Problem 94 - Algebra and Trigonometry

Question

In Exercises 94–97, use a graphing utility with a viewing rectangle large enough to show end behavior to graph each polynomial function. $$f(x)=x^{3}+13 x^{2}+10 x-4$$

Step-by-Step Solution

Answer

The end behavior of the polynomial function \(f(x)=x^{3}+13 x^{2}+10 x-4\) as \(x\) approaches infinity is that it also is infinity (going upwards), as \(x\) approaches negative infinity, \(f(x)\) approaches negative infinity (going downwards).

Step 1: Identify Leading term

The leading term of the function \(f(x)=x^{3}+13 x^{2}+10 x-4\) is \(x^3\). This is the term of highest degree - the \(x^3\) term.

Step 2: Determine end behavior

For polynomial functions, the end behavior is determined by the leading term. Since the coefficient of the leading term \(x^3\) is positive, as \(x\) approaches infinity (\(x\) gets larger and larger), \(f(x)\) also approaches infinity. Meanwhile, as \(x\) approaches negative infinity (\(x\) gets smaller and smaller), \(f(x)\) approaches negative infinity. In simpler words, as \(x\) moves to the right, the graph goes up, and as \(x\) moves to the left, the graph goes down.

Step 3: Graphing the polynomial

Using a graphing utility, input the function \(f(x)=x^{3}+13 x^{2}+10 x-4\). Ensure that the viewing rectangle is large enough to show the end behaviors. You should observe that as we predicted in Step 2, as \(x\) gets larger, the function also gets larger and as \(x\) gets smaller, the function gets smaller. Therefore, the graph should emerge from the bottom left, rise and then fall, before rising again to the top right.

Key Concepts

End Behavior

Understanding the end behavior of a polynomial function is crucial for graphing. End behavior describes what happens to the value of a function as the variable approaches positive or negative infinity. For the polynomial function \(f(x) = x^3 + 13x^2 + 10x - 4\), which is a cubic function, the end behavior is dictated by its leading term. Given our leading term \(x^3\) is positive, as \(x\) moves towards infinity, \(f(x)\) also goes towards infinity.
Conversely, as \(x\) moves towards negative infinity, \(f(x)\) dips to negative infinity. This means, on a graph, the left end of the graph will trend downwards while the right end will trend upwards. This characteristic swooping action is signature to cubic polynomials.

Leading Term

The leading term of a polynomial function plays a pivotal role in determining various characteristics of the graph, such as the end behavior and the degree of the polynomial. For the function \(f(x) = x^3 + 13x^2 + 10x - 4\), the leading term is \(x^3\).
This term is key since it establishes the degree of the polynomial, which is 3 in this case. It also informs us about the number of possible turning points, which is generally one less than the degree, so possibly up to 2 turning points.

• In polynomial functions, the leading term determines how quickly the function values increase or decrease.
• A positive leading coefficient means the function rises as \(x\) increases and falls as \(x\) decreases.

Therefore, the leading term heightens both the expectations of the function's direction and the range of output values.

Graphing Polynomial Functions

Graphing polynomial functions requires considering their end behavior, critical points, and most importantly the overall shape dictated by the degree and leading terms. When graphing \(f(x) = x^3 + 13x^2 + 10x - 4\), ensure that you use a graphing utility to visualize the function accurately.
Before graphing, identify where turning points might occur by finding the derivative and solving for zero, or by analyzing the function's factors. When using a graphing calculator, make sure the view screen covers a wide range to see the full picture of the end behavior:

• The graph should start from the bottom left and proceed upwards, reflecting the behavior explained by the cubic leading term.
• Look for turning points where the function might change direction, something typical for cubic graphs.

Engage with the graph to discover intercepts and point of inflections which add further detail to understanding the polynomial's structure.

Cubic Functions

Cubic functions represent third-degree polynomial functions, characterized by the standard form \(ax^3 + bx^2 + cx + d\). Our function \(f(x) = x^3 + 13x^2 + 10x - 4\) is a perfect example.
These functions often have an 'S' shaped curve graph, reflecting one or two turning points. Important features of cubic functions include:

• The inflection point, where the graph changes concavity, is a distinguishing feature of cubic graphs.
• As with most polynomial functions of odd degree, they have one root on average and cross the x-axis at least once.

When analyzing the function, it's the changes in direction and curvature that define these functions and make them distinct and important in mathematical modeling. The cubic model is versatile, appearing in physics, engineering, and economics to depict real-world scenarios that require accounting for multiple direction shifts.