Function could presented as a graph. 

Examples of some graph of functions:


Getting Information from the Graph of a Function

We can get domain and range form graph:

Also graphs could help us for solving Equations and Inequalities Graphically


Increasing and Decreasing Functions

Function is increasing on an interval if f(x2)>f(x1), where x2>x1

Function is decreasing on an interval if f(x2)<f(x1), where x2>x1

Finding Intervals on Which a Function Increases or Decreases

(a) Sketch a graph of the function f(x)=12x2+4x3-3x4

(b) Find the domain and range of f.

(c) Find the intervals on which f is increasing and on which f is decreasing.

(a)First, you could use calculator to do it:

We use a graphing calculator to sketch the graph.

(b) The domain of f is R because f is defined for all real numbers. Using the trace feature on the calculator "Trace", we find that the highest value for f(2) =32 and the range of f is (- ,32].

(c) From the graph we see that f is increasing on the intervals (- ,-10 and (0,2) and is decreasing on (-1,0) and (2,.)

Local Maximum and Minimum Values of a Function

Finding Local Maxima and Minima from a Graph

Need to find the local maximum and minimum values of the function f(x)=x3-8x + 1, rounded to three decimal places.

There appears to be one local maximum between x =-2 and x=-1, and one local minimum between x =1 and x = 2. 

Let’s find the coordinates of the local maximum point first. We zoom in to enlarge the area near this point, as shown in graph. Using the trace feature on the graphing device, we move the cursor along the curve and observe how the y-coordinates change. The local maximum value of y is 9.709, and this value occurs when x is -1.633, correct to three decimal places.

We locate the minimum value in a similar fashion. By zooming in to the viewing rectangle shown in Figure below, we find that the local minimum value is about -7.709, and this value occurs when x ≈ 1.633.

The maximum and minimum commands on a TI-83 or TI-84 calculator provide another method for finding extreme values of functions.


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