Problem 68 - College Algebra

Question

In Exercises 68-69, graph each of the functions in the same viewing rectangle. Describe how the graphs illustrate the Binomial Theorem. \(f_{1}(x)-(x+2)^{3} \quad f_{2}(x)-x^{3}\) \(f_{3}(x)-x^{3}+6 x^{2} \quad f_{4}(x)-x^{3}+6 x^{2}+12 x\) \(f_{s}(x)-x^{3}+6 x^{2}+12 x+8\) Use a \([-10,10,1]\) by \([-30,30,10]\) viewing rectangle.

Step-by-Step Solution

Answer

The graphs of the functions \(f_{1}, f_{2}, f_{3}, f_{4}, and f_{5}\) embody the intermediate steps of the Binomial theorem. As we graph each function, we're essentially adding the successive terms of the binomial expansion. This clearly depicts the construction of the binomial expansion through the theorem when \(x+2\) is cubed.

Step 1: Identify the Functions to be Plotted

In this case, there are five functions to be plotted: \(f_{1}(x)=(x+2)^3\), \(f_{2}(x)=x^{3}\), \(f_{3}(x)=x^3 + 6x^2\), \(f_{4}(x)=x^3 + 6x^2 + 12x\), and \(f_{5}(x)=x^3 + 6x^2 + 12x + 8\).

Step 2: Recognize the Relation to Binomial theorem

The Binomial theorem allows us to expand any power of \(x+2\) as a sum of the form \(\sum {3 \choose k} . x^k . 2^{3-k}\). This gives us \(x^3 + 3.2x^2 + 3.4x + 8\). As we can see, the functions \(f_{1}(x), f_{2}(x), f_{3}(x), f_{4}(x)\), and \(f_{5}(x)\) correspond to the summation order of this expansion.

Step 3: Plot the Functions in the same Viewing Window

Using a graphing tool, plot each of the functions within the viewing rectangle specified by the problem: \([-10,10,1]\) by \([-30,30,10]\). This will allow for comparison of the function in the same viewport.

Step 4: Describe the link to the Binomial theorem

By observing the graphs, one sees the functions as visual representations of different intermediate steps of the binomial expansion of \((x+2)^3\). Each subsequent graph adds a term from the binomial expansion, demonstrating the construction of the binomial expansion through the theorem.

Key Concepts

Function Graphing

Graphing functions is a fundamental part of understanding mathematics, allowing us to visualize equations and their behaviors. In this exercise, the task is to graph five different functions in a coordinated view. This helps to see their relationship and development as they fit into the Binomial Theorem.

When graphing functions, each one will be plotted on a coordinate plane, allowing us to see intersections, changes in shape, and overall comparison between the functions. Here we are using a specific 'viewing rectangle' from [-10, 10] for x-values and [-30, 30] for y-values.

The key points for graphing include:

• Identifying key features like extrema and intercepts.
• Recognizing symmetry or asymptotic behavior.
• Utilizing a consistent scale to compare multiple graphs efficiently.

Graphing allows us to see the process of building (functions) step-by-step. From simple powers of x to more elaborate polynomial functions.

Polynomial Functions

Polynomial functions are expressions of variables involving terms in the form of powers. In this exercise, we have a series of polynomial functions, gradually building complexity as we graph them.

Each of the functions presented, like \( f_1(x) = (x+2)^3 \), starts as a simple monomial or binomial. As you progress, terms are added or expanded, turning these into higher order polynomial functions.

• \( f_2(x) = x^3 \) shows a basic cubic function.
• \( f_3(x) = x^3 + 6x^2 \) introduces an additional quadratic term.
• \( f_4(x) = x^3 + 6x^2 + 12x \) adds a linear term.
• \( f_5(x) = x^3 + 6x^2 + 12x + 8 \) introduces a constant term.

Understanding polynomial functions helps to explain how the y-values respond to changes in x. The graph's shape delineates factors like growth rate through slopes and turning points.

Algebraic Expansions

Algebraic expansions allow us to break down expressions into their individual terms. This process is crucial for simplifying equations or understanding polynomial functions.

The Binomial Theorem is a powerful tool for expanding binomial expressions. In this exercise, it's used to transform \((x+2)^3\) into a sum.

The general form of expanding via binomial theorem is: \\[ (x + a)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k}a^k \] Here, \(n = 3\) and \(a = 2\), generating a sequence that builds upon previous terms.

• The first expanded term is \(x^3\), a direct representation involving no additional factors.
• Subsequent terms incorporate increasing powers of the constant (2 in this case) in accord with Pascal's triangle coefficients.
• This systematic addition of terms is shown in the functions as each graph includes more terms.

Algebraic expansions are not just a way to simplify expressions but also to appreciate the deeper relationships between terms and their graphical interpretations.