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 $f(x) = 2x^2 – 4x + 1$
- Vertex: Using $x = -frac{b}{2a}$, we get $x = frac{4}{4} = 1$. So, the vertex is at $(1, f(1)) = (1, -1)$
- Axis of Symmetry: The axis of symmetry is $x = 1$
- 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.