Quadratic function

Definition

A quadratic function is a polynomial function of degree two, typically expressed in the standard form $$f(x) = ax^2 + bx + c$$, where $$a$$, $$b$$, and $$c$$ are constants and $$a \neq 0$$. The graph of a quadratic function is a parabola, which opens either upward or downward depending on the sign of the coefficient $$a$$. Understanding the characteristics of quadratic functions is essential for graphing them accurately and analyzing their properties.

5 Must Know Facts For Your Next Test

1. The general form of a quadratic function is $$f(x) = ax^2 + bx + c$$, with $$a$$ affecting the width and direction of the parabola.
2. If $$a > 0$$, the parabola opens upwards, while if $$a < 0$$, it opens downwards.
3. The vertex of the parabola can be found using the formula $$x = -\frac{b}{2a}$$ to determine the axis of symmetry.
4. Quadratic functions can have zero, one, or two real roots, which correspond to the x-intercepts of the graph.
5. The y-intercept of a quadratic function occurs when $$x = 0$$ and can be directly found as $$c$$ in the standard form.

Review Questions

• How does changing the coefficient 'a' in a quadratic function affect its graph?
  • Changing the coefficient 'a' in a quadratic function significantly affects the shape and orientation of its graph. If 'a' is positive, the parabola opens upward, creating a U-shape. If 'a' is negative, it opens downward, resulting in an inverted U-shape. Additionally, a larger absolute value of 'a' makes the parabola narrower, while a smaller absolute value makes it wider.
• What is the process for finding the vertex of a quadratic function given in standard form, and why is this important?
  • To find the vertex of a quadratic function in standard form $$f(x) = ax^2 + bx + c$$, use the formula $$x = -\frac{b}{2a}$$ to locate the x-coordinate. Then substitute this x-value back into the function to find the corresponding y-coordinate. The vertex is crucial because it represents either the maximum or minimum value of the function, which helps in graphing and understanding its behavior.
• Evaluate how understanding quadratic functions can aid in solving real-world problems involving projectile motion.
  • Understanding quadratic functions is vital in solving real-world problems such as projectile motion because they accurately model trajectories. When an object is thrown or projected, its height over time can be described by a quadratic function. By analyzing this function's properties—like finding its vertex for maximum height or its roots for when it hits the ground—one can predict and understand the object's motion effectively. This application demonstrates how mathematics can be used to model and solve practical challenges.

Related terms

Parabola: A symmetric curve that represents the graph of a quadratic function, characterized by its vertex, axis of symmetry, and direction of opening.

Vertex:

The highest or lowest point on the graph of a quadratic function, where the parabola changes direction.

Axis of Symmetry: A vertical line that divides the parabola into two mirror-image halves, passing through the vertex.