Problem 39 - 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 \(f\). \(g (x) = 3(x - 2)^3 \)

Step-by-Step Solution

Answer

The parent function is \(f(x) = x^3\). \(f(x)\) transforms into \(g(x)\) by scaling vertically by a factor of 3 and shifting 2 units to the right. \(g(x) = 3f(x-2)\).

Step 1: Identify the Parent Function

The parent function, denoted by \(f(x)\), serves as the simplest form of the type of function \(g\) is. We can identify \(f(x) = x^3\) as the parent function here, as this form is present in the equation of \(g(x) = 3(x - 2)^3\).

Step 2: Describe the Transformations

On comparing \(f(x) = x^3\) and \(g(x) = 3(x - 2)^3\), it can be seen that \(f(x)\) is transformed into \(g(x)\) by scaling vertically by a factor of 3, and shifting 2 units to the right.

Step 3: Sketch the Graph

Start with the basic graph of \(f(x) = x^3\) which is a cubic graph starting from negative infinity and rising to positive infinity. Then, make the necessary transformations to obtain \(g\). Scale vertically by a factor of 3 means stretching the graph of \(f(x)\) by 3 times along y-axis. Shift 2 units to the right means moving every point on the graph of \(f(x)\) 2 units to the right.

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

In order to write \(g(x)\) in terms of \(f(x)\), we need to replace \(x\) in \(f(x)\) with \((x-2)\), and multiply the whole equation by 3. Hence, \(g(x) = 3f(x-2)\)

Key Concepts

Transformations

Understanding transformations is key to grasping how a function can change its shape and position. Essentially, transformations are modifications applied to the graph of a parent function to form a new function. They can involve:

• Vertical Stretching/Compressing: If a function is multiplied by a constant factor greater than 1, like in our equation where it is multiplied by 3, it stretches the graph vertically. Compare this to when it is between 0 and 1, which compresses the graph.
• Horizontal Shifting: This occurs when the function's variable, usually denoted as "x," is adjusted by adding or subtracting a constant. In our case, "+2" shifts the graph 2 units to the right, moving every point in this direction.

By understanding these transformations, it becomes easier to manipulate and visualize how functions behave under such changes.

Graph Sketching

Graph sketching involves drawing the basic visual representation of the function after applying all necessary transformations. Here is a step-by-step approach:

• Start with the Parent Function: Begin by sketching the graph of the parent function, which is the simplest version. For example, if the parent function is a cubic function like \(f(x) = x^3\), it starts from negative infinity and extends to positive infinity through the origin.
• Apply Vertical Stretch/Compression: Multiply the y-values of every point on the parent graph by the vertical stretch factor—in this case, 3—and redraw. This stretches all points away from the x-axis.
• Shift Horizontally: Move every modified point horizontally according to the shift. For example, a transformation of \((x-2)\) implies moving each point 2 units to the right.

Through these steps, you create an accurate sketch of the transformed function, reflecting its new shape and position.

Function Notation

Function notation provides a precise and concise way to express transformations and dependencies between variables.
It often uses a general structure like \(g(x) = ...\) to define how one function is related to another.
In this context, function notation is especially useful for expressing transformed functions:

• Express in Terms of Parent Function: Using the original function \(f(x)\), define \(g(x)\) in terms of \(f\) and specific transformations. For instance, if \(f(x) = x^3\), then \(g(x) = 3(x-2)^3\) can be written as \(g(x) = 3f(x-2)\), showing a clear relationship.
• Highlight Transformations: This notation helps readily identify the sequence of transformations applied, such as shifts or scaling, enhancing the understanding of the function's behavior.

Understanding and using function notation is crucial for clearly defining and manipulating functions in mathematics.