Download Semester 2 Unit 5: Radical Functions Notes: Throughout units

Semester 2
Unit 5: Radical Functions
Notes: Throughout units, reference the Algebra 2 Wiki for supplemental resources. Consider teaching application problems as a unified section instead of
separately in each individual section.
Total Suggested Number of Days for the Unit: 19 days
Topic
Operations
on
Functions
Inverse
Functions
Square
Root
Functions
and
Inequalities
Nth Roots
Operations
on Radicals
Standards
I can…
 Find the sum, difference,
product, and quotient of
functions
 Identify restrictions in the
domain for a quotient of
functions
 Find the composition of functions
FBF.04 Find inverse functions.
 Find the inverse of a function or
a. Solve an equation of the form f(x) = c for a simple function f that
relation
has an inverse and write an expression for the inverse. For
 Determine whether two
example, f(x) = 2x3 or f(x) = (x+1)/(x-1) for x 1.
functions or relations are
inverses
FIF.07 Graph functions expressed symbolically and show key
 Graph radical functions and
features of the graph, by hand in simple cases and using technology
inequalities
for more complicated cases.
 Identify the domain and range of
b. Graph square root, cube root, and piecewise-defined functions,
a radical function
including step functions and absolute value functions
 Identify transformations on
radical functions
ASSE.02 Use the structure of an expression to identify ways to
 Simplify nth roots with variable
rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it
expressions (Focus on perfect
as a difference of squares that can be factored as (x2 – y2)(x2 + y2).
nth roots in this section)
 Approximate radicals
ASSE.02 Use the structure of an expression to identify ways to
 Simplify radical expressions
rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it  Add, subtract, multiply, and
as a difference of squares that can be factored as (x2 – y2)(x2 + y2).
divide radical expressions
 Rationalize a denominator
(focus on square roots)
FBF.01 Write a function that describes a relationship between two
quantities.
b. Combine standard function types using arithmetic operations.
For example, build a function that models the temperature of a
cooling body by adding a constant function to a decaying
exponential, and relate these functions to the model.
Resources
Time
Frame
Glencoe 6.1
2 days
Glencoe 6.2
2 days
Glencoe 6.3
1 day
Glencoe 6.4
1 day
Glencoe 6.5
3 days
Rational
Exponents
NRN.01 Explain how the definition of the meaning of rational
exponents follows from extending the properties of integer
exponents to those values, allowing for a notation for radicals in
terms of rational exponents. For example, we define 51/3 to be the
cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3
must equal 5.
 Convert expressions into radical
or exponential form
 Perform operations on
expressions with rational
exponents
Glencoe 6.6
3 days
 Solve an equation radicals
 Check answers for extraneous
solutions
Glencoe 6.7
2 days
NRN.02 Rewrite expressions involving radicals and rational
exponents using the properties of exponents
Solving
Radical
Equations
AREI.02 Solve simple rational and radical equations in one variable,
and give examples showing how extraneous solutions may arise.
AREI.11 Explain why the x-coordinates of the points where the
graphs of the equations y = f(x) and y = g(x) intersect are the
solutions of the equation f(x) = g(x); find the solutions
approximately, e.g., using technology to graph the functions, make
tables of values, or find successive approximations. Include cases
where f(x) and/or g(x) are linear, polynomial, rational, absolute
value, exponential, and logarithmic functions.
Semester 2
Unit 6: Exponential and Logarithmic Functions
Notes: Throughout units, reference the Algebra 2 Wiki for supplemental resources. Students are not required to memorize application formulas. Try to embed
the necessary formulas within the problems.
Total Suggested Number of Days for the Unit: 23 days
Topic
Standards
Graphing
FIF.07 Graph functions expressed symbolically and show key
Exponential features of the graph, by hand in simple cases and using technology
for more complicated cases
Functions
e. Graph exponential and logarithmic functions, showing intercepts
and end behavior, and trigonometric functions, showing period,
midline, and amplitude.
I can…
 Graph exponential growth
functions and exponential decay
functions
Resources
Suggested
Time
Frame
Glencoe 7.1
3 days
Glencoe 7.1, 7.2, 7.7,
7.8
3 days
(e, growth
models,
interest)
 Determine whether a relation
demonstrates exponential
growth or decay based on a
table of values
 Identify intercepts, asymptotes,
end behavior, the domain, and
the range of exponential
functions
 Identify transformations on
exponential functions
Exponential ACED.01 Create equations and inequalities in one variable and use
Modeling
them to solve problems. Include equations arising from linear and
quadratic functions, and simple rational and exponential functions.
