Problem 56 - College Algebra

Question

Graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$f(x)=x^{3}-27$$

Step-by-Step Solution

Answer

The x-intercept is at (3, 0), the y-intercept at (0, -27). As x → ±∞, f(x) → ±∞.

Step 1: Enter the Function into Calculator

First, input the function \( f(x) = x^3 - 27 \) into the graphing calculator. Navigate to the y= section and type "x^3-27".

Step 2: Observe the Graph

View the graph displayed by the calculator. Notice how the graph behaves as it extends towards positive and negative infinity and where it crosses the axes.

Step 3: Find the X-intercept

The x-intercept is a point where the graph crosses the x-axis. Observe the graph to identify the value of \( x \) where the function equals zero. \( f(x) = 0 \) implies \( x^3 - 27 = 0 \). Solve for \( x \) to find \( x = 3 \). The x-intercept is \((3, 0)\).

Step 4: Find the Y-intercept

The y-intercept occurs where the graph crosses the y-axis. This happens when \( x = 0 \). Substitute \( x = 0 \) into \( f(x) \): \( f(0) = 0^3 - 27 = -27 \). The y-intercept is \((0, -27)\).

Step 5: Determine End Behavior

Consider the leading term of the polynomial, \( x^3 \). As \( x \to +\infty \), \( f(x) \to +\infty \), and as \( x \to -\infty \), \( f(x) \to -\infty \). This indicates the function rises to the right and falls to the left.

Key Concepts

Intercepts

In the world of polynomial functions, intercepts tell us where our function crosses the axes. This can be extremely helpful in understanding what the graph of a polynomial looks like. Let's discuss how we find them for our example, the polynomial \(f(x) = x^3 - 27\).

• X-Intercept: The x-intercept is a point where the graph of a function crosses the x-axis. At this point, the function value is zero. For our function, solve \(x^3 - 27 = 0\) to find the x-intercept. The solution is \(x = 3\), so the x-intercept is at \((3, 0)\).
• Y-Intercept: The y-intercept occurs where the graph crosses the y-axis. This happens when \(x = 0\). Substitute \(x = 0\) into the function: \(f(0) = 0^3 - 27 = -27\). Thus, the y-intercept is at \((0, -27)\).

These intercept points provide a great starting framework for sketching or understanding the graph's basic shape.

End Behavior

To fully grasp a polynomial graph's end behavior, you need to focus on the leading term of the polynomial function. This tells you what happens to the graph as \(x\) moves towards positive and negative infinity.

In the polynomial \(f(x) = x^3 - 27\), the leading term is \(x^3\). This leading term is the most influential in determining end behavior. Here's how:

• As \(x \to +\infty\), \(f(x) = x^3 - 27\) tends towards \(+\infty\). This means the graph rises as it moves towards positive infinity.
• As \(x \to -\infty\), \(f(x) = x^3 - 27\) tends towards \(-\infty\). This means the graph falls as it moves towards negative infinity.

Understanding these limits gives you a clearer picture of how the graph behaves at its extremes, showcasing its overall "rising right and falling left" nature.

Graphing Calculators

Graphing polynomial functions can be very engaging, particularly when you have a graphing calculator at hand. This tool accomplishes several essential tasks in understanding polynomials, such as sketching the graph and identifying key properties like intercepts and end behavior.

Here's a quick guide to using a graphing calculator for our polynomial function \(f(x) = x^3 - 27\):

• Input Function: Begin by entering the function into the calculator. On most calculators, you'll navigate to the y= section and type in \(x^3 - 27\). After entering the equation, prompt the calculator to display the graph.
• Observe the Graph: Watch closely as the calculator illustrates the graph. Notice the points where the graph intersects the axes and the general slope of the function, reinforcing visual understanding of intercepts and end behavior.

Using these steps, graphing calculators enhance the learning experience by combining visual representation with computational accuracy, empowering you to grasp polynomial behavior effectively.