Exponential Functions Topic 3: Applications of Exponential Functions.
Exponential Functions
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Transcript of Exponential Functions
Exponential Functions
• The domain is the set of real numbers, and the range is the set of positive real numbers
• if b > 1, the graph of y = bx rises from left to right and intersects the y-axis at (0, 1). As x decreases, the negative x-axis is a horizontal asymptote of the graph.
• If 0 < b < 1, the graph of y = bx falls from left to right and intersects the y-axis at (0, 1). As x increases, the positive x-axis is a horizontal asymptote of the graph.
The equation y = bx is an exponential function provided that b is a positive number other than 1. Exponential functions have variables as exponents.
Generalizations about Exponential Functions
Graphs of Exponential Functions
Let’s look at the graph of y = 2x
x 2x -3 -2 -1 0 1 2 3
.125 .25 .5 1 2 4 8
Let’s look at the graph of y = (½)x
x (½)x -3 -2 -1 0 1 2 3
8 4 2 1 .5 .25 .125
That was easy
Comparing Graphs of Exponential Functions
What happens to the graph of y = bx as the value of b changes?
x 2x 4x 6x
-3-2-10123
.125.25.51248
.01563.0625.25141664
.00463
.02778.16667
1636216
Let’s look at some tables of values.
Now, let’s look at the graphs.
I think I see a pattern here.
Let’s Look at the Other Side
What happens to the graph of y = bx when b < 1 and the value of b changes?
x (½)x (¼)x -3 -2 -1 0 1 2 3
8 4 2 1 .5 .25 .125
64 16 4 1
.25 .0625 .01563
Let’s look at some tables of values.
Now, let’s look at the graphs.
I knew there was going to be a pattern!
Let’s Shift Things Around
Let’s take another look at the graph of y = 2x
Now, let’s compare this to the graphs ofy = (2x)+3
andy = 2(x+3)
x -3 -2 -1 0 1 2 3
(2x)+3 3.125 3.25 3.5 4 5 7 11
2(x+3) 1 2 4 8 16 32 64
Translations of Exponential Functions
The translation Th, k maps y = f(x) to y = f(x - h) + kHey, that rings a bell!
Since y and f(x) are the Sam Ting,
It looks like that evaluating functions stuff.
Remember if f(x) = x2, then f(a - 3) = (a - 3)2
It’s all starting to come back to me now.
We can apply this concept to the equation y = bx
Let’s take a closer look
If the translation Th, k maps y = f(x) to y = f(x - h) + k
If y = bx and y = f(x), then f(x) = bx
This is a little confusing, but I’m sure it gets easier.
Then the translation Th, k maps y = bx to y = b(x - h) + k
This will be easier to understandif we put some numbers in here.
Now we have a FormulaThe translation Th, k maps y = bx to y = b(x - h) + k
Let’s try a translation on ourbasic exponential equation
Let’s apply the transformation T3, 1 to the equation y = 2x
The transformed equation would be y = 2(x - 3) + 1
I’m not ready to push the easy button yet.Let’s look at some other examples first.
Let’s look at some graphs
Let’s start with the graph of y = 2x
Let’s go one step at a time.When the
transformation T3, 1 is applied to the equation y = 2x we gety = 2(x - 3) + 1
Step 1 y = 2(x - 3)
Step by = 2(x - 3) + 1
What happenedto the graph?
What happenedto the graph now?
What conclusions can we make from this example?
Let’s look at some other graphs
Let’s start with the graph of y = 2x
Let’s go one step at a time.When the
transformation T-4, -2 is applied to the equation y = 2x we gety = 2(x + 4) - 2
Step 1 y = 2(x + 4)
Step b y = 2(x + 4) - 2
What happenedto the graph?
What happenedto the graph now?
What conclusions can we make from this example?
Let’s Summarize Translations
The translation Th, k maps y = bx to y = b(x - h) + k
Positive k shifts thegraph up k units
Negative k shifts thegraph down k unitsPositive h shifts thegraph left h units
Negative h shifts thegraph right h units
This translation stuff sounds pretty shifty, but don’t let it scare you.
This exponential equation stuff is
pretty easy.I feel like jumping for joy!
Oh my! I think I’ll just push the easy button.
That was easy