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The quadratic formula is a crucial tool when dealing with polynomials, especially quadratics. It provides a simple way to find the roots of any quadratic equation in the standard form of \(ax^2 + bx + c = 0\). The formula itself is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula derives from completing the square on the generic quadratic equation.
- **a, b, c**: These are constants where \(aeq 0\). - **Discriminant**: The part under the square root symbol \(b^2 - 4ac\) is called the discriminant. It can tell us about the nature of the roots:
- If it's positive, there are two distinct real roots. - If it's zero, there is one real root (a repeated root). - If it's negative, the roots are complex numbers.
By using this formula, we found the roots of the polynomial \(p(x) = x^2 - 3x - 4\) to be \(x = 4\) and \(x = -1\). These roots correspond to the x-intercepts of the graph.

X-intercepts are points where the graph of a function crosses the x-axis. This means the y-value at these points is zero. For a quadratic function, the x-intercepts are also known as the roots or solutions of the equation.
To find these points for the function \(p(x) = x^2 - 3x - 4\), we solved the equation \(x^2 - 3x - 4 = 0\) using the quadratic formula, leading to two solutions: \(x = 4\) and \(x = -1\). These solutions tell us where the graph of the polynomial touches or crosses the x-axis, resulting in the intercepts at points \((4, 0)\) and \((-1, 0)\).
It's important to graph these points accurately as they help in shaping the curve of the parabola accurately.

A stationary point on a graph occurs where the derivative of a function equals zero. At this point, the graph has a horizontal tangent and represents a local maximum, minimum, or a saddle point. For the polynomial \(p(x) = x^2 - 3x - 4\), we found the derivative to be \(p'(x) = 2x - 3\).
Setting the derivative equal to zero, \(2x - 3 = 0\), we solved for \(x\) and found \(x = \frac{3}{2}\). Evaluating the polynomial at this point gives us \(p\left(\frac{3}{2}\right) = -\frac{25}{4}\), so the stationary point is \(\left(\frac{3}{2}, -\frac{25}{4}\right)\).
This point is essential when sketching the graph, as it provides information about the turning point of the curve, helping to define whether the point is a minimum or maximum. In this case, since the parabola opens upwards, \(\left(\frac{3}{2}, -\frac{25}{4}\right)\) is a local minimum.

While studying quadratic polynomials, it's important to note the absence of inflection points. Inflection points are where a curve changes its concavity, from concave up to concave down, or vice versa.
The presence of inflection points is determined by the second derivative of a function. If the second derivative changes its sign at a particular point, then that point is an inflection point. For the polynomial \(p(x) = x^2 - 3x - 4\), we find that the second derivative is \(p''(x) = 2\), a constant.
Since the second derivative does not change sign, there are no inflection points for this polynomial. This tells us that the graph of a quadratic function is entirely concave up or concave down, never changing in between, making it a smooth, symmetrical curve.