Reading a Sign Chart

A sign chart shows at a glance where a product is positive or negative. Mark its roots on a number line: for $\left(x-1\right)\left(x+3\right)$ they are -3 and 1. They cut the line into three intervals, and inside each one the sign never changes, because a factor can only change sign by passing through 0.

So test one number in each interval. Left of -3, try -4: the factors are -5 and -1, and the product is 5, positive. Between the roots, try 0: the factors are -1 and 3, and the product is -3, negative. Right of 1, try 2: the factors are 1 and 5, and the product is 5, positive. The chart reads plus, minus, plus.

The parabola $y = {x}^{2}+2x-3$, which is the same product multiplied out, agrees. It opens up, so it dips below the x-axis between its roots and rises above it outside them. Opening up always means negative between the roots; opening down, with a minus in front, means positive between them.