ALGEBRA 2
Texas Essential Knowledge and Skills
Annotated by TEA for Pre-AP rigor
Introduction
As the committee began its examination of the Texas Essential Knowledge and Skills (TEKS), we were often surprised by what was included or left
out of courses that preceded or followed those that we normally teach.
"Do they really expect eighth graders to be able to do that?"
"Where are the sequences and series that we used to do in Algebra II?"
Ultimately, we agreed that all of the concepts and skills necessary to prepare students for success in AP* Statistics and AP Calculus would be
covered if the TEKS were interpreted in a particular way. Due to time constraints, we were reluctant to add any additional topics to the TEKS, though
a teacher might choose to do so.
The problem is particularly acute at the middle school level when all of the TEKS for grades 6-8 are often covered in only two years in order for
students to take Algebra I in grade 8. Having students just skip over a year of elementary or middle school mathematics is a dangerous proposition
that can have serious repercussions in subsequent courses. A well-planned and instructed Pre-AP* middle school program combines, streamlines, and
collapses the material in such a way that all of the TEKS are addressed at a deeper and more complex level.
At one point, someone on the committee said, "The problem is not that the TEKS are incomplete; it is that all of these things are treated equally.
Some of these TEKS are three-minute topics, and some of them are three-week topics." That gave us our idea for the structure of the charts in this
section. We went through the TEKS and sorted them into three groups.

The TEKS in regular font are topics with which students already have some familiarity due to previous instruction and which are being
revisited through the spiraling curriculum or are topics that can be covered in minimal time. These topics might provide foundational
knowledge (such as definitions) that will be used for future topics throughout the course.

The TEKS typed in italics are topics that might be addressed throughout the course on multiple occasions or might be addressed to greater
depth than the previous topics.

The TEKS in a bold, slightly larger, font are those that merit greater time commitment and greater depth of understanding for the
Pre-AP student. These topics should be taught with a particular emphasis toward preparing students for AP Calculus or AP
Statistics.
After categorizing the TEKS, we looked for problems or activities that would exemplify those TEKS in the third group and included them in the
second column as examples of what we felt were good Pre-AP mathematics problems and activities. Remember that these are only examples;
students will have to do many more than the few problems that we were able to include here in order to be well-prepared for AP Statistics and AP
Calculus. These are meant to give you ideas and get you started in understanding what makes a good Pre-AP mathematics problem. You will also
find in the second column additional comments about the TEKS or sample problems that we felt might be important.
TEKS: Algebra II
TEKS
111.33 ALGEBRA II (ONE-HALF TO ONE
CREDIT) Foundations for functions: knowledge and
skills and performance descriptions.
(2A.1) The student uses properties and attributes of
functions and applies functions to problem situations.
(A) identify the mathematical domains and ranges of
functions and determine reasonable domain and range
values for continuous and discrete situations.
(B) collect and organize data, make and interpret
scatter plots, fit the graph of a function to the data,
interpret the results, and proceed to model, predict, and
make decisions and critical judgements.
(2A.2) The student understands the importance of the
skills required to manipulate symbols in order to solve
problems and uses the necessary algebraic skills
required to simplify algebraic expressions and solve
equations and inequalities in problem situations.
(A) use tools including factoring and properties of
exponents to simplify expressions and to transform and
solve equations.
(B) use complex numbers to describe the solutions of
quadratic equations.
(2A.3) The student formulates systems of equations and
inequalities from problem situations, uses a variety of
methods to solve them, and analyzes the solutions in
terms of the situations.
(A) analyze situations and formulate systems of
equations in two or more unknowns or inequalities in
two unknowns to solve problems.
(B) use algebraic methods, graphs, tables, or matrices,
to solve systems of equations or inequalities.
Examples
Commentary
(C) interpret and determine the reasonableness of
solutions to systems of equations or inequalities for
given contexts.
Algebra and geometry: Knowledge and skills and
performance descriptions.
(2A.4) The student connects algebraic and geometric
representations of functions.
(A) identify and sketch graphs of parent functions,
including linear (f(x) = x), quadratic (f(x) = x2),
exponential (f(x) = ax), and logarithmic (f(x) = logax)
functions, absolute value of x (f(x) = |x|), square root of
x (f(x) = √x), and reciprocal of x(f(x) = 1/x).
