Problem 108 - 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. \(f(x)=x^{3}-4 x\) \(g(x)=f(x+4)\)

Step-by-Step Solution

Answer

The graph of \(g(x)\) is a shift of the graph of \(f(x)\) four units to the left. The standard form of \(g(x)\) is \(g(x) = x^3 + 12x^2 + 44x + 48\).

Step 1: Graphing Functions

Start by graphing both \(f(x) = x^{3} - 4x\) and \(g(x) = f(x+4)\) on the same graph. We already have the equation of \(f(x)\), but we need to substitute \(x+4\) for \(x\) in the equation of \(f\) to obtain the equation of \(g\). Doing so gives \(g(x) = (x + 4)^{3} - 4(x + 4)\). Now you can graph these two equations.

Step 2: Interpret the Graphs

By looking at the graphs of \(f(x)\) and \(g(x)\), we can see that \(g(x)\) is a shift of \(f(x)\) to the left by four units. This is in line with our knowledge that adding a constant to \(x\) in a function results in a horizontal shift.

Step 3: Using Binomial Theorem

To write \(g(x) = (x + 4)^3 - 4(x + 4)\) in standard form, we can use the Binomial Theorem, which says: \((a + b)^n = a^n + \binom{n}{1} a^{n-1}b + \binom{n}{2} a^{n-2}b^2 + ... + b^n\). Applying this theorem to \(g(x)\) gives us: \(g(x) = x^3 + 3*4x^2 + 3*4^2x + 4^3 - 4x - 16\). Simplifying, we get: \(g(x) = x^3 + 12x^2 + 48x + 64 - 4x - 16 = x^3 + 12x^2 + 44x + 48\). We've now written \(g(x)\) in standard form.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that involve sums of powers of one or more variables. Typically, these functions are written in the standard form, which orders the terms from the highest power to the lowest. For example, in this exercise, the polynomial function \(f(x) = x^3 - 4x\) represents a cubic polynomial with degrees arranged from the highest to the lowest. This function is not in complete standard form, as it is missing some intermediary powers, but it still follows the ordering principle from high to low degree terms.
Understanding polynomial functions is vital because they provide the foundation for understanding more complex mathematical concepts, such as calculus. Key features of polynomial functions include their degree, determining the number of roots or solutions, and their symmetry properties, which can be established based on even or odd powers.
In our exercise, the task was to shift the polynomial \(f(x)\) to derive \(g(x)\) by horizontally altering the input, demonstrated through \(g(x) = f(x+4)\). Polynomial functions like these are essential building blocks in algebra and are used extensively in modeling real-world scenarios, optimization problems, and data fitting strategies.

Graphing Utilities

Graphing utilities are tools, such as graphing calculators or software programs, that allow us to visualize mathematical functions easily. By inputting functions like \(f(x) = x^3 - 4x\) and \(g(x) = f(x+4)\), students can immediately see the effect of transformations and relationships between graphs.
Using graphing utilities for this exercise is essential as they help verify how transformations, such as horizontal shifts, affect the graph's shape and position. When we graph \(f(x)\) and \(g(x)\) together, the graphing utility shows how \(g(x)\) shifts horizontally compared to \(f(x)\). The visual alignment confirms predictions made by algebraic manipulation, such as shifting ideas from the Binomial Theorem in this context.
Overall, employing graphing utilities improves comprehension of abstract concepts, helps check manual calculations, and provides a method to experiment with mathematical functions and transformations interactively.

Horizontal Shifts

Horizontal shifts in functions occur when a constant is added or subtracted from the input variable within the function's argument. This operation moves a graph left or right along the x-axis, depending on whether the constant is positive or negative. For the function \(g(x) = f(x+4)\) in the exercise, the addition of 4 inside the function indicates a horizontal shift to the left by 4 units.
Understanding horizontal shifts is crucial for analyzing transformations in functions and linking them to real-world applications, like signal processing or economic models where translating functions can represent changes over time or scenarios.
When interpreting horizontal shifts, remember:

• Adding a constant moves the graph left if positive.
• Subtracting a constant moves the graph right if negative.

As demonstrated, \(f(x) = x^3 - 4x\) leads to the altered \(g(x) = x^3 + 12x^2 + 44x + 48\) via a horizontal shift, confirming the theoretical understanding through practical application.