How do you check if a graph ever touches the x-axis?

At the x-axis, y is always zero at any point. If you want to fnd out whether or not a graph reaches the x-axis, then substitute y=0 into the equation and find the value of x. If it solves, then we know it does touch the x-axis.

E.g. say we have an equation of y = 2x²-6x+4. Substituting y for 0 gives 2x²-6x+4=0. Factorising gives (2x-2)(x-2)=0, so x = 2 or 1. We know here that it touches the axis twice as we have two solutions.

Conversely, if we had y = x²+4, then substituting y for 0 gives x²+4=0 and therefore x = SquareRoot(-4). As we know, squareroots of negatives is not possible so x has no solutions, which means that this graph does not touch the x-axis.