(B) extend parent functions with parameters such as Let f be the function given by f ( x)  x 3  6 x 2  p , where p is an arbitrary
a in f(x) = a|x and describe the effects of the
constant.
parameter changes on the graph of parent functions.
a) For what values of the constant p does f have 3 distinct real roots? Explain
your reasoning.
(C) describe and analyze the relationship between a
function and its inverse.
(2A.5) The student knows the relationship between the
geometric and algebraic descriptions of conic sections.
(A) describe a conic section as the intersection of a
plane and a cone.
(B) sketch graphs of conic sections, to relate simple
parameter changes in the equation to corresponding
changes in the graph.
(C) identify symmetries from graphs of conic sections.
(D) identify the conic section from a given equation.
(E) use the method of completing the square.
Quadratic and square root functions:
(2A.6) The student understands that quadratic functions
can be represented in different ways and translates
among their various representations.
This question is based on the 1997
AB4 AP* Calculus question.
(A) determine the reasonable domain and range values
of quadratic functions, as well as interpret and
determine the reasonableness of solutions to quadratic
equations and inequalities.
(B) relate representations of quadratic functions, such
as algebraic, tabular, graphical, and verbal
descriptions.
(C) determine a quadratic function from its roots or a
graph.
(2A.7) The student interprets and describes the effects
of changes in the parameters of quadratic functions in
applied and mathematical situations.
(A) use characteristics of the quadratic parent function
to sketch the related graphs and connect between the y
= ax2 + bx + c and the y = a(x - h)2 + k symbolic
representations of quadratic functions.
(B) use the parent function to investigate, describe, and
predict the effects of changes in a, h, and k on the
graphs of y = a(x - h)2 + k form of a function in applied
and purely mathematical situations.
(2A.8) The student formulates equations and
inequalities based on quadratic functions, uses a variety
of methods to solve them, and analyzes the solutions in
terms of the situation.
This question is based on the 1996
AB6 AP Calculus question.
(A) analyze situations involving quadratic functions
and formulate quadratic equations or inequalities to
solve problems.
a) Write the equation of the line shown in the figure.
b) Write the equation for the parabola shown in the figure.
c) Suppose the graph of the parabola shown in the figure represents a hill. There is
a 50-foot tree growing vertically at the top of the hill. Does a spotlight at point P
directed along line shine on any part of the tree? Explain your reasoning.
(B) analyze and interpret the solutions of quadratic
equations using discriminants and solve quadratic
equations using the quadratic formula.
(C) compare and translate between algebraic and
graphical solutions of quadratic equations.
(D) solve quadratic equations and inequalities using Consider the curve defined by x 2  xy  y 2  27 . Solve for y in terms of x and
graphs, tables, and algebraic methods.
use your calculator to graph the curve. Determine the intercepts algebraically and
verify graphically.
Graph the region given by the quadratic system 25 x 2  9 y 2  225 , x ≥ 0 and
x ≤ 8 on a coordinate plane. Rotate the region about the x-axis and sketch the
resulting three-dimensional solid.
(2A.9) The student formulates equations and
inequalities based on square root functions, uses a
variety of methods to solve them, and analyzes the
solutions in terms of the situation.
(A) use the parent function to investigate, describe, and
predict the effects of parameter changes on the graphs
of square root functions and describe limitations on the
domains and ranges.
(B) relate representations of square root functions, such
as algebraic, tabular, graphical, and verbal
descriptions.
(C) determine the reasonable domain and range values
of square root functions, as well as interpret and
determine the reasonableness of solutions to square root
equations and inequalities.
(D) determine solutions of square root equations using
graphs, tables, and algebraic methods.
(E) determine solutions of square root inequalities using
graphs and tables.
This question is based on the 1994
AB3 AP Calculus question.
Express the equation as a quadratic in
terms of y, like this (y2 + xy + (x2 27) = 0). Use the quadratic b=x and
c=x2 - 27.
This introduces the calculus concept
of solids of revolution.
Also addresses TEKS 2A.8B and
2A.8C.
To start this activity you will need a stopwatch, a piece of string at least 1.7 meters
(F) analyze situations modeled by square root
functions, formulate equations or inequalities, select long, a ruler, and a weight.
a method, and solve problems.
1. Tie the weight to the end of your string and then measure off 1.5 meters.
2. Have one person hold the pendulum. Use your stopwatch to time, in seconds,
how long it takes the pendulum to complete ten periods. Divide this time by 10 to
estimate the time it takes for the pendulum to complete one period.
