Showing posts with label algebra. Show all posts
Showing posts with label algebra. Show all posts

Friday, July 26, 2013

Graphing Logarithmic Functions

One way to graph logarithmic functions is to first graph its inverse exponential. Then use your knowledge about the symmetry of inverses to graph the logarithm. Recall that inverses are symmetric about the line y = x. For example,
Step 1: Find some points on the exponential f(x). The more points we plot the better the graph will look.
 
Step 2: Switch the x and y values to obtain points on the inverse.
 
Step 3: Determine the asymptote.
 
In practice, we use a combination of techniques to graph logarithms.  We can use our knowledge of transformations, techniques for finding intercepts, and symmetry to find as many points as we can to make these graphs. General guidelines follow:

1. Graph the vertical asymptote. All logarithmic functions of the form  
have a vertical asymptote at x = h.

2. Find the x- and y-intercepts if they exist. To find x-intercepts set y = f(x) to zero and to find y-intercepts set x = 0.

3. Plot a few more points and graph it.

Graph the following logarithmic functions. State the domain and range.
 
 
In the previous solved problem, make a note of the rigid transformations.  If we start with the basic graph y = log(x) then the first translation is a shift to the left 3 units y = log(x+3).  Next we see a vertical shift up 2 units y = log(x+3)+2 .
 
 
 
 
In the above problem there was a reflection about the x-axis as well as a shift to the left 3 units.
 
 
 
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Tuesday, July 23, 2013

Relations, Graphs, and Functions

In this section we will study functions in more depth and begin by defining a relation as any set of ordered pairs.
An example of a relation might look like { (2,3), (2,5), (0,1) }. Here we have a relation consisting of three ordered pairs or points on the Cartesian coordinate system. We are familiar with ordered pairs and usually see them denoted as (x, y).  Typically the x-value (the first component) will be the independent variable or input and the y-value (the second component) is the dependent variable or output.
The example { (2,3), (2,5), (0,1) } is NOT a function because the x-value 2 is assigned more than one y-value, namely 3 and 5. For every x-value there can be only one y-value. Next we define the domain as the set of x-values and the range as the set of y-values for which the relation is defined.
Example:
Tip: When looking at a list of ordered pairs, if there are repeating x-values then the relation is not a function.  This usually indicates that there is an input with multiple outputs. (This does not apply to the y-values)

Determine if the relations are functions.  If so state the domain and range.
 
 
Relations can consist of an infinite number of ordered pairs, in which case, making a big list would be impossible. A graph can represent a relation by considering it as a big list of ordered pairs.  Each dot on the graph represents an ordered pair (x, y).
 
Notice that if we can draw a vertical line that intersects the graph twice we will be able to identify one x-value with two corresponding y-values.  Therefore, it can not be a function.
  
Alternatively, if any vertical line crosses the graph only once then it does represent a function.

Use the vertical line test to determine whether or not the graph represents a function.
 
Be prepared to state the domain and range given the graph.  Remember to think of the graph as an infinite set of ordered pairs (x, y).  From the graph, determine the x-values and y-values. First shade in the domain and range then convert to interval notation.
 
It should be clear, at this point, that the circle shown is not a function.  Nonetheless, we still can determine the domain and range of the relation that it represents.


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