91Ó°ÊÓ

Graphing a polynomial helps us understand its overall behavior, such as its intercepts and end behavior. For cubic functions like \( f(x) = x^3 - 0.01x \), this typically results in an `S` shaped curve. This occurs because of the highest power of \( x \), which in this case is 3. The graph of this cubic polynomial will show how the function behaves over a range of \( x \) values by crossing the axes at certain points called intercepts. To visually explore this function, you can use a graphing calculator which allows you to see this behavior clearly. Make sure you set a window that shows the entire important parts of the graph, a starting point could be \( [-10, 10] \) for both \( x \) and \( y \) values. This gives you a complete and detailed perspective of how the polynomial looks.

X-intercepts are the points where the graph crosses the x-axis. For our function \( f(x) = x^3 - 0.01x \), these intercepts can be found by solving the equation \( f(x) = 0 \). This involves setting the polynomial equal to zero and solving for \( x \).

Looking at the polynomial \( x^3 - 0.01x = 0 \), you can factor it to \( x(x^2 - 0.01) = 0 \). This shows us that the x-intercepts are found when \( x = 0 \), \( x = \sqrt{0.01} \), and \( x = -\sqrt{0.01} \). Simplified, we have intercepts at the points \( (0, 0), (0.1, 0), \text{and} (-0.1, 0) \).

These intercepts indicate where the polynomial will touch or cross the x-axis, offering insights into the roots of the equation.

The y-intercept is a specific point where the graph crosses the y-axis. It's important because it tells you the function's value when \( x = 0 \). For the function \( f(x) = x^3 - 0.01x \), determining the y-intercept involves substituting \( x = 0 \) directly into the function.

Performing this substitution, we find that \( f(0) = 0^3 - 0.01(0) = 0 \). Therefore, the y-intercept is at the point \( (0, 0) \).

In other words, the point where the graph meets the y-axis is at the origin, providing a foundational reference for the graph's position.

Understanding the end behavior of a polynomial is essential for grasping its long-term behavior as you move towards positive or negative infinity along the x-axis. For the polynomial \( f(x) = x^3 - 0.01x \), the leading term \( x^3 \) dictates this behavior.

The term \( x^3 \) causes a distinct cubic characteristic. As \( x \to \infty \) (moving to the right infinitely), \( f(x) \to \infty \), meaning the graph goes upwards. Conversely, as \( x \to -\infty \) (moving to the left infinitely), \( f(x) \to -\infty \), thus the graph tends downwards.

These observations define the end behavior: the graph ascends to the right and descends to the left, giving it the signature shape of most cubic polynomials.