Problem 48 - Precalculus: Functions and Graphs

Question

(a) Graph the cubic polynomials $$ \begin{array}{l} f(x)=-3 x^{3} \\ g(x)=-3 x^{3}-x^{2}+1 \\ h(x)=-3 x^{3}+x^{2}-1 \end{array} $$ and $$ k(x)=-3 x^{3}-2 x^{2}+2 x $$ in the same coordinate plane, using each of the following viewing rectangles: (1) \([-2,2]\) by \([-2,2]\) (2) \([-10,10]\) by \([-10,10]\) (3) \([-50,50,10]\) by \([-5000,5000,1000]\) (4) \([-100,100,10]\) by \(\left[-5 \times 10^{5}, 5 \times 10^{5}, 10^{5}\right]\) (b) As the viewing rectangle increases in size, how do the graphs of the four functions compare? (c) Which term has the greatest influence on each function's value when \(|x|\) is large?

Step-by-Step Solution

Answer

As the viewing rectangle increases, graphs look more similar; \(-3x^3\) is dominant at large \(|x|\).

Step 1: Understand the Functions

We have four cubic functions to graph: \(f(x) = -3x^3\), \(g(x) = -3x^3-x^2+1\), \(h(x) = -3x^3 + x^2 - 1\), and \(k(x) = -3x^3-2x^2+2x\). All these functions have a dominant cubic term, \(-3x^3\), which will shape the overall behavior of the functions when \(|x|\) is large.

Step 2: Select the Viewing Rectangle (1)

Use the viewing rectangle \([-2, 2]\) by \([-2, 2]\). In this small viewing window, you can observe the local changes and differences close to the origin due to linear and quadratic terms. Graph these functions using graphing software or by plotting key points manually within this range.

Step 3: Select the Viewing Rectangle (2)

Use the viewing rectangle \([-10, 10]\) by \([-10, 10]\). This larger scale will allow you to see the general shape of each cubic polynomial as they extend away from the origin. Notice how the cubic term \(-3x^3\) begins to dominate the behavior of the graph.

Step 4: Select the Viewing Rectangle (3)

Use the viewing rectangle \([-50, 50]\) by \([-5000, 5000]\). At this scale, each function will appear more similar, as the influence of the highest degree term, \(-3x^3\), becomes prominent and the lower degree terms contribute less to the overall shape.

Step 5: Select the Viewing Rectangle (4)

Use the viewing rectangle \([-100, 100]\) by \([-500000, 500000]\). At even larger scales, the cubic term \(-3x^3\) completely dominates the overall graph shape. Differences between the functions are minimized due to diminishing effects of the lower degree terms.

Step 6: Analyze the Effect of Larger Viewing Rectangles

As the viewing rectangle size increases, the graphs of \(f(x), g(x), h(x),\) and \(k(x)\) all appear more similar because the cubic term \(-3x^3\) dominates their behavior at large \(|x|\) values.

Step 7: Identify the Dominant Term

For all functions, when \(|x|\) is large, the cubic term \(-3x^3\) has the greatest influence on the function values because it grows much faster than linear or quadratic terms.

Key Concepts

Graphing Polynomial Functions

When graphing polynomial functions, especially cubic ones, understanding the behavior of each term is crucial. Cubic polynomials like \(f(x) = -3x^3\) and others given in the exercise typically have three types of terms: constant, linear (\(x\)), and quadratic (\(x^2\)). However, it's the cubic term (\(x^3\)) that significantly shapes the graph as \(x\) becomes large in magnitude.

To effectively graph these functions, consider utilizing a graphing tool or plotting key points by hand. This involves choosing specific values for \(x\), plugging them into each function, and then plotting these pairs \((x, f(x))\). By connecting the dots, the curve begins to form.

Defining the appropriate viewing rectangle is vital to gain insight into different graph aspects. Smaller ranges may highlight differences due to lower degree terms, whereas larger ranges will showcase how the cubic term dominates.

Dominant Term in Polynomials

In polynomial functions, the dominant term is the one with the highest degree, which influences the graph's shape most significantly as \(|x|\) increases. For the cubic functions \(f(x) = -3x^3\), \(g(x) = -3x^3-x^2+1\), and the rest, the cubic term \(-3x^3\) is the dominant term.

As \(|x|\) grows, this term's influence becomes more apparent due to its faster growth compared to the linear or quadratic terms. This is because its degree is higher, making it increase or decrease at a much quicker rate.

• Constant terms (e.g., \(1\) or \(-1\)) do not vary with \(x\) and are overshadowed as \(x\) grows.
• Linear terms (e.g., \(2x\)) influence the slope but are outpaced by higher-degree terms.
• Quadratic terms (e.g., \(x^2\), \(-2x^2\)) affect the curve's shape but still yield to cubic terms when \(x\) is large.

Therefore, always identify the highest-degree term to understand how a polynomial behaves at larger scales.

Effect of Viewing Window on Graphs

The choice of viewing window dramatically affects how we perceive the graph of polynomial functions. Different window sizes can either highlight subtle details or show the dominant pattern.

1. **Small Viewing Windows (e.g., \([-2, 2]\))** help us focus on the origins of the graph where linear and quadratic terms might show significant changes.

2. **Medium Viewing Windows (e.g., \([-10, 10]\))** afford a wider view, showcasing the general behavior of the cubic term and starting to minimize the lower degree term effects.

3. **Large Viewing Windows (e.g., \([-100, 100]\))** emphasize the dominance of the cubic term, making graphs of different cubic functions appear similar. It's here that lower degree terms become almost negligible in comparison.

Adjusting the viewing window is a powerful tool in understanding which term in a polynomial function truly dictates the graph's behavior as \(x\) ranges broadly. Always try different window sizes to see all behaviors – from intricate local variations to grand overarching shapes.