Presentation on theme: "2.1 Describing Graphs of Functions. If we examine a typical graph the function y = f(x), we can observe that for an interval throughout which the function."— Presentation transcript:
If we examine a typical graph the function y = f(x), we can observe that for an interval throughout which the function is defined, that the function might be increasing, decreasing or neither.
We say that a function is increasing on an interval if x1 and x2 are in the interval such that x1 < x2 and we have f(x1) < f(x2). Further, we say that f(x) is increasing at x = c provided that f(x) is increasing in some open interval on the x-axis that contains c.
We say that a function is decreasing on an interval if x1 and x2 are in the interval such that x1 f(x2). Further, we say that f(x) is decreasing at x = c provided that f(x) is decreasing in some open interval on the x-axis that contains c.
We note that the slope of graph I is increasing while the slope of graph II is decreasing. In application, we would say that the debt per capita depicted in graph I is rising at an increasing rate. From graph II, we observe that the population is increasing at a declining rate.
An inflection point is a point on the graph of a function at which the function is continuous and the concavity of the graph changes, i.e., goes from concave up to concave down, or concave down to concave up.
Use the terms defined earlier to describe the graph.
For x < 3, f(x) is increasing and concave down. Relative maximum at the point x = 3. For 3 < x < 4, f(x) is decreasing and concave down. Inflection point at x = 4. For 4 < x < 5, f(x is decreasing and concave up. Relative minimum at x = 5. For x > 5, f(x) is increasing and concave up.
We have previously discussed the idea of intercepts. Recall that The x-intercept is a point at which a graph intersects the x-axis. (x,0) The y-intercept is a point at which the graph intersects the y-axis. (0,y)
Graphs sometimes straighten out and approach some straight line as x increases (or decreases). Theses straight lines are called asymptotes. Asymptotes of a graph may be horizontal, vertical or diagonal.
The horizontal asymptotes of a graph may be determined by calculating the limits and If either limit exists, then the value of the limit determines a horizontal asymptote.
We often expect the graph of a function f(x), at a value x that would result in division by zero, to have a vertical asymptote.
We now have six categories for describing the graph of a function 1.Intervals in which the finction is increasing or decreasing, relative maximum/minimum points 2.Maximum/minimum values 3.Intervals in which a function is concave up or concave down, inflection points. 4.X-intercepts, y-intercepts 5.Undefined points. 6.Asymptotes