Quadratic function

Definition

A quadratic function is a type of polynomial function that can be expressed in the standard form $$f(x) = ax^2 + bx + c$$, where $$a$$, $$b$$, and $$c$$ are constants and $$a$$ is not equal to zero. This function forms a parabola when graphed, which can open either upwards or downwards depending on the value of $$a$$. Understanding the properties of quadratic functions helps in analyzing their behavior, such as finding their vertex, determining their maximum or minimum values, and exploring their increasing and decreasing intervals.

5 Must Know Facts For Your Next Test

1. The graph of a quadratic function is a parabola, which can either open upwards (if $$a > 0$$) or downwards (if $$a < 0$$).
2. The vertex form of a quadratic function is given by $$f(x) = a(x - h)^2 + k$$, where (h,k) is the vertex of the parabola.
3. The axis of symmetry for a quadratic function can be found using the formula $$x = -\frac{b}{2a}$$.
4. The roots or x-intercepts of a quadratic function can be determined using the quadratic formula: $$x = \frac{-b \pm \sqrt{D}}{2a}$$, where D is the discriminant.
5. Quadratic functions are continuous and smooth; they do not have any breaks or sharp corners in their graphs.

Review Questions

• How does the value of the coefficient 'a' affect the graph of a quadratic function?
  • The coefficient 'a' in the quadratic function affects both the direction and the width of the parabola. If 'a' is positive, the parabola opens upwards and has a minimum point at its vertex. Conversely, if 'a' is negative, the parabola opens downwards and has a maximum point at its vertex. Additionally, larger absolute values of 'a' result in a narrower parabola, while smaller absolute values make it wider.
• Discuss how to find the vertex of a quadratic function and why it is important.
  • To find the vertex of a quadratic function in standard form $$f(x) = ax^2 + bx + c$$, you can use the formula for the x-coordinate: $$x = -\frac{b}{2a}$$. Once you have this x-value, substitute it back into the original equation to find the y-coordinate. The vertex is important because it represents the highest or lowest point of the parabola, which helps identify whether the function has a maximum or minimum value, impacting optimization problems.
• Evaluate how understanding quadratic functions aids in solving real-world problems involving projectile motion.
  • Understanding quadratic functions is crucial when analyzing projectile motion since their trajectories can be modeled by quadratic equations. For instance, when an object is thrown into the air, its height as a function of time can often be represented as a quadratic function. By applying knowledge about vertices, roots, and intercepts, one can predict maximum heights, time to reach ground level, and distances traveled. This makes quadratic functions not only essential in mathematics but also highly applicable in physics and engineering.

Related terms

Vertex:

The vertex of a parabola is the highest or lowest point on the graph of a quadratic function, depending on whether it opens upward or downward.

Discriminant: The discriminant is part of the quadratic formula given by $$D = b^2 - 4ac$$, and it determines the nature of the roots of the quadratic equation.

Axis of symmetry: The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves and passes through the vertex.