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Factoring is a critical technique in working with polynomial functions. It simplifies polynomials, making it easier to analyze and solve problems associated with them. Let's consider the polynomial function given in the exercise: \(y = x^3 - 9x\). Our first step is to identify any common factors. Here, each term of the polynomial contains \(x\), so we can factor out \(x\), resulting in:

• \(y = x(x^2 - 9)\)

After removing the common factor, we notice the expression \(x^2 - 9\) can be recognized as a "difference of squares." The difference of squares rule states that \(a^2 - b^2 = (a-b)(a+b)\). Applying this rule:

• \(y = x(x-3)(x+3)\)

The polynomial is now fully factored. This form reveals potential roots, critical for finding intercepts and graphing the function. Understanding factoring is fundamental, as it turns complex expressions into simpler, actionable forms.

Intercepts are the points where the graph of a function crosses the axes. These points are important because they provide information about the roots and behavior of the function. To find the \(x\)-intercepts, set \(y = 0\) in the factored equation, as intercepts occur where the function equals zero:

• From \(x(x - 3)(x + 3) = 0\), we derive that \(x = 0, x = 3, x = -3\).

Hence, the \(x\)-intercepts are \((-3, 0), (0, 0), (3, 0)\). Notice that \((0, 0)\) is both an \(x\)-intercept and a \(y\)-intercept. For the \(y\)-intercept, set \(x = 0\) in the original equation:

• \(y = 0^3 - 9(0) = 0\)

Thus, the \(y\)-intercept is also \((0, 0)\). Identifying these intercepts provides key points which aid in graphing the function accurately.

Graphing polynomial functions involves several steps and considerations that paint a complete picture of the function's behavior. Start by plotting the intercepts found: \((-3,0), (0,0), (3,0)\). These intercepts serve as a basic framework for sketching the graph. Next, consider the general shape of the polynomial. Since the highest degree term in the polynomial is \(x^3\), this function is cubic, meaning it will have an 'S' shape — increasing, decreasing, or both. Remember, cubic functions can change direction up to twice. As you sketch, anticipate smooth curves because polynomial functions are continuous.

• Notice how the curve should pass through the intercepts smoothly.
• The cubic term indicates there should be a turning point somewhere between each pair of intercepts.

This process emphasizes visualization of the fundamental properties of the polynomial by connecting each sequential intercept with smooth turns.

End behavior describes what happens to the values of a polynomial function as \(x\) approaches either infinity or negative infinity. It is crucial in understanding the bigger picture of how a polynomial behaves over large intervals. For our function \(y = x^3 - 9x\):

• The term \(x^3\) dominates the behavior for large values of \(x\).

When \(x\) goes to positive infinity, \(x^3\) also heads towards infinity, making \(y\) very large and positive.Conversely, when \(x\) goes to negative infinity, \(x^3\) becomes very negative, dragging \(y\) down to negative infinity.

• This means that the ends of the graph extend upwards right and downwards left.

By observing the function's end behavior, we can predict how the graph stretches and positions, which complements the plot of the intercepts, providing a thorough understanding of the function's structure.