Problem 110 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. What is the relationship between the two graphs? Use the Binomial Theorem to write the polynomial function \(g\) in standard form. \(f(x)=-x^{4}+4 x^{2}-1\) \(g(x)=f(x-3)\)

Step-by-Step Solution

Answer

The relationship between the two graphs is a horizontal shift to the right by 3 units. Using the Binomial theorem, the polynomial function \(g\) can be written in standard form as \(g(x) = -x^4 + 12x^3 - 36x^2 + 46x - 36\).

Step 1: Graph the functions

Plot the function \(f(x)=-x^{4}+4 x^{2}-1\) using a graphing tool and do the same with the function \(g(x)=f(x-3)\), to better understand their relationship visually.

Step 2: Observe and Compare the Graphs

Observation of the two graphs should indicate that the graph of \(g(x)\) is identical to the graph of \(f(x)\) but is shifted 3 units to the right. Therefore, the relationship between the two graphs is a horizontal shift.

Step 3: Rewrite g(x) using the Binomial Theorem

To get the polynomial function \(g\) in standard form, apply the Binomial theorem to expand \(f(x-3)\). This means replacing all instances of \(x\) in the function \(f\) with \(x-3\). As a result, \(g(x) = -(x-3)^4 + 4(x-3)^2 -1\). Expanding and simplifying yields \(g(x) = -x^4 + 12x^3 - 36x^2 + 46x - 36\).

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that involve variables raised to whole number powers, combined with coefficients. Their general form can be written as:

• Polynomial: \[ a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \]

The highest power in the polynomial is known as its degree. For example, the function given in the exercise, \( f(x) = -x^{4} + 4x^{2} - 1 \), is a fourth-degree polynomial because the highest power of \( x \) is 4.
Typically, these functions are continuous and smooth and can take on a wide variety of shapes. They can be classified based on their degree, such as linear, quadratic, cubic, and so on. This classification helps understand their behavior on a graph, like how many times they cross the x-axis or how their shape changes with variations in coefficients.
Understanding polynomials is crucial as they form the foundation for more complex mathematical concepts, including calculus and numerical methods.

Graphing Transformations

Graphing transformations involve altering a graph's position or shape according to mathematical rules. When a transformation is applied to a function, the graph changes, but its general form and features might remain consistent.
The main types of transformations include translations, reflections, stretches, and compressions:

• Translations: Shifting the graph horizontally or vertically. A horizontal shift moves the graph left or right, while a vertical shift moves it up or down.
• Reflections: Flipping the graph over a line, such as the x-axis or y-axis.
• Stretches and Compressions: Altering the graph's size either horizontally or vertically without changing its shape.

For example, in our exercise, the function \( g(x) \) is a horizontally shifted version of \( f(x) \), which is a specific type of transformation. Understanding these transformations helps in predicting how a graph will change with different function inputs.

Horizontal Shift

A horizontal shift in a graph occurs when the entire graph of a function moves left or right on the coordinate plane. This is achieved by altering the input variable \( x \) within the function.
In the provided exercise, the function \( g(x) = f(x-3) \) represents a horizontal shift. Here, every point of \( f(x) \) moves 3 units to the right. This is because the transformation inside the function is \( x-3 \). In general, if \( y=f(x) \), the function \( y=f(x-c) \) shifts the graph of \( f \) \( c \) units to the right if \( c \) is a positive number.
Understanding horizontal shifts is essential because they allow for the modification of functions without altering their core characteristics. They are commonly used in real-world applications where a positional change is required without impacting the function's overall output.