91Ó°ÊÓ

A quartic function is a type of polynomial that has a degree of four. This means the highest power of the variable, typically denoted as \( x \), is 4. Quartic functions are interesting because they have a variety of shapes and features, including the possibility of having up to four x-intercepts and three turning points or local peaks and valleys in their graphs.

The general form of a quartic function is \( ax^4 + bx^3 + cx^2 + dx + e \), where \( a eq 0 \). The coefficient \( a \) determines the basic shape or stretch of the graph, while \( e \) is often the y-intercept or where the graph crosses the y-axis. In this exercise, the quartic function is expressed as \( \frac{1}{4}(x+1)^3(x-3) \), reflecting a more factored form. This form can help identify key features like roots or intercepts directly from the factorized equation.

The end behavior of a polynomial function is how the graph behaves as \( x \) approaches positive or negative infinity. This behavior is crucial for understanding the long-term trend of the graph. For quartic functions, the end behavior is largely influenced by the leading term, in this case, \( \frac{1}{4}x^4 \).

Since our quartic function has a positive leading coefficient \( \frac{1}{4} \) and even degree, the graph will rise on both sides. This means:

• As \( x \to -\infty \), \( P(x) \to \infty \).
• As \( x \to \infty \), \( P(x) \to \infty \).

This consistent upward trend as \( x \) moves away from zero in either direction is typical for servings of quartic functions with a positive coefficient for the term with the highest power.

X-intercepts are points where the graph of a function crosses the x-axis. These occur when the function's output is zero, or in mathematical terms, \( P(x) = 0 \). For our specific function \( P(x) = \frac{1}{4}(x+1)^3(x-3) \), the intercepts occur at points derived from the roots of the equation.

To find the x-intercepts:

• Solve \( (x+1)^3 = 0 \) which gives the intercept \( x = -1 \) with multiplicity 3. This indicates a point of inflection, where the graph merely touches and then bends away from the x-axis.
• Solve \( (x-3) = 0 \) which provides the intercept \( x = 3 \) with multiplicity 1. Here, the graph crosses the x-axis, signifying a change in the sign of the function values.

Each x-intercept tells us where the polynomial equals zero and how the graph interacts with the x-axis at those points.

The polynomial degree is a fundamental characteristic, quickly revealing the highest power of the variable \( x \) in the polynomial expression. It essentially determines the general shape and behavior of the graph. For the polynomial \( P(x) = \frac{1}{4}(x+1)^3(x-3) \), we calculate its degree by adding up the exponents associated with each factor:

• The degree of \((x+1)^3\) is 3.
• The degree of \((x-3)\) is 1.

Adding these together gives a total degree of 4, confirming as a quartic function. The degree tells us several important aspects about the polynomial, including:

• The number of x-intercepts it can have (up to 4 for a quartic).
• The potential for three turning points, as per the general rule of \( n-1 \), where \( n \) is the degree.

Understanding the degree helps in sketching the graph and predicting the shape and turning behavior of the polynomial.