key term - Quartic function

Definition

A quartic function is a polynomial function of degree four, expressed in the form $$f(x) = ax^4 + bx^3 + cx^2 + dx + e$$ where 'a', 'b', 'c', 'd', and 'e' are constants and 'a' is not equal to zero. Quartic functions can have up to four real roots and exhibit a variety of shapes in their graphs, including up to three turning points. They play a significant role in understanding polynomial behavior and the characteristics of their graphs.

5 Must Know Facts For Your Next Test

1. The general form of a quartic function is $$f(x) = ax^4 + bx^3 + cx^2 + dx + e$$ with 'a' being non-zero, meaning it defines its highest degree term.
2. Quartic functions can have either 0, 2, or 4 real roots based on the discriminant and their specific coefficients.
3. The graph of a quartic function can have up to three turning points, which are critical for analyzing its shape and behavior.
4. The end behavior of quartic functions is determined by the leading coefficient: if 'a' is positive, the graph rises on both ends; if negative, it falls on both ends.
5. Quartic functions can exhibit various symmetry properties, such as being even or odd functions depending on their terms.

Review Questions

• How do you determine the number of real roots a quartic function may have, and what role does the discriminant play in this process?
  • The number of real roots for a quartic function can be determined using the discriminant, which evaluates the relationships between its coefficients. By analyzing the quartic's factors and applying Descartes' Rule of Signs alongside the discriminant, we can predict whether there are 0, 2, or 4 real roots. The nature of these roots is also influenced by how the graph behaves at turning points and intercepts with the x-axis.
• Discuss how to find the turning points of a quartic function and why they are important for graphing.
  • To find the turning points of a quartic function, you start by calculating its first derivative and setting it equal to zero to identify critical points. Once you have these points, you can use the second derivative test to determine whether each point is a maximum or minimum. Turning points are essential for graphing as they help define the shape of the curve, allowing you to visualize where the function increases or decreases.
• Evaluate how changing the leading coefficient in a quartic function affects its graph, particularly regarding end behavior and symmetry.
  • Changing the leading coefficient in a quartic function directly impacts its end behavior and overall shape. If the leading coefficient is positive, the graph will rise on both ends; if negative, it will fall on both ends. Additionally, quartic functions with even powers generally exhibit symmetry about the y-axis if only even-degree terms are present. This understanding helps in predicting how changes in coefficients will transform the graph's characteristics.

Related terms

polynomial function:

A mathematical expression consisting of variables raised to whole number exponents and their coefficients, which includes functions of degree zero or higher.

roots: The values of 'x' for which a polynomial function equals zero, indicating the points where the graph intersects the x-axis.

turning points: Points on the graph of a function where it changes direction from increasing to decreasing or vice versa, which can be determined by the function's derivatives.