91Ó°ÊÓ

A quadratic function is a polynomial function of degree two. It typically takes the form \(y = ax^2 + bx + c\), where \(a, b,\) and \(c\) are constants and \(a eq 0\). Quadratic functions graph as parabolas, which are U-shaped curves that can open upwards or downwards depending on the sign of \(a\). If \(a > 0\), the parabola opens upwards; if \(a < 0\), it opens downwards.

The x-intercepts of a quadratic function are the points where the graph crosses the x-axis, meaning the value of \(y\) at these points is zero. To find the x-intercepts, you set the quadratic function equal to zero and solve for \(x\).

In the given example, the function is \(y = 4x - 4x^2 - 1\). This can be rearranged into standard form: \(-4x^2 + 4x - 1\). To find the x-intercepts, we need to solve \(-4x^2 + 4x - 1 = 0\).

The quadratic formula is a powerful tool to find the solutions (or roots) of any quadratic equation. The formula is derived from the process of completing the square and provides the solutions to \(ax^2 + bx + c = 0\). The quadratic formula is given by: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Using the quadratic formula involves:

• Identifying the coefficients \(a, b,\) and \(c\) from the quadratic equation.
• Calculating the discriminant \(b^2 - 4ac\).
• Substituting the values of \(a\), \(b\), and the discriminant into the formula to find the values of \(x\).

For the equation \(-4x^2 + 4x - 1 = 0\), we identified \(a = -4\), \(b = 4\), and \(c = -1\). Substituting these into the quadratic formula, we find the x-intercepts as \(x = \frac{-4 \pm \sqrt{0}}{2(-4)}\), which simplifies to \(x = \frac{1}{2}\).

The discriminant is a key part of the quadratic formula found under the square root symbol, \(b^2 - 4ac\). It helps determine the nature and number of roots of the quadratic equation. The value of the discriminant can be:

• Positive: If the discriminant is greater than 0, the quadratic equation has two distinct real roots.
• Zero: If the discriminant is exactly 0, the quadratic equation has one real root (also called a repeated or double root).
• Negative: If the discriminant is less than 0, the quadratic equation has no real roots but two complex roots.

In our example, the discriminant is calculated as \(4^2 - 4(-4)(-1) = 16 - 16 = 0\). Since the discriminant is 0, this means our quadratic equation \(-4x^2 + 4x - 1 = 0\) has exactly one real root, which is \(x = \frac{1}{2}\). This single root indicates that the parabola touches the x-axis at one point only, making this point the sole x-intercept.