent functions is crucial for identifying and applying transformations effectively. Parent functions are the simplest form of functions in various families, such as quadratic, linear, or cubic. These basic functions act as a template or starting point. By knowing their basic shapes and properties, you can predict how transformations will alter them.

A cubic parent function, represented as \(f(x) = x^3\), features a unique, smooth curve that originates from the origin. It increases in both directions, displaying a point of inflection at \((0,0)\) where the function transitions from concave up to concave down.

In the context of transformations, parent functions serve as the foundation. Once you identify them, you can easily understand how various algebraic modifications will impact the graph. Recognizing that the parent function of \(g(x)\) is cubic helps us anticipate that the graph will retain its cubic characteristics after transformations are applied.

Cubic Functions

Cubic functions are polynomial functions of degree three, characterized by their polynomial form \(f(x) = ax^3 + bx^2 + cx + d\). In this particular case, we focus on the simplest cubic function, \(f(x) = x^3\), which acts as a parent function.

The graph of \(x^3\) exhibits specific features:

• It is symmetric with respect to the origin, implying that making the input negative results in the output becoming negative as well.
• Its domain and range include all real numbers, indicating that no input or output is restricted.
• It has one point of inflection at the origin \((0,0)\), where the graph's curvature changes.

Transformations alter the cubic function's graph while maintaining its core properties. For example, shifting the cubic function horizontally or vertically changes its position on the coordinate plane but not its overall behavior or end-to-end symmetry.

Graph Sketching

Graph sketching involves understanding the shape and position of a graph after transformations. Let's break down how to sketch a transformed cubic function step by step.

First, sketch the parent function \(f(x) = x^3\). This graph should display a gently curved line passing through the origin, going upward to the right and downward to the left.

For function \(g(x) = (x + 1)^3 - 10\), apply the transformations in the right order:

• Horizontal Shift: Shift every point of the parent function \(f(x)\) to the left by 1 unit. This is due to the \((x+1)\) inside the cubic power.
• Vertical Shift: Next, move the entire graph down by 10 units, corresponding to the \(-10\) outside of the cubic.

Understanding these shifts enables you to visualize how the graph moves across the plane. Each point on \(f(x) = x^3\) gets adjusted according to these directions, allowing a complete sketch of \(g(x) = (x + 1)^3 - 10\) to be drawn accurately.