Family Functions: Increasing and Decreasing End Behavior

We read graphs from left to right, just like a book!

Increasing or Decreasing? Both!

Notation We write where our graphs are Increasing or Decreasing in interval notation – similar to writing the Domain and Range. However, we do not use the brackets “[ ]” because at the point of change, it is neither increasing or decreasing. If we have more than one increasing/decreasing section of our graph, we combine the intervals with a “U”, which represents the union of the two intervals.

Increasing Start from the left and look at the sections of the graph that are increasing (y-values getting larger). {-∞, -3) U (1, 4)

Decreasing Start from the left and look at the sections of the graph that are decreasing (y-values getting smaller). (-3, 1)

Constant Start from the left and look at the sections of the graph that are flat – neither increasing or decreasing (y-values stay the same). (4, ∞}

(-5, -3) U (-1, 1) U (4, 5) Increasing: Decreasing: Constant: {-∞, -5) U (1, 4) (-3, -1) U (5, ∞}

End Behavior We can look at the ends of our graphs and determine what the rest of the graph will look like. We say, “As x  ∞, f(x)  ? x DOES NOT have to “go to” infinity. If the graph ends in point rather than arrows, then it is “as x goes to (the point)”

7 As x  9, f(x) _____ As x  -∞, f(x)  _____ 7 9 -∞