Single-variable calculus from first principles
Functions behave like vectors. You already know you can add two arrows and stretch an arrow by a number. You can do the very same two things to functions, and almost everything you know about vectors carries straight over.
To add two functions, you add them pointwise: at every input x, the new function's output is just the sum of the two outputs. To scale a function by a number c, you multiply every output by c. Those two operations are exactly what makes something a "vector space."
Think of two audio tracks playing at once: a bassline and a melody. To mix them you add the two waveforms moment by moment, exactly like adding functions pointwise. And turning one track's volume knob to 70% is just scaling that function by 0.7 at every instant. Mixing and volume are addition and scaling, the two moves that make functions behave like vectors.