How could you sketch a graph for y=x^2-10x+21?

This is a quadratic equation. We can recall that this means the graph will have a parabolic shape. Next we need to do a little bit of manipulation to get our final bits of information. So we want to know where the graph will cross the x axis, to do this we need to know what the values of x will be when y is 0. This is because at every point on the x axis, y is always 0. So we need to solve 0=x^2-10x+21. We can do this by factorising 0=(x-7)(x-3) with x=7 and x=3 7 and 3 multiply to give 21, if they're both negative we still get positive 21 but they also add together to give -10, which is what this expression requires. So the graph crosses the x axis at positive 7 and positive 3. Finally, the coefficient of x^2 is positive, so the graph's shape will be like a smiley face, not a sad face. With this, we have all the information we need to sketch the graph.