Download exponential functions

Warm Up Thursday – have each group
do one of the following
1. Make a problem using the zero exponent
rule that simplifies to 12x6
2. Make a problem using the power of a
quotient that simplifies to 12x6
3. Make a problem using the power of a
product rule that simplifies to 12x6
4. Make a problem using the power of a power
rule that simplifies to 12x6
5. Make a problem using negative exponents
that simplifies to 12x6
Solving Simple Exponential Equations
• When we “solve” an equation, we are really
setting two equations equal to each other and
finding the value of x that satisfies both
equations. We could then substitute the solution
(the x) into either equation and find the “y” of
that point.
• Make up two equation, set them equal to each
other, and show what “equal” means
geometrically on geo sketch pad
• Solve as an equality, then “guess and check” to
see which side of the equality is the solution.
Solving Simple Exponential Equalities
One-to-One Property:
• For a > 0 and a ≠ 1, ax = ay iff x = y
• Example: if 2x = 29, then x = 9
NOTE: we must have the same base.
What if we do not have the same base?
Solve for x: 3 x  2  9 2 x  9
Practice - Thursday
• Packet page 6
Warm Up for Friday 2/10
• Use each power rule (seven of them) to write
an expression that reduces to 16x6 and
identify the power rule used.
• Explain why 00 ≠ 1
• Explain what the domain restrictions to x-1
are.
Solving Simple Exponential Inequalities
• Go back and review what solving an algebraic
equation really means geometrically.
• When we are solving an inequality, we want to
know the values of “x” that makes the
inequality true. We need to know the values
of “x” that makes “y” value of one equation
greater or less than the other equation.
Solving Simple Exponential Inequalities
• To solve an inequality:
1. Solve the equation as an equality,
2. then “guess and check” to see which side of
the equality satisfies the original equation.
Practice - Friday
• Packet pages 8, make all the evens less than
(<)
• Make all the odds greater than (>)
Warm Up Monday 2/13
• Create an expression that reduces to 716 by
using each exponential property (there are
seven of them)
Reduce & Solve Exponent Review
• Coach, lesson 11, page 77
• American Choice, lesson 2, page 5 & 6
Review of Exponent Rules
• Am Ch Lesson 1
• Coach Lesson 4 Properties of Exponents, pg 27
- 30
• Coach Lesson 11 Solving Exponential
Equations, pg 73 - 77
• Coach Lesson 12 Solving Exponential
Inequalities, pg 78 - 82
Warm up Wednesday 2/15
Simplify:
2x  x y 
3 3
2
4 x 
2
7 0
Standards: MM2A2. Students will
explore exponential functions.
(Book 4.4, 4.5, 4.6)(Coach 9 – 13)(Am Ch 1 – 7)
b. Investigate and explain characteristics of
exponential functions, including domain and
range, asymptotes, zeros, intercepts,
intervals of increase and decrease, rates of
change, and end behavior.
c. Graph functions as transformations of
f x   b
x
• Basic exponential functions (good)
• http://www.onlinemathlearning.com/expone
ntial-functions.html
Graphing exponential functions
(Khanacademy.com)(six minutes to make one
graph)
• http://www.khanacademy.org/video/graphing
-exponential-functions?topic=developmentalmath-3
Exponential Functions – Just Changing
the Base (Wednesday)
• Investigate various exponential functions
using the web page:
• http://www.mathopenref.com/graphfunctions
.html
• Use 3 functions: f(x) = a^x, g(x) = b^x, h(x) =
c^x, y from -1 to 9, and x from -10 to 10
• Emphasize we are taking a positive number to
an exponent. We are NOT playing with
stretch, reflection, or translations. We will do
those later.
Exponential Functions – Just Changing
the Base
• Look at the graph as “b” increases and
decreases
• What are the similarities?
– The basic shape remains the same
– Y-intercept is always 1
• Explain why that would be so
– All curves have an asymptote of y = 0
– Domain: All real numbers
– Range: y > 0
Exponential Functions – Just Changing
the Base
• What are the differences?
– The curve reacts faster, i.e., gets steeper when “b”
gets farther from 1
• Explain why
– The curve raises to the right when b > 1
• As x increases, y increases
• Explain why
– The curve raises to the left when 0 < b < 1.
• As x increases, y decreases
• Explain what is happening
– The graph is a straight line of y = 1 when b = 1
• Why?
Make a “T” chart, graph on the same
sheet, and explain what happens as
the base changes
•
•
•
•
•
•
f(x) = 3x
g(x) = 2x
h(x) = 1.4x
i(x) = 0.7x
j(x) = 0.5x
k(x) = 0.3x
Describe the transformation from:
f(x) = x2
• g(x) = -3(x – 4)2 + 7
• The 4 moves (translates) the graph 4 units to
the right
• The 3 stretches the graph vertically by a factor
(multiplies) by 3
• The negative in front of the 3 can be thought
of as “times -1”, and it reflects the graph over
the x-axis.
• The 7 moves (translates) the graph up 7 units.
Exponential Functions –
Transformations
• REMEMBER VERTEX FORM OF QUADRATIC:
2
f ( x)  a( x  h)  k
• a – vertical stretch
• If there is a negative in front of the a, reflects
across the x-axis
• h – horizontal transformation (shift)
• k – vertical transformation (shift)
Exponential Functions –
Transformations
• Investigate various exponential functions
using the web page:
• http://www.mathopenref.com/graphfunctions
.html
• Use: f(x) = a*b^(x-c)+d, y from -10 to 10, and x
from -10 to 10
• We are now using the function:
 xh 
f ( x)  a  b
k
Exponential Functions –
Transformations
• Look at the graph as a, h, and k are varied
• What happens?
– All constants perform the same transformation as
in a quadratic
– The “a” stretches the function
– The “h” is “inside the house” so it translates the
function horizontally
– The “k” is “outside the house” so it translates the
function vertically, shifting the asymptote from
y=0
Exponential Functions –
Transformations
• Look at the graph as a, h, and k are varied
• What happens?
– If there is a negative in front of the a, the graph is
reflected across the x-axis
– If there is a negative in front of the x, the graph is
reflected across the y-axis (this is new for you)
Exponential Functions –
Transformations
• How does that change the domain and range?
• Domain: is always all real numbers
• Range:
– y > asymptote if a > 0
– y < asymptote if a < 0
Describe the transformation from
f(x) = 3x and give the domain, range,
y-intercept end conditions
•
•
•
•
g(x) = 3*2(x – 2)
h(x) = 2-x
i(x) = -2x + 3
j(x) = -3*2(-x + 4) - 7
Make a “T” chart, graph f(x) or f’(x) on
each graph, and explain what happened
from f(x) or f’(x) as a result of changing a,
b, h, and/or k, or making a or x negative
f(x) = a*b(x – h) + k
•
•
•
•
•
•
f(x) = 1.3x
g(x) = 1.3(-x)
h(x) = 2*1.3x
i(x) = 1.3(x + 3)
j(x) = 1.3x - 4
k(x) = -1*1.3x
•
•
•
•
•
•
f’(x) = 0.7x
g’(x) = 0.7(-x)
h’(x) = 0.5*0.7x
i’(x) = 0.7(x - 3)
j’(x) = 0.7x + 4
k’(x) = -1*0.7x