U32 Quadratic Functions and Graphs: Essential Formulas

General Form

The standard representation of a quadratic function is $y = ax^2 + bx + c$, where $a \neq 0$. The coefficient $a$ determines the direction of opening: if $a > 0$, the parabola opens upwards; if $a < 0$, it opens downwards. The constant $c$ is the $y$-intercept, the point where the graph crosses the $y$-axis.

Vertex Form

By completing the square, the general form can be rewritten into the vertex form: $y = a(x - h)^2 + k$. Here, $(h, k)$ are the coordinates of the vertex, the maximum or minimum point of the parabola. The axis of symmetry is the vertical line $x = h$.

The process of completing the square for $y = ax^2 + bx + c$ is: $$ y = a\left(x^2 + \frac{b}{a}x\right) + c = a\left[\left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right] + c $$ This leads to the vertex coordinates: $h = -\frac{b}{2a}$ and $k = c - \frac{b^2}{4a}$.

The Discriminant

For the quadratic equation $ax^2 + bx + c = 0$, the discriminant $\Delta$ is defined as $\Delta = b^2 - 4ac$. It determines the number and type of $x$-intercepts (roots) of the quadratic function $y = ax^2 + bx + c$.

• If $\Delta > 0$: Two distinct real roots. The parabola cuts the $x$-axis at two points.
• If $\Delta = 0$: One repeated real root (a double root). The parabola touches the $x$-axis at the vertex.
• If $\Delta < 0$: No real roots. The parabola does not intersect the $x$-axis.

Vertex Coordinates Formula

The vertex $(h, k)$ of the parabola $y = ax^2 + bx + c$ can be found directly using formulas derived from completing the square. The $x$-coordinate of the vertex is also the equation of the axis of symmetry.

The axis of symmetry is the vertical line: $$ x = -\frac{b}{2a} $$

Roots of a Quadratic Equation

The roots (or $x$-intercepts) of the quadratic function $y = ax^2 + bx + c$ are the solutions to $ax^2 + bx + c = 0$. They can be found using the quadratic formula, which involves the discriminant.