Problem 60 - College Algebra

Question

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$ f(x)=x^{3}-0.01 x $$

Step-by-Step Solution

Answer

X-intercepts: -0.1, 0, 0.1; Y-intercept: 0; End behavior: Up on right, down on left.

Step 1: Input Function into Calculator

Use a graphing calculator or software. Enter the polynomial function \( f(x) = x^3 - 0.01x \) as the function to graph. Ensure the graphing window is set to display both the x and y axes adequately.

Step 2: Analyze the Graph for Intercepts

From the graph, identify points where the graph intersects the axes. The x-intercepts occur where the graph crosses the x-axis, and the y-intercept is where the graph crosses the y-axis.

Step 3: Determine the X-intercepts

For \( f(x) = x^3 - 0.01x \), set \( f(x) = 0 \) to find the x-intercepts. Solve \( x^3 - 0.01x = 0 \), which factors to \( x(x^2 - 0.01) = 0 \). Thus, the x-intercepts are \( x = 0, \pm\sqrt{0.01} = \pm0.1 \).

Step 4: Identify the Y-intercept

The y-intercept occurs at \( x = 0 \). Substitute \( x = 0 \) into the function: \( f(0) = 0^3 - 0.01(0) = 0 \). Thus, the y-intercept is \( (0, 0) \).

Step 5: Examine End Behavior

Analyze the graph to determine how the ends of the graph behave as \( x \to \infty \) and \( x \to -\infty \). A cubic function such as \( x^3 \) implies that as \( x \to \infty \), \( f(x) \to \infty \); as \( x \to -\infty \), \( f(x) \to -\infty \). So the ends point up on the right and down on the left.

Key Concepts

Graphing Calculator

A graphing calculator is a handy tool for visualizing polynomial functions like cubic functions. By inputting the equation into the calculator, you can quickly generate a graph that displays the relationship between different values of the function and the x-axis. The graphing calculator allows you to see the overall shape of the polynomial and identify important features like intercepts and end behavior.

• Ensure you are entering the polynomial function correctly; for example, in this scenario, you would input it as \( f(x) = x^3 - 0.01x \).
• Set the calculator window to appropriately display the necessary part of the graph, ensuring visibility of relevant parts of the function.
• Utilize zoom functions if needed to better see both the x and y intercepts and to understand the way the parabola stretches.

By familiarizing yourself with a graphing calculator’s functions, you can quickly interpret and analyze polynomial graphs.

Intercepts

Intercepts are key points where the function meets the x-axis and the y-axis. Finding these allows you to understand where the function equals zero and how it behaves around that point.For the given function \( f(x) = x^3 - 0.01x \):

• X-Intercepts: Set \( f(x) = 0 \) and solve for \( x \). This particular equation solves to \( x(x^2 - 0.01) = 0 \), making the x-intercepts occur at \( x = 0, \pm0.1 \).
• Y-Intercept: This is found by substituting \( x = 0 \) into the function. With \( f(0) = 0^3 - 0.01(0) = 0 \), the y-intercept is at the origin, \( (0, 0) \).

Understanding intercepts is crucial in sketching and interpreting polynomial graphs, as they offer a starting point for exploring the behavior of the function.

End Behavior

End behavior helps describe how a function behaves as the input values become very large or very small. The behavior at each "end" of the function can be observed using a graph or calculated by examining leading terms.For cubic functions like \( f(x) = x^3 - 0.01x \), it's essential to understand:

• As \( x \to \infty \): The function \( f(x) \to \infty \), meaning the graph rises to the right.
• As \( x \to -\infty \): The function \( f(x) \to -\infty \), indicating it falls to the left.

This is typical for cubic functions, where the leading term \( x^3 \) dictates this pattern of end behavior.

Cubic Function

Cubic functions are polynomial functions where the highest exponent of the variable \( x \) is three, giving it the form \( f(x) = ax^3 + bx^2 + cx + d \). In the function \( f(x) = x^3 - 0.01x \), the dominant term is \( x^3 \), which significantly influences its graph and characteristics:

• Shape: Cubic functions generally have an "S"-shaped curve due to the odd power of the leading term.
• Roots: They can have up to three real roots, depending on how they interact with the x-axis.
• Critical Points: They may have inflection points where the curvature changes direction.

Understanding cubic functions is crucial because their unique traits and characteristics are foundational in calculus and higher-level math courses.