The graph of a function consists of all points with coordinates (a,b), where b is the value of the response (or dependent) variable that corresponds to the value a of the explanatory (or independent) variable. In many cases the values of the independent variable can take on infinitely many values in a given interval, so the graph would be continuous, rather than isolated points.
In the graph of a function, since each value of the explanatory variable is paired with exactly one of the response variable, a vertical line drawn anywhere on the graph will intersect the graph is no more than one point. So the vertical line test helps determine whether a graph represents a function.
A function is increasing if the values of the response variable increase when the corresponding values of explanatory variable increase.
A function is decreasing if the values of the response variable decrease when the corresponding values of explanatory variable increase.
Indicate where on the graph the function is increasing and where it is decreasing.
The largest and smallest values of the response variable are the highest and lowest points on the graph of a function, and they are called the (absolute) maximum and (absolute) mininimum, respectively.
Indicate on the above graph of a function the (absolute) maximum and (absolute) mininimum.
The "turning points" of a graph, where the function changes from increasing to decreasing, or vice-versa, are of interest as well. The function has a relative maximum when it changes from increasing to decreasing, and a relative minimum when it changes from decreasing to increasing. Note that the right endpoint of the above graph would be an (absolute) minimum of the function, but not a relative minimum, since the function isn't changing from decreasing to increasing.
Indicate on the above graph of a function all the relative maximum and relative mininimum points.
The average rate of change of y per unit change in x is 
If the function is increasing, then 
and 
are both positive (when 
), and so the rate of change is positive. In this case a steeper graph would
mean the rate of change of the function is greater.
If the function is decreasing, then
is negative while is positive (when
), and so the rate of change is
negative.
The way a graph is curved, upward or downward, indicates whether the rate of change of the function is increasing or decreasing. When the curve is "bent upward" the function is concave up, and when it is curved downward it is concave down.
(a) 
(b) 
(c) 
(d) 
For each of the four graphs above, indicate whether the function is increasing or decreasing and whether it is concave-up or concave-down.