Throughout my 20+ years of teaching mathematics, I have somewhat dreaded this question, but have always made sure I had a good answer: be it the trig functions modelling tides and seasons, or quadratics to represent projectile motion. If there was no direct application, I would pull the “exercise for the brain, just like going to the gym and doing leg extensions” card. You never need to sit down with a heavy weight on your ankles then lift your calves in “real life”, but nobody ever questions the “need” for that exercise.
More recently, however, thanks mostly to the IB MYP mathematics program, I have not been at all concerned about this question coming up, nor have I noticed it coming up. This is for two main reasons, as far as I can see.
The MYP program mandates that content has to be taught within a “context” so there has to automatically be a “reason” for learning the mathematics. The context might be from daily life, such as budgeting, or interest and percentages etc. It might also be within a context of human development, such as gaining an appreciation for how logarithms were used to simplify complex calculations before calculators were commonplace. Also, as a result, the mathematics tends to be enjoyable and engaging so it is appreciated and the big question does not come up.
Assessment in the MYP mathematics program has four criteria for evaluating student work. One of the criteria for much of the assignment work is Criterion D: Reflection. This criterion requires students to consider the accuracy of their results and whether their results make sense. But it also demands students to consider the significance of their results beyond the classroom, in society, in life. So, rather than the students asking us about the usefulness of a topic in mathematics, we are asking them.