Reading a Graph as Faster or Slower Change (F 601)

A straight line climbs at a constant rate, so its steepness never changes. A curve that bends upward gets steeper as you move right, which means the rate itself is increasing. That is the difference between 'getting bigger' and 'getting bigger faster.' Picture a car's distance-time graph: a straight line means constant speed, but a curve bending upward means the car is accelerating, covering more distance in each successive second.

Steepness at one single point is the slope of the line that just grazes the curve there, called the tangent line. You cannot get that from two far-apart points, because that only gives an average rate over a stretch. To compare two curves at the same instant, look at which one is tilted more steeply at that exact x-value, not which one is higher up. Height tells you the total accumulated so far; steepness tells you how fast it is changing right now. A curve can sit below another yet be climbing faster, and it will eventually overtake.

This is the graph-reading half of the functions strand: the other half builds the actual rules, like linear y = mx + b for constant rate or exponential y = a·b^x for rates that scale with size. Reading steepness is how you check a rule's story against its picture, and it sets up comparing average rate of change over an interval, which is just total change divided by the width of the interval.