Problem 68 - Algebra and Trigonometry

Question

Graph each of the functions in the same viewing rectangle. Describe how the graphs illustrate the Binomial Theorem. $$f_{1}(x)=(x+2)^{3} \quad f_{2}(x)=x^{3}$$ $$f_{3}(x)=x^{3}+6 x^{2} \quad f_{4}(x)=x^{3}+6 x^{2}+12 x$$ $$f_{5}(x)=x^{3}+6 x^{2}+12 x+8$$ Use a \([-10,10,1]\) by \([-30,30,10]\) viewing rectangle.

Step-by-Step Solution

Answer

Functions \( f_1(x) \), \( f_2(x) \), \( f_3(x) \), \( f_4(x) \) and \( f_5(x) \) when graphed sequentially illustrate how each term in the expansion of \( (x+2)^3 \) through the Binomial Theorem progressively modifies the graph. The complete Binomial expansion is shown by the final function \( f_5(x) \).

Step 1: Graph the first function

Graph the function \( f_1(x)=(x+2)^3 \) in the viewing rectangle. This is a cubic function graph with its vertex displaced 2 units to the left.

Step 2: Graph the second function

Now graph the function \( f_2(x)=x^{3} \). This function coincides with the first term of the expansion \( f_1(x)=(x+2)^3 = x^3 + \ldots \). Plotting \( f_2(x) = x^{3} \) results in a cubic graph.

Step 3: Graph the third function

Plot the function \( f_3(x)=x^{3}+6x^{2} \). This function matches the first two terms of the expansion \( f_1(x)=(x+2)^3 = x^3 + 3*2*x^2 + \ldots \). Therefore, this graph is based on the second one but modified due to the additional \( 6x^2 \) term.

Step 4: Graph the forth function

Next, graph \( f_4(x)=x^{3}+6x^{2}+12x \). Now, this function includes up to three terms of the Binomial Theorem, so it's the same as the third function but further modified due to the additional \( 12x \) term.

Step 5: Graph the final function

Finally, plot \( f_5(x)=x^{3}+6x^{2}+12x+8 \). At this point, all terms of the Binomial Theorem are considered, so this graph is the complete expansion of \( f_1(x) = (x+2)^3 \).

Key Concepts

Cubic Functions

Cubic functions are a type of polynomial whose highest degree is three. In a cubic function, the general form can be expressed as \(f(x) = ax^3 + bx^2 + cx + d\). The leading coefficient \(a\) determines the function's behavior as \(x\) approaches infinity and negative infinity. Cubic functions have:

• One real root or three real roots (which can include a combination of distinct or repeated roots),
• Multiple turning points,
• S-shaped curve, often referred to as a cubic curve.

In the exercise, each function \(f_1(x) = (x + 2)^3\), \(f_2(x) = x^3\), and so on, is a cubic function. These functions illustrate how the roots and shape of cubic functions evolve as additional terms are added or altered.

Polynomial Expansion

Polynomial expansion is a process of expressing a polynomial that involves a power into a sum of polynomials of smaller powers. The Binomial Theorem is a common method used for polynomial expansions that involves binomials raised to a power. The theorem provides a quick way to expand such expressions: \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]Each function in the exercise was derived from the expansion of \((x+2)^3\):

• \(f_2(x) = x^3\) represents the first term,
• \(f_3(x) = x^3 + 6x^2\) adds the second term,
• \(f_4(x) = x^3 + 6x^2 + 12x\) includes up to the third term,
• \(f_5(x) = x^3 + 6x^2 + 12x + 8\) completes the expansion with the fourth term.

Each term corresponds to successive terms of \((x+2)^3\) taught in the Binomial Theorem, revealing the gradual buildup in the polynomial expansion.

Graphing Functions

Graphing functions visually represents how outputs are affected by different inputs. When graphing cubic functions, the shape typically showcases a curve that may resemble a flattened 'S'. This is because they often have points of inflection, where the curve changes concavity. In the exercise, each graph \(f_1(x)\) through \(f_5(x)\) modifies the curve by adding terms from the polynomial expansion, showing how additional terms change the graph:

• \(f_1(x) = (x+2)^3\) shows a simple cubic graph shifted to the left,
• \(f_3(x)\) through \(f_5(x)\) illustrate how adding \(6x^2, 12x\), and finally \(8\) impacts the graph by changing steepness and intercepts.

Using a defined viewing rectangle like \([-10,10,1]\) by \([-30,30,10]\) helps in systematically viewing these changes.

Coordinate Plane

A coordinate plane is a two-dimensional plane formed by two perpendicular lines called axes. These lines are usually referred to as the x-axis (horizontal) and y-axis (vertical). The point where the axes intersect is called the origin, marked as \((0,0)\). Coordinates on this plane are ordered pairs \((x,y)\) that tell the location of points relative to the axes. In graphing:

• Cubic functions can stretch across multiple quadrants,
• The viewing rectangle \([-10,10,1]\) by \([-30,30,10]\) sets the bounds for where to view the graph to comprehend its full behavior.

Graphing on the coordinate plane shows not just positions but also the relationship between variables expressed by these functions, helping to recognize patterns or changes in the curve's shape as a result of different polynomial expansions.