Quadratic equation - Parabolic Graph and Factorization

Graphical and Geometric Interpretations The Shape of Quadratic Functions: The Parabola When you graph any quadratic function of the form $f(x) = ax^2 + bx + c$, you always get the same distinctive shape: a parabola. This is a smooth, U-shaped curve that's one of the most important graphs you'll encounter in algebra. The first thing to understand is that the coefficient $a$ (the coefficient of $x^2$) completely determines which way the parabola opens. If $a > 0$, the parabola opens upward, like a cup that could hold water. If $a < 0$, it opens downward, like an arch or bridge. This matters because it tells you whether the function has a minimum value or a maximum value. When the parabola opens upward ($a > 0$), it has a minimum point—the lowest point on the graph where the function reaches its smallest value. When it opens downward ($a < 0$), it has a maximum point—the highest point where the function reaches its largest value. This special turning point is called the vertex. Finding the Vertex The vertex is crucial because it gives you the most important information about the quadratic function's behavior. Finding the vertex doesn't require guessing or graphing—there's a formula. The $x$-coordinate of the vertex is: $$x{\text{vertex}} = -\frac{b}{2a}$$ Once you've found this $x$-value, substitute it back into the original function $f(x)$ to find the corresponding $y$-coordinate. Together, these give you the complete vertex. Example: For $f(x) = x^2 - 2x - 2$, we have $a = 1$ and $b = -2$, so: $$x{\text{vertex}} = -\frac{-2}{2(1)} = 1$$ Substituting $x = 1$ into the function: $f(1) = (1)^2 - 2(1) - 2 = -3$ So the vertex is at $(1, -3)$, which is the minimum point of this parabola. X-Intercepts and Roots The $x$-intercepts of a graph are the points where the curve crosses the $x$-axis. These occur when $f(x) = 0$. Here's the key insight: the $x$-intercepts of the parabola are exactly the real roots of the quadratic equation $f(x) = 0$. This connection is fundamental. If you can find where the parabola crosses the $x$-axis, you've found the solutions to the quadratic equation. Conversely, if you solve the equation $ax^2 + bx + c = 0$, you're finding the $x$-intercepts of the parabola. A parabola can have: Two $x$-intercepts (two distinct real roots)—the parabola crosses the $x$-axis at two points One $x$-intercept (one repeated real root)—the parabola touches the $x$-axis at exactly one point (the vertex) No $x$-intercepts (no real roots)—the parabola doesn't touch the $x$-axis at all The Relationship Between Factors and Roots Here's a powerful algebraic fact: if $r$ is a root of the quadratic equation $f(x) = ax^2 + bx + c = 0$, then $(x - r)$ is a factor of the polynomial $ax^2 + bx + c$. This means you can write: $$ax^2 + bx + c = a(x - r1)(x - r2)$$ where $r1$ and $r2$ are the two roots. This factored form is incredibly useful because it shows the roots explicitly and makes it easy to see how the function behaves. Example: The quadratic $f(x) = x^2 - 3x + 2$ has roots $x = 1$ and $x = 2$ (you can verify this by setting it equal to zero). This means we can factor it as: $$x^2 - 3x + 2 = (x - 1)(x - 2)$$ The Special Case: Double Roots When the quadratic equation has exactly one solution (a double root), something special happens with the factorization. Instead of two different factors, the polynomial becomes a perfect square: $$ax^2 + bx + c = a(x - r)^2$$ where $r$ is the double root. This occurs when the discriminant $\Delta = b^2 - 4ac$ equals zero. Geometrically, this means the parabola touches the $x$-axis at exactly one point (the vertex), rather than crossing through it at two points. Example: The quadratic $f(x) = x^2 - 2x + 1$ can be factored as $(x - 1)^2$, meaning $x = 1$ is a double root. The parabola just touches the $x$-axis at $x = 1$ and nowhere else.