key term - Quadratic

Definition

A quadratic is a polynomial function of the second degree, where the highest exponent of the variable is 2. Quadratics are characterized by their distinctive U-shaped or parabolic graphs and have important applications in various fields, including physics, engineering, and economics.

5 Must Know Facts For Your Next Test

1. Quadratic functions can be expressed in the standard form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers, and $a \neq 0$.
2. The graph of a quadratic function is a parabola, which can open upward or downward depending on the sign of the coefficient $a$.
3. The vertex of a parabola represents the minimum or maximum value of the quadratic function, and its coordinates can be found using the formula $x = -b/(2a)$.
4. The discriminant, $b^2 - 4ac$, determines the nature of the solutions to the quadratic equation: if it is positive, the equation has two real solutions; if it is zero, the equation has one real solution; if it is negative, the equation has two complex solutions.
5. Quadratic functions have many real-world applications, such as modeling the motion of projectiles, the cost of production, and the revenue generated by a business.

Review Questions

• Explain the relationship between the standard form of a quadratic function and the graph of the function.
  • The standard form of a quadratic function, $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$, directly relates to the characteristics of the graph of the function. The coefficient $a$ determines the orientation of the parabola, with $a > 0$ resulting in an upward-opening parabola and $a < 0$ resulting in a downward-opening parabola. The coefficient $b$ affects the horizontal shift of the parabola, and the constant term $c$ affects the vertical shift. Understanding this relationship between the standard form and the graph is crucial for analyzing and sketching quadratic functions.
• Describe how the vertex of a parabola can be used to determine the maximum or minimum value of a quadratic function.
  • The vertex of a parabola represents the point where the quadratic function changes from increasing to decreasing or vice versa, corresponding to the maximum or minimum value of the function. The coordinates of the vertex can be found using the formula $x = -b/(2a)$, where $a$ and $b$ are the coefficients in the standard form of the quadratic function, $f(x) = ax^2 + bx + c$. Knowing the location of the vertex allows you to determine the maximum or minimum value of the function, which is essential for optimizing various applications involving quadratic functions, such as projectile motion, cost-benefit analysis, or revenue maximization.
• Analyze how the discriminant of a quadratic equation can be used to determine the nature of the solutions to the equation.
  • The discriminant of a quadratic equation, $b^2 - 4ac$, where $a$, $b$, and $c$ are the coefficients in the standard form $ax^2 + bx + c = 0$, is a crucial factor in determining the nature of the solutions to the equation. If the discriminant is positive, the equation has two real, distinct solutions; if the discriminant is zero, the equation has one real, repeated solution; and if the discriminant is negative, the equation has two complex, conjugate solutions. Understanding the relationship between the discriminant and the solutions to a quadratic equation allows you to analyze the behavior of quadratic functions, such as the number and type of roots, which is essential for solving a wide range of problems involving quadratic relationships.

Related terms

Parabola:

The graph of a quadratic function, which has a distinctive U-shaped or curved appearance.

Vertex:

The point on a parabola where the graph changes from increasing to decreasing or vice versa, representing the minimum or maximum value of the function.

Discriminant:

A value calculated from the coefficients of a quadratic equation that determines the nature of the solutions, whether they are real, imaginary, or repeated.