AI Tutor
Every time a basketball launches into the air, its flight path traces a symmetrical curve called a parabola.
You do not need to plot fifty coordinate pairs to draw this curve accurately; you only need to pinpoint four essential features.
In the general quadratic form (or standard polynomial form), a quadratic function is written as y=ax2+bx+cy = ax^2 + bx + c, where aa, bb, and cc are constants with a≠0a \neq 0.
The sign of the leading coefficient aa dictates global concavity: if a>0a > 0, the parabola opens upwards with a minimum turning point; if a<0a < 0, it opens downwards with a maximum turning point.
The easiest feature to find is the y-intercept, the point where the curve crosses the vertical y-axis by setting x=0x = 0, giving y=cy = c.
Where does the curve cross the horizontal floor where height is zero?

Finding the X-Intercepts

The x-intercepts (or roots) are the locations where the curve hits the x-axis, which means setting y=0y = 0 and solving ax2+bx+c=0ax^2 + bx + c = 0.
When a quadratic factorizes into (xβˆ’p)(xβˆ’q)=0(x - p)(x - q) = 0, the zero-product principle states that at least one factor must equal zero, giving roots at x=px = p and x=qx = q.
If it does not factor easily, the quadratic formula x=βˆ’bΒ±b2βˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} finds the exact crossing points every time.
Performing a discriminant analysis with Ξ”=b2βˆ’4ac\Delta = b^2 - 4ac reveals how the curve interacts geometrically with the x-axis.
If Ξ”>0\Delta > 0, there are two distinct real x-intercepts; if Ξ”=0\Delta = 0, the turning point is tangent to the x-axis (one repeated root); if Ξ”<0\Delta < 0, there are no real roots and the curve lies strictly above or below the axis.