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Evaluate Exp. Function Example 1.x = x = 2.x = -35. x = 3.x = x = 3.784
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Exponential Functions
4.3 **You might want
graph paper**
Vocabulary• An exponential function has the form
y = abx, where a ≠ 0 and the base b is a positive number other than 1.
• If a > 0 and b > 1, the function y = abx is an exponential growth function– and b is called the growth factor or
multiplier.
Evaluate Exp. Function
• Example
1.x = 1.4 4. x = 2.x = -3 5. x = 3.x = 1.68 6. x = 3.784
x
xy b2xy
2
Graphing Exp. Functions
• Parent
• Plot the points ( 0, 1) and ( 1, base)• The asymptote will be a horizontal line y = ? and it is always the shift up or down
xy b
Make a table of values.STEP 1
STEP 2 Plot the points from the table.
Graph y = 2x.
Graph the following on your graph paper.
1
4
1. 2
2. 2
3. 2
4. 2 3
5. 2
6. 2 3
x
x
x
x
x
x
y
y
y
y
y
y
Graph the function.
SOLUTION
Plot and (1, 2) .Then,
from left to right, draw a curve that begins just above the x-axis, passes through the two points, and moves up to the right.
0, 12
a.
1a. 42
xy
Graph the function.
b. y = – 52
x
Plot (0, –1) and .
Then,from left to right, draw a curve that begins just below the x-axis, passes through the two points,and moves down to the right.
b. 1, – 52
SOLUTION
Graph
SOLUTION
14 2 3.xy
• Begin by sketching the graph y = 4 ● 2x
• Then translate the graph: y = 4 ● 2x – 1 – 3
•Domain?
•Range?
•Asymptote?
Your TurnGraph each function below. State the
Domain, Range and Asymptote.
• y = 5x
• f(x) = 1/3 ● 4x
• y = 2x + 1 – 1
Your Turn: Answers
Domain: all real #sRange: y > 0Asym.: y = 0
Domain: all real #sRange: y > 0Asym.: y = 0
Domain: all real #sRange: y > -1Asym.: y = -1
Vocabulary
• An exponential decay function has the form y = abx, where a > 0 and 0 < b < 1.
• The base b of an exponential decay function is called the decay factor.
Example 1
Graph y = State the domain and range. Give the equation for the asymptote.
12
x
Example 2• Graph y = -3(1/4)x. State the domain and range.
Give the equation for the asymptote.
Example 3• Graph y = 2(2/3)x – 1 + 1. State the domain and
range. Give the equation for the asymptote.
When b>1, f(x) = bx represents GROWTH
When 0<b<1, f(x) = bx represents DECAY
Growth vs. Decay
Use Functions Involving e• The number denoted by the letter e is
called the natural base e or the Euler number. – Think of e like you do π– e is approximately 2.718281828.
• A function of the form f(x) = aerx is called a natural base exponential function.
• Graph the function. Graph the parent first.
• State the domain and range.a) y = 4e-x + 5
xy e
• Graph the function. State the domain and range.
b) y = e2(x+1) - 2
Exponential Equations
• Exponential equations are equations in which variable expressions occur as exponents.
• Property of Equality for Exponential Equations:If b is a positive number other than 1, then bx = by if and only if x = y.
Examples
1 3 1 11. 3 81 2. 5 25x x x
2 2 4 2 3 13. 9 3 4. 44
x x x
2 235. 8 4 6. 5 25x x x x
21 17. 4 8. 2 16 4
2
xx x x
2 25 43
19. 10.x x x xe e e ee
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