+ exponential growth functions how do we graph exponential growth functions? m2 unit 5a: day 5
TRANSCRIPT
+Example:Graph the function
y 5x
1
1
x -1 0 1
y
15
1 5
Draw a smooth curve from left to right just above the x-axis that moves up and to the right.
We will call functions like this “parent functions” because they haven’t been translated
Translations in Exponential Functions….
yabx h k•h moves the function to the right or left•k moves the function up or down
Example: The graph of is translated up 3 units.
What is the equation of the translation?
Example: The graph of is translated left 2 units
and down 5 units. What is the equation of the
translation?
y5x
y4(2x )
y5x 3
y4(2x2 ) 5
The graph of is translated
left 3 units and down 1 unit. What is the
equation of the translation?
34
2
x
yæö÷ç= × ÷ç ÷çè ø
33
4 12
x
y+æö÷ç= -÷ç ÷çè ø
You Try:
+Asymptotes and y-intercepts
y-intercept:
To find the y-intercept, plug in zero for x. In an exponential growth function, your y-intercept is always a when your function is in the form
Asymptote:
a line that a graph approaches more and more closely. Exponential functions have a horizontal asymptote at y=k when your function is in the form .
.xy ab=
x hy ab k-= +
+Example: What are the asymptote and y-intercept for the function on the left?
Since this is a “parent function”, it’s asymptote is y = 0.
The y-intercept is always a in a “parent function”, so the y-intercept is 1.
+Domain and Range
The domain of an exponential function will always be ALL REAL NUMBERS
The range of an exponential function will depend on where the asymptote is
+Example: Graph y = 3x. Analyze the graph.
“Parent functions” of exponential growth function have a horizontal asymptote at y = 0.
The DOMAIN is all real numbers and the RANGE is y > 0
x -1 0 1
y 1/3 1 31
1
Example: Graph . Find the asymptote, domain, and range.
y23x 2 2
Start by sketching the graph of .
This is the “parent function”.
y23x
Then, translate the graph __________ 2 units and __________ 2 units.
rightdown
This graph has an asymptote at the line ______________.
The domain is ____________ and the range is ________________.
y = -2
All Real #s y > -2
x -1 0 1
y 2 6
23
+Graph the function . Find the asymptote, domain, and range.
16
4xy = ×
x -1 0 1
y 1/4 3/21/24
The DOMAIN is all real numbers and the RANGE is y > 0
“Parent functions” have a horizontal asymptote at y = 0.
You Try:
+Reflections of Exponential FunctionsSometimes, you may see an exponential
growth function in which a<0. When this occurs, the graph is reflected over the x-axis.Example:
Graph 4 3xy =- -x -1 0 1
y Dow
n 3
x -1 0 1
y
-1/4 -1 -4
-3 ¼ -4 -7
+End BehaviorAfter you graph your function, decide what it is doing as x goes to -∞(to the left) and ∞(to the right)
Ex: Describe the end behavior
As , ( ) and as x - , ) 3(x f x f x® ¥ ® ® ¥ ®¥As , ( ) __ and as x - , ( ) __x f x f x® ¥ ® ® ¥ ®
+
a. As x - ∞, f(x) 0; as x ∞, f(x) - ∞
b. As x - ∞, f(x) 0; as x ∞, f(x) ∞
c. As x - ∞, f(x) ∞; as x ∞, f(x) 0
d. As x - ∞, f(x) - ∞; as x ∞, f(x) ∞
Describe the end behavior of the following graph.
+average rate of change
Average Rate of Change – the “slope”
The most steep part of the graph has the highest rate of change (ROC)
2 1
2 1
y yx x
--
+ROC
Where would the rate of change be highest for this function:
A. Between -6 and -4
B. Between -4 and -2
C. Between -2 and 0
D. Between 0 and 2
+ROC
Where would the rate of change be highest for this function:
A. Between 6 and 8
B. Between 4 and 6
C. Between 2 and 4
D. Between 0 and 2
+
Asymptote:
Y-intercept:
Domain:
Range:
Describe the translation:
Describe the End Behavior:
-æö÷ç ÷= +ç ÷ç ÷çè ø
132 12xy
y = 1
All Real #s
x -1 0 1
y
x 0 1 2
y1
13
2 31
23
3 4
y > 1
As x - ∞, f(x) 1; as x ∞, f(x) ∞
Graph
Right 1, Up 1
2
+Graph and analyze the function.
Asymptote:
Y-intercept:
Domain:
Range:
Describe the translation:
Describe the End Behavior:
12 3xy +=- -
As x - ∞, f(x) ___; as x ∞, f(x) __
+Graph and analyze the function.
Asymptote
Y-intercept
Domain:
Range:
Describe the translation:
Describe the End Behavior:
53 3xy -= ×
As x - ∞, f(x) ___; as x ∞, f(x) __