Learning Resources
Lesson
Investigation 10: Looking for More Patterns
Another pattern will emerge in this investigation that will permit you to graph the exponential directly from the equation without the mapping rule or table of values required. This is done by further examination of the function 
; however, keep in mind that it applies to all exponential functions of the form 
with the pattern identified for specific bases.
In Step A, recall that the focal point for all exponential functions of the form is (0,1) since for x = 0 we know that b0 = 1. Answer the questions making special note of the pattern of movement for determining points on the curve.
For Step B repeat the questions using the exponential in the form 
. The focal point changes as a result of the constant a. Again make note of the pattern.
These patterns can be used when graphing with transformations rather that relying on the mapping rule.
Example
Determine the focal point and two other points for the curve drawn from 
Solution
For x = 0 the value of y is ½. Therefore, the focal point is (0,½)
To find successive points on the graph of , move 1 unit to the right of the focal point and 1 unit up or one unit to the left of the focal point and ½ unit down. Since this equation has a vertical stretch of ½, move 1 unit right of the focal point and ½ x 1 = ½ unit up to obtain the point (1, 1); move 1 unit to the left of the focal point and ½ x ½ = ¼ unit down for the point (-1, ¼). You could plot these three points and sketch a rough curve of the function.
Complete investigation question 5 before continuing.
Focus E: More Graphing with Transformations
Read and study Method 2: Directly from Equation to Graph as outlined in this focus. Following the steps presented you be able to graph any exponential equation following this pattern.
Consider, as an extra example, how to graph 
. First note the various transformations
- reflection in the x-axis ( the negative y value indicates this )
- Vertical translation of -4
- Horizontal stretch of ½
- Horizontal asymptote y = -4
Click on the step ahead button and view the steps in graphing this function:
Activity
Investigation Questions p.147. Complete 5
Check Your Understanding p.148 - 155. Complete 6 to 27 inclusive
When you have completed these questions, ask your on-site teacher to get the solutions for you from the Teacher's Resource Binder and check them against your answers. After you do this, if there is something you had trouble with and still do not understand, contact your on-line teacher for help.
After you do the assigned activities, continue on to the Test Yourself section below for a quick quiz on this lesson.
Test Yourself
Using the equation 
- Write its transformational form.
- Describe the transformations with respect to
- State its horizontal asymptote.
- Sketch by using its focal point and two other points.
Click here for a suggested solution.