Functions - Quadratic Functions

General Form $f(x) = ax^2 + bx + c$: The graph of a quadratic function is a symmetrical curve called a parabola. If the coefficient $a$ is positive ($a > 0$), the parabola opens upwards like a 'U' shape, creating a minimum point. If $a$ is negative ($a < 0$), the parabola opens downwards like an inverted 'U', creating a maximum point. The constant $c$ is the $y$-intercept, representing the point $(0, c)$ where the curve crosses the vertical axis.

Vertex Form $f(x) = a(x - h)^2 + k$: This form is particularly useful for identifying the vertex $(h, k)$, which is the turning point of the parabola. Visually, the graph is a transformation of the parent function $y = x^2$, shifted horizontally by $h$ units and vertically by $k$ units. The vertical line $x = h$ acts as the axis of symmetry, splitting the parabola into two identical mirror images.

Factored (Intercept) Form $f(x) = a(x - p)(x - q)$: This form highlights the $x$-intercepts, also known as the roots or zeros of the function, which occur at $(p, 0)$ and $(q, 0)$. On a graph, these are the points where the curve crosses the horizontal $x$-axis. The axis of symmetry is always located exactly midway between these two points at $x = \frac{p + q}{2}$.

The Discriminant $\Delta = b^2 - 4ac$: The discriminant determines the nature of the roots and how the graph interacts with the $x$-axis. If $\Delta > 0$, the parabola crosses the $x$-axis at two distinct points. If $\Delta = 0$, the parabola's vertex touches the $x$-axis at exactly one point (a repeated root). If $\Delta < 0$, the entire parabola floats above or below the $x$-axis and never touches it, meaning there are no real roots.

Axis of Symmetry and Turning Point: Every parabola has a vertical line of symmetry with the equation $x = -\frac{b}{2a}$. The vertex always lies on this line. To find the $y$-coordinate of the vertex (the maximum or minimum value), substitute $x = -\frac{b}{2a}$ back into the original function. Visually, this turning point is the peak of a hill ($a < 0$) or the bottom of a valley ($a > 0$).

Vertical Dilations and Reflections: The value of $a$ controls the 'steepness' of the curve. As $|a|$ increases ($|a| > 1$), the parabola undergoes a vertical stretch, making it look thinner or narrower. When $0 < |a| < 1$, the parabola undergoes a vertical compression, making it appear wider or flatter. A negative sign in front of $a$ reflects the graph across the $x$-axis.