Showing posts with label asymptote. Show all posts
Showing posts with label asymptote. 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.
 
 
 
YouTube Videos:


---

Friday, May 10, 2013

Interactive: Hyperbolas


Interactive Instructions: Move any of the green points. Also, you can change the orientation below.

Hyperbola: Horizontal Vertical


[ NOTESHyperbolas ]


Tuesday, May 7, 2013

Hyperbolas

An hyperbola is the set of all points whose distances to two fixed points subtract to the same constant. In this section, we will focus on the equation of hyperbolas.
   
The axis that contains the vertices is called the transverse axis and the axis that does not contain the vertices is called the conjugate axis. Also notice that the shape is very different than that of a parabola, hyperbolas are asymptotic to the following lines:
We will make use of these asymptotes to sketch the graphs of hyperbolas.
[ Interactive: Hyperbolas ]
To easily graph the asymptotes of a hyperbola use the following process.

Step 1: Identify the center (h, k) from standard form.
Step 2: Plot points a units left and right from center.
Step 3: Plot points b units up and down from center.
Step 4: Draw the rectangle defined by these points.
Step 5: The lines though the corners of this rectangle are the asymptotes!
Finally use the equation to determine if the parabola is vertical or horizontal.
   
We knew to draw the above hyperbola opening to the left and right because the x^2 term was positive.  If the x^2 term is negative then the hyperbola opens up and down.
   
Compare this hyperbola to the previous and note the difference in standard form. The rectangle and asymptotes are not actually part of the graph. We use these to obtain a more accurate sketch.  When graphed on a graphing utility the result looks like this.
Graph the hyperbola and give the equations of the asymptotes.

  
  
  
The next examples require us to complete the square to obtain standard form.  Remember to factor the leading coefficient out of each variable grouping before using (B/2)^2 to complete the square.

Graph the hyperbola, x- and y- intercepts, and give the equations of the asymptotes.
   
   
When we completed the square, notice that we added 64 and subtracted 225 to balance the equation.  It may appear, at first glance, that we added 16 and 25 to the left side. But in reality, after we distribute the 4 and -9 we added and subtracted larger values.
 
   
When the transverse axis is vertical be careful with the center.  A common error is to use (-2, 1) for the center in the above example because the y variable comes first and we are used to reading these from left to right.  This would be incorrect so take care to use the correct h and k.

Example: Find the equation of the hyperbola centered at (-3, 1) where a = 2, b = 3, and with a vertical transverse axis.  Graph it.
Example: Find the equation of the hyperbola with vertices (3,0) and (-3,0), horizontal transverse axis, and conjugate axis of length 4 units.
Example: Find the equation of the hyperbola with center (2, -5), vertical transverse axis measuring 10 units, and conjugate axis measuring 6 units.
YouTube Videos: Click problem to see it worked out.



---