key term - Repeated Root

Definition

A repeated root is a root of a polynomial equation that occurs more than once. In the context of graphing quadratic functions, a repeated root represents a point where the graph of the function touches or intersects the x-axis more than once, indicating that the function has a local maximum or minimum at that point.

5 Must Know Facts For Your Next Test

1. A repeated root of a quadratic function occurs when the discriminant, $b^2 - 4ac$, is equal to 0.
2. When a quadratic function has a repeated root, the graph of the function will touch the $x$-axis at that point, forming a local maximum or minimum.
3. The multiplicity of a repeated root indicates the number of times the root occurs. For example, a root with multiplicity 2 is a repeated root.
4. Repeated roots can be used to determine the number of $x$-intercepts of a quadratic function, as well as the location and type of the vertex.
5. Understanding repeated roots is crucial for accurately graphing quadratic functions and analyzing their key features, such as the number of $x$-intercepts, the location of the vertex, and the behavior of the function near the repeated root.

Review Questions

• Explain how the presence of a repeated root affects the graph of a quadratic function.
  • When a quadratic function has a repeated root, the graph of the function will touch the $x$-axis at that point, forming a local maximum or minimum. This is because a repeated root indicates that the function has a factor of the form $(x - a)^2$, where $a$ is the repeated root. Geometrically, this means the graph will have a point where it changes from decreasing to increasing (or vice versa), which corresponds to the vertex of the parabola.
• Describe the relationship between the discriminant of a quadratic function and the presence of a repeated root.
  • The discriminant of a quadratic function, $b^2 - 4ac$, is a key factor in determining the presence of repeated roots. If the discriminant is equal to 0, then the quadratic function has a repeated root. This is because the discriminant represents the value of the expression under the square root in the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. When the discriminant is 0, the square root term becomes 0, and the two roots of the quadratic function are equal, resulting in a repeated root.
• Analyze how the multiplicity of a repeated root affects the behavior of the graph of a quadratic function near that point.
  • The multiplicity of a repeated root, which indicates the number of times the root occurs, directly impacts the behavior of the graph of a quadratic function near that point. If the multiplicity is 2, the graph will have a local maximum or minimum at the repeated root, and the graph will touch the $x$-axis at that point. If the multiplicity is higher, the graph will have a flatter appearance near the repeated root, indicating a higher-order point of tangency with the $x$-axis. Understanding the multiplicity of a repeated root is crucial for accurately sketching the graph of a quadratic function and predicting its behavior.

Related terms

Quadratic Function:

A polynomial function of degree two, written in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers.

Roots of a Quadratic Function: The values of $x$ for which the quadratic function $f(x) = 0$, also known as the $x$-intercepts of the graph.

Vertex of a Quadratic Function: The point on the graph of a quadratic function where the function changes from decreasing to increasing (or vice versa), representing the local maximum or minimum of the function.