What is the best way to solve this problem within ~1 minute? (given SAT time limit)
(April 2019 QAS Section 4 #24)
In the xy-plane, the graph of a linear equation of the form y=mx+b and the graph of an exponential equation of the form y=ab^x both contain points (1,3) and (2,4). If the point (r,s) is on the graph of the linear equation and the point (r,t) is on the graph of the exponential equation, where 0<r<4 and s>t, which of the following must be true?
To get this REALLY fast, you have to know the general shapes of both graphs. The fact that the line has (1, 3) and (2, 4) means it has a positive slope, and the fact that the exponential function also contains those points means its increasing.
Shape-wise, that means the exponential function will be above the line before and after those two intersections, and below it only between the intersections. By asking about when s > t, the question is asking when the line will be above the exponential function, and that’s only between the intersections, when x is between 1 and 2.
Does that help?