This activity introduces the student to
data collection, curve fitting, and
experimental design—all topics of the
AP Statistics curriculum.
Does the weight of the pendulum
affect the regression equation? Give
each group a different weight and
compare the results.
3. Repeat the process in part 2 several times, each time shortening the length of
your string by 15 cm. Continue to collect your data (length of string, time/period).
4. Draw a sketch of the data, the time of a period as a function of the length of the
pendulum.
5. If the graph appears linear, write an equation of the regression line that best
models the data. If the graph does not appear to be linear, you will need to perform
a transformation to straighten the data. What model and what transformation seem
most appropriate? Write the equation of this model.
6. Predict t(.9 meters), that is, the time of one period if the length of the pendulum
is .9 meters.
7. Predict t(2 meter), that is, the time of one period if the length of the pendulum is
2 meters.
8. Which of the two values t(.9) or t(2) do you feel is more accurate? Explain why.
9. What type of function did you determine was the best model for your data?
Explain the process you had to use to re-express your data to be able to write an
equation for this model.
(G) connect inverses of square root functions with
quadratic functions.
(2A.10) The student formulates equations and
inequalities based on rational functions, uses a variety
of methods to solve them, and analyzes the solutions in
terms of the situation.
(A) use quotients of polynomials to describe the
graphs of rational functions, predict the effects of
parameter changes, describe limitations on the
domains and ranges, and examine asymptotic
behavior.
Let f be the function given by f ( x) 
x
x 4
2
a) Find the domain of f.
b) Write an equation for each vertical asymptote to the graph of f.
c) Write an equation for each horizontal asymptote to the graph of f.
This question is based on the 1989
AB4 AP Calculus question.
(B) analyze various representations of rational
functions with respect to problem situations.
(C) determine the reasonable domain and range values
of rational functions, as well as interpret and determine
the reasonableness of solutions to rational equations
and inequalities.
(D) determine the solutions of rational equations using
graphs, tables, and algebraic methods.
(E) determine solutions of rational inequalities using
graphs and tables.
(F) analyze a situation modeled by a rational
function, formulate an equation or inequality
composed of a linear or quadratic function, and
solve the problem.
The city of Katy, Texas, wants to enclose a 3000 square foot rectangular region as This relates to the calculus concept of
a park. The city plans to build a brick fence along 3 sides of the park that will cost optimization.
$25 per linear foot. A wooden fence that will cost $10 per linear foot will enclose
the fourth side of the park. Find the minimum cost of the fence.
(G) use functions to model and make predictions in
problem situations, involving direct and inverse
variation.
(2A.11) The student formulates equations and
inequalities based on exponential and logarithmic
functions, uses a variety of methods to solve them, and
analyzes the solutions in terms of the situation.
(A) develop the definition of logarithms by exploring
and describing the relationship between exponential
functions and their inverses.
(B) use the parent functions to investigate, describe, Let f be the function given by f ( x)  2 xe2 x Use your graphing calculator to do
and predict the effects of parameter changes on the the following:
graphs of exponential and logarithmic functions,
describe limitations on the domains and ranges, and a) find all horizontal asymptotes of f(x)
examine asymptotic behavior.
b) locate the absolute minimum value of f
c) determine the domain and range of f
d) Examine the family of functions defined by y  bxebx where b is a nonzero
constant. What do all of these graphs have in common? Why?
This problem is based on the 1998
AB/BC2 AP Calculus exam. This is a
good problem to tackle as a class.
Every student can be assigned a
different value of b.
(C) determine the reasonable domain and range values
of exponential and logarithmic functions, as well as
interpret and determine the reasonableness of solutions
to exponential and logarithmic equations and
inequalities.
(D) determine solutions of exponential and logarithmic
equations using graphs, tables, and algebraic methods.
(E) determine solutions of exponential and logarithmic
inequalities using graphs and tables.
(F) analyze a situation modeled by an exponential
function, formulate an equation or inequality, and
solve the problem.
The rate of consumption of cola in the United States is given by S(t) = Ce kt, where This question is based on the 1996
S is measured in billions of gallons per year and t is measured in years from the
AB/BC3 AP Calculus exam.
beginning of 1980.
a) The consumption rate doubles every 5 years and the consumption rate at the
beginning of 1980 was 6 billion gallons per year. Find C and k.
b) When did the consumption rate surpass 50 billion gallons per year?