upgrade

📘intermediate algebra review

Y = ax² + bx + c

Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025

Definition

The equation y = ax² + bx + c is the general form of a quadratic function, where 'a' represents the coefficient of the squared term, 'b' represents the coefficient of the linear term, and 'c' represents the constant term. This equation is fundamental in describing the behavior and properties of parabolic curves.

5 Must Know Facts For Your Next Test

  1. The coefficient 'a' in the equation y = ax² + bx + c determines the orientation of the parabola, with a positive 'a' resulting in an upward-facing parabola and a negative 'a' resulting in a downward-facing parabola.
  2. The coefficient 'b' in the equation y = ax² + bx + c determines the horizontal shift of the parabola, with a positive 'b' shifting the parabola to the left and a negative 'b' shifting the parabola to the right.
  3. The constant term 'c' in the equation y = ax² + bx + c determines the vertical shift of the parabola, with a positive 'c' shifting the parabola upward and a negative 'c' shifting the parabola downward.
  4. The discriminant, $b^2 - 4ac$, determines the number and nature of the real roots of the quadratic equation. If the discriminant is positive, the equation has two real roots; if the discriminant is zero, the equation has one real root; if the discriminant is negative, the equation has no real roots.
  5. Quadratic functions represented by the equation y = ax² + bx + c can be used to model a variety of real-world phenomena, such as the path of a projectile, the growth of a population, and the supply and demand curves in economics.

Review Questions

  • Explain how the coefficients 'a', 'b', and 'c' in the equation y = ax² + bx + c affect the shape and position of the parabola.
    • The coefficient 'a' determines the orientation of the parabola, with a positive 'a' resulting in an upward-facing parabola and a negative 'a' resulting in a downward-facing parabola. The coefficient 'b' determines the horizontal shift of the parabola, with a positive 'b' shifting the parabola to the left and a negative 'b' shifting the parabola to the right. The constant term 'c' determines the vertical shift of the parabola, with a positive 'c' shifting the parabola upward and a negative 'c' shifting the parabola downward. Together, these coefficients define the shape, orientation, and position of the parabolic curve represented by the equation y = ax² + bx + c.
  • Describe how the discriminant, $b^2 - 4ac$, can be used to determine the number and nature of the real roots of the quadratic equation.
    • The discriminant, $b^2 - 4ac$, is a key feature of the quadratic equation y = ax² + bx + c that determines the number and nature of the real roots. If the discriminant is positive, the equation has two real roots; if the discriminant is zero, the equation has one real root; if the discriminant is negative, the equation has no real roots. This information is crucial for understanding the behavior and properties of the parabolic curve represented by the equation y = ax² + bx + c, as the real roots correspond to the points where the curve intersects the x-axis.
  • Identify and explain potential real-world applications of the quadratic function represented by the equation y = ax² + bx + c.
    • The quadratic function represented by the equation y = ax² + bx + c has a wide range of real-world applications. It can be used to model the path of a projectile, such as a ball or a rocket, where the height of the object is represented by the y-coordinate and the horizontal distance is represented by the x-coordinate. It can also be used to model population growth, where the population size is represented by the y-coordinate and time is represented by the x-coordinate. Additionally, the equation y = ax² + bx + c can be used to model supply and demand curves in economics, where the price is represented by the y-coordinate and the quantity is represented by the x-coordinate. These diverse applications demonstrate the versatility and importance of understanding the properties and behavior of the quadratic function.