Lighthouse Initiative for Texas Classrooms

TEKS: Algebra II

Read an introduction to Texas Essential Knowledge and Skills charts.

TEKS Examples Commentary

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.

(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 a in f(x) = a|x and describe the effects of the parameter changes on the graph of parent functions.

Let f be the function given by f(x) = x3 - 6x2 + p, where p is an arbitrary constant.

For what values of the constant p does f have 3 distinct real roots? Explain your reasoning.

This question is based on the 1997 AB4 AP* Calculus question.

(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.

(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.

 

 

 

 

 

 

(A) analyze situations involving quadratic functions and formulate quadratic equations or inequalities to solