What is a Quadratic Function?

A quadratic function is a type of polynomial function that can be represented by the equation:
$f(x) = ax^2 + bx + c$
where $a$, $b$, and $c$ are constants, and $a
eq 0$

Key Characteristics

Parabolic Shape

The graph of a quadratic function is a parabola. It can either open upwards (if $a > 0$) or downwards (if $a < 0$).

Vertex

The vertex is the highest or lowest point on the parabola, depending on whether it opens downwards or upwards. The vertex can be found using the formula:
$x = -frac{b}{2a}$

Axis of Symmetry

The parabola is symmetric about a vertical line called the axis of symmetry, which passes through the vertex. The equation for the axis of symmetry is:
$x = -frac{b}{2a}$

Roots or Zeros

These are the values of $x$ for which $f(x) = 0$. They can be found using the quadratic formula:
$x = frac{-b pm sqrt{b^2 – 4ac}}{2a}$

Example

Let’s consider the quadratic function $f(x) = 2x^2 – 4x + 1$

1. Vertex: Using $x = -frac{b}{2a}$, we get $x = frac{4}{4} = 1$. So, the vertex is at $(1, f(1)) = (1, -1)$
2. Axis of Symmetry: The axis of symmetry is $x = 1$
3. Roots: Using the quadratic formula, $x = frac{-(-4) pm sqrt{(-4)^2 – 4 cdot 2 cdot 1}}{2 cdot 2} = frac{4 pm sqrt{16 – 8}}{4} = frac{4 pm 2sqrt{2}}{4} = 1 pm frac{sqrt{2}}{2}$

Conclusion

Understanding quadratic functions is crucial because they appear in various real-world scenarios, such as projectile motion and economics. Recognizing their properties, such as the vertex, axis of symmetry, and roots, allows us to solve complex problems more easily.

Citations

1. 1. Khan Academy – Quadratic functions & equations
2. 2. Math is Fun – Quadratic Equation
3. 3. Purplemath – Quadratic Functions