Problem 58 - 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 \(f\).$$g(x)=-(x+3)^{3}-10$$.

Step-by-Step Solution

Answer

The parent function is \( f(x) = x^3 \). The sequence of transformations from \( f \) to \( g \) is: horizontal shift to the left by 3 units, reflection in x-axis, and then vertical shift down by 10 units. The function \( g(x) \) in terms of parent function \( f \) is \( g(x) = -f(x+3) - 10 \).

Step 1: Identify the parent function

The parent function is the cubic function \( f(x) = x^{3} \). This is determined by identifying the basic function present in \( g(x) \). Here, we notice that the function involves cubing of \( x \). Therefore, the parent function, \( f \), is \( x^{3} \).

Step 2: Describe the sequence of transformations

The given function \( g(x)= -(x+3)^3 - 10 \), is a transformation of the parent function. It includes a horizontal shift to the left by 3 units, a reflection in the x-axis and a vertical shift downwards by 10 units.

Step 3: Sketch the graph

To sketch the graph, start by drawing the cubic function \( f(x) = x^3 \). Then, shift the entire graph to left by 3 units. Next, reflect the graph over the x-axis, which means flip it upside down. Finally, shift the graph downwards by 10 units. This will give us the graph of the function \( g(x) \).

Step 4: Define g in terms of f

The transformed function \( g(x) \) can be expressed in terms of the parent function \( f(x) \) as \( g(x) = -f(x+3) - 10 \). This represents all the transformations that have been applied to \( f(x) \) to get \( g(x) \).

Key Concepts

Parent function

A **parent function** is the simplest, most basic form of a type of function. It serves as the foundation upon which more complex functions are built through transformations.
In the context of many math classes, you will deal with several parent functions such as:

• Linear: \( f(x) = x \)
• Quadratic: \( f(x) = x^2 \)
• Cubic: \( f(x) = x^3 \)
• Exponential: \( f(x) = b^x \), where \( b > 0 \)
• Square Root: \( f(x) = \sqrt{x} \)
• Absolute Value: \( f(x) = |x| \)

In our exercise, the parent function is the cubic function, \( f(x) = x^3 \). This function begins its journey as a simple power of 3, and can be transformed into more complex functions like \( g(x) = -(x+3)^3 - 10 \) with various transformations.

Cubic function

The **cubic function** is a crucial member of the family of power functions, represented mathematically as \( f(x) = x^3 \). This function's graph has a distinctive elongated S-shape, which passes through the origin (0,0). The cubic curve rises steeply in both directions — stretching upwards as \( x \) increases, and downward as \( x \) decreases.

Its increasing and decreasing nature is dictated by the change in sign from positive to negative when multiplied by any coefficients or constants, as seen in transformations. The cubic function's properties make it an excellent candidate to demonstrate shifts, reflections, and stretches — all of which are hallmark transformation operations in algebra. In our given function \( g(x) = -(x+3)^3 - 10 \), it starts with the cubic form and undergoes the series of changes.

Graph sketching

**Graph sketching** refers to the visual representation of a function's equation on a coordinate plane. This process helps us understand the behavior of the function, providing insights into its key features such as intercepts, asymmetries, and overall shape.

To sketch the graph of a cubic transformation like \( g(x) = -(x+3)^3 - 10 \), one should follow these steps:

• Begin with the basic cubic graph \( f(x) = x^3 \), typically drawn as a curve starting at the origin.
• Apply the horizontal transformation by shifting the curve 3 units to the left (result of \( x+3 \)).
• Reflect the graph over the x-axis to achieve the flipping effect (due to the negative sign in front of the \( x^3 \)).
• Finally, execute a vertical shift moving the graph downward by 10 units.

This ordered approach allows one to systematically transform the parent graph, capturing all changes that lead to function \( g(x) \).

Function notation

**Function notation** is a compact and precise way of expressing functions and their transformations. It is essential in indicating the input-output relationship of a function.
In mathematical expressions, function notation is typically represented as \( f(x) \), where \( f \) is the function name and \( x \) is the input variable.

When dealing with transformations, function notation becomes a powerful tool. In our scenario, the function \( g(x) = -(x+3)^3 - 10 \) can be neatly expressed in terms of the parent function \( f(x) = x^3 \) using function notation as \( g(x) = -f(x+3) - 10 \).
This notation succinctly captures each transformation step:

• Shift involves changing the argument from \( x \) to \( x+3 \).
• Reflection is indicated by the negative sign in front of \( f(x+3) \).
• Vertical shift is represented by subtracting 10 from the expression.

Such notation enables clear communication and understanding across various mathematical contexts.