Here is a graph of \(f(x)=x^2\):
Let there be a straight line that goes through the parabola at two points. One at \(x=a\) and \(x=b\), where \(a\) is a negative number and \(b\) is positive:
Let's think about where this line crosses the y-intercept (in terms of \(a\) and \(b\)). Since this is a straight line, let's define it like this:
The line \(g(x)\) crosses the parabola at two points: \((a, f(a))\) and \((b, f(b))\). This means we can express the gradient as:
Since \(f(x)=x^2\):
This means:
Since \(f(x)\) and \(g(x)\) intersect at \(x=a\), then \(g(a)=f(a)\):
This means if you draw any straight line goes through the parabola at \(x=a\) and \(x=b\), then we can use the y-intercept to find the product \(-ab\). Remeber, \(a\) is negative, so \(-ab\) would be positive.
