Quartic

Definition

A quartic is a polynomial function of degree four, meaning it has the general form $f(x) = ax^4 + bx^3 + cx^2 + dx + e$, where $a$, $b$, $c$, $d$, and $e$ are real numbers. Quartic functions are important in the study of polynomial functions and their graphical representations.

5 Must Know Facts For Your Next Test

1. The graph of a quartic function can have up to four real zeros, which correspond to the $x$-intercepts of the graph.
2. Quartic functions can have a maximum of three critical points, which are the points where the derivative of the function is equal to zero.
3. The behavior of a quartic function, such as the number and location of local extrema and inflection points, can be determined by analyzing the signs of the first and second derivatives.
4. Quartic functions can exhibit a variety of shapes, including parabolas, S-curves, and functions with multiple local extrema.
5. The factorization of a quartic function can provide insights into the location and multiplicity of its real zeros.

Review Questions

• Explain the general form of a quartic function and how the coefficients affect the shape of the graph.
  • A quartic function has the general form $f(x) = ax^4 + bx^3 + cx^2 + dx + e$, where $a$, $b$, $c$, $d$, and $e$ are real numbers. The coefficient $a$ determines the overall concavity of the graph, with $a > 0$ resulting in a U-shaped curve and $a < 0$ resulting in an inverted U-shaped curve. The coefficients $b$, $c$, $d$, and $e$ affect the location and number of critical points, local extrema, and $x$-intercepts of the graph, leading to a variety of possible shapes for the quartic function.
• Describe the relationship between the critical points of a quartic function and the behavior of its graph.
  • The critical points of a quartic function, where the derivative is equal to zero, are crucial in determining the function's behavior. A quartic function can have up to three critical points, which correspond to the values of $x$ where the graph may change concavity or have local extrema. Analyzing the signs of the first and second derivatives at these critical points can provide information about the number and location of local maxima, local minima, and inflection points on the graph of the quartic function.
• Explain how the factorization of a quartic function can provide insights into the location and multiplicity of its real zeros.
  • The factorization of a quartic function, $f(x) = a(x - x_1)(x - x_2)(x - x_3)(x - x_4)$, where $x_1$, $x_2$, $x_3$, and $x_4$ are the real zeros of the function, can reveal important information about the graph. The number of distinct real zeros and their multiplicities (single or repeated) can be determined from the factored form. This, in turn, helps to understand the shape of the graph, the location of $x$-intercepts, and the behavior of the function near its zeros.

Related terms

Polynomial Function: A polynomial function is an expression that combines variables and constants using only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

Degree of a Polynomial: The degree of a polynomial is the highest exponent of the variable in the polynomial.

Critical Points: Critical points of a function are the values of the independent variable where the derivative of the function is zero or undefined.