 Apply growth and decay models
and compound interest formulas
(including continuously
compounding)
ACED.02 Create equations in two or more variables to represent
relationships between quantities; graph equation on a coordinate
axis with labels and scales.
AREI.11 Explain why the x-coordinates of the points where the
graphs of the equations y = f(x) and y = g(x) intersect are the
solutions of the equation f(x) = g(x); find the solutions
approximately, e.g., using technology to graph the functions, make
tables of values, or find successive approximations. Include cases
where f(x) and/or g(x) are linear, polynomial, rational, absolute
value, exponential, and logarithmic functions
 Simplify expressions with e
 Solve for time using graphing
technology
FBF.01 Write a function that describes a relationship between two
quantities.
b. Combine standard function types using arithmetic operations.
For example, build a function that models the temperature of a
cooling body by adding a constant function to a decaying
exponential, and relate these functions to the model..
Logarithms
and
Logarithmic
Functions
FBF.04 Find inverse functions.
a. Solve an equation of the form f(x) = c for a simple function f that
has an inverse and write an expression for the inverse. For
example, f(x) = 2x3 or f(x) = (x+1)/(x-1) for x 1.
 Use properties of inverse
relations to develop the concept
of a logarithmic function and its
graph
FIF.07 Graph functions expressed symbolically and show key
features of the graph, by hand in simple cases and using technology
for more complicated cases
e. Graph exponential and logarithmic functions, showing intercepts
and end behavior, and trigonometric functions, showing period,
midline, and amplitude.
 Graph logarithmic functions and
identify transformations,
asymptotes, and domain and
range
**Consider introducing common
logs and natural logs
simultaneously.
Glencoe 7.3, 76, 7.7
3 days
 Condense and expand
logarithmic expressions
(incorporate common and
natural logs)
 Solve an exponential equation
 Solve a logarithmic equation
Glencoe 7.5
2 days
Glencoe 7.2, 7.4
4 days
Continuous FIF.08 Write a function defined by an expression in different but
Exponential equivalent forms to reveal and explain different properties of the
function.
Growth
and Decay
FLE.04 For exponential models, express as a logarithm the solution
to abct = d where a, c, and d are numbers and the base b is 2, 10, or
e; evaluate the logarithm using technology.
 Use logarithms to solve
application questions involving
continuous exponential growth
and decay
Glencoe 7.8
2 days
Geometric
sequences
and series
 Find the sum of a geometric
series
 Derive the formula for the sum
of a finite geometric series
Glencoe 10.3
Properties
of
Logarithms
ASSE.02 Use the structure of an expression to identify ways to
rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it
as difference of squares that can be factored as (x2 – y2)(x2 + y2).
Solving
FLE.04 For exponential models, express as a logarithm the solution
ct
Logarithmic to ab = d where a, c, and d are numbers and the base b is 2, 10, or
e; evaluate the logarithm using technology.
and
Exponential
Equations
ASSE.04 Derive the formula for the sum of a finite geometric series
(when the common ratio is not 1), and use the formula to solve
problems. For example, calculate mortgage payments.
Application
Worksheet 7.8
2 days
Semester 2
Unit 7:
Rational Functions
Notes: If time allows, you are strongly encouraged to teach adding and subtracting rational functions.
Total Suggested Number of Days for the Unit: 10 days
Standards
I can…
Simplifying
Rational
Expressions
AAPR.07 (+) Understand that rational expressions form a system
analogous to the rational numbers, closed under addition,
subtraction, multiplication, and division by a nonzero rational
expression; add, subtract, multiply, and divide rational expressions
 Simplify, multiply and divide
rational expressions
 Add and subtract rational
expressions (*HIGHLY
RECOMMENDED, but topic
won’t be assessed on
semester exam)
Glencoe 8.1
2 days
+ 3 days for
add/subtract
if time
Graphing
reciprocal
and
rational
functions
FIF.05 Relate the domain of a function to its graph and, where
applicable, to the quantitative relationship it describes. For
example, if the function h(n) gives the number of person-hours it
takes to assemble n
engines in a factory, then the positive
integers would be an appropriate domain for the function.5
 Identify transformations with
reciprocal functions
Glencoe 8.3
3 days
FBF.03 Identify the effect on the graph of replacing f(x) by f(x) + k, k
f(x), f(kx), and f(x + k) for specific values of k (both positive and
negative); find the value of k given the graphs. Experiment with
cases and illustrate an explanation of the effects on the graph using
technology. Include recognizing even and odd functions from their
graphs and algebraic expressions for them.
FIF.07 Graph functions expresses symbolically and show key
features of the graph, by hand in simple cases and using technology
for more complicated cases.
d.(+) Graph rational functions, identifying zeroes when suitable
factorizations are available, and showing end behavior.
Resources
Time
Frame
Topic
Glencoe 8.4
 Identify the domain and range
of a reciprocal function
 Identify vertical and horizontal
asymptotes
 Find a vertical asymptote
algebraically
 Sketch a graph of a reciprocal
and rational function that
correct placement of
asymptotes
Solve a
rational
equation
AREI.02 Solve simple rational and radical equations in one variable,
and give examples showing how extraneous solutions may arise.
Solve a rational equation using
proportional reasoning
Glencoe 8.6
Supplement
problems solving and
cross multiplying to
solve
2 days
Semester 2
Unit 8:
Trigonometry
Notes : Throughout units, reference the Algebra 2 Wiki for supplemental resources.
Total Suggested Number of Days for the Unit: 24 days
Topic
Unit Circle
Standards
I Can….
F.TF.1 Understand radian measure of an angle as the length of the
arc on the unit circle subtended by the angle.
• Define radian, define degree,
explain the need for radian
measure.
 Define and use Initial side,
terminal side, vertex, standard
position, positive angle,
negative angles, coterminal,
central angles.
 Sketch angles in standard
position.
 Label the unit circle using
degrees, radians and ordered
pairs.
 Find coterminal angles
 Find reference angle
 Convert angles from radians to
degrees
 Convert angles from degrees to
radians
F.TF.2 Explain how the unit circle in the coordinate plane enables
the extension of trigonometric functions to all real numbers,
interpreted as radian measures of angles traversed
counterclockwise around the unit circle.
Converting
between
degrees and
radians
F.TF.1 Understand radian measure of an angle as the length of the
arc on the unit circle subtended by the angle.
Resources
Glencoe 12.2
Suggested
Time
Frame
3 days
Glencoe 12.3
String Activity
Unit Circle Handout
Glencoe 12.2
1 day
 Find arc length, radius and
theta using the arc length
formula.
 Evaluate trig ratios on the unit
circle
Arc Length
F.TF.1 Understand radian measure of an angle as the length of the
arc on the unit circle subtended by the angle.
Glencoe 12.2
2 days
Derive
trigonometric
ratios on the
unit circle
using special
right triangle
F.TF.2 Explain how the unit circle in the coordinate plane enables
the extension of trigonometric functions to all real numbers,
interpreted as radian measures of angles traversed
counterclockwise around the unit circle.
Glencoe 12.1
2 days
Determine
trig ratios off
the unit circle
given either a
point or a trig
ratio.
Pythagorean
Identity
F.TF.2 Explain how the unit circle in the coordinate plane enables
the extension of trigonometric functions to all real numbers,
interpreted as radian measures of angles traversed
counterclockwise around the unit circle.
 Evaluate trig ratios off the unit
circle
Glencoe 12.3
May need to
supplement for
assignment
2 days
F.TF.8 Prove the Pythagorean identity sin^2(theta)+cos^2(theta)=1
and use it to find sin(theta), cos(theta), or tan(theta), given
sin(theta), cos(theta), or tan(theta), and the quadrant of the angle.
Glencoe 13.1
2 days
Graphing
sine and
cosine
functions
F.TF.5 Choose trigonometric functions to model periodic
phenomena with specified amplitude, frequency, and midline.
 Use the Pythagorean Identity to
find trig ratios.
 Determine when an equation is
a manipulation of the
Pythagorean Identity.
 Graph a Trig Function
 Identify transformations
 Identify the period, amplitude
and phase shift of the trig
graphs
Glencoe 12.6
8 days
Unit Circle Discovery
Packet
Glencoe 12.7
Supplement as
needed