Is it possible to combine a linear equation with a sine curve equation
to produce a single equation that describes an oscillating line that
is increasing over time? This is essentially the 'Keeling Curve' --
the typical sine curve graph that runs a along a line that is angled
at about 45 degrees from horizontal. I have both equations but don't
know how to go about 'combining' them into a single one. Basically,
the limit of my high school math (calculus) of 30 years ago has been
reached but I have a 16-year old facing this daunting question in a
juniot pre-calc course. Thanks.

Add them together.

The Keeling curve in particular demonstrates the superposition of a
periodic (annual) cycle and a "secular" (long term) trend.

Equation of a line:

  y = mx + b

Equation of a sinusoidal curve:

  y = r sin(ax + c)

By "superposition" is meant what racecar-ga points out, the addition
of the two functions:

  y = mx + b + r sin(ax + c)

or in the simplest case perhaps y = x + sin(x).

Given the audience of a junior pre-calc class, this is most likely the
intended interpretation.  A different approach might be to take the
sine curve and rotate it about the origin by 45 degrees.  This might
better fit the description of a "typical sine curve graph that runs a
along a line that is angled at about 45 degrees from horizontal."  It
would be the graph of a function, but not the same function as y = x +
sin(x).

regards, mathtalk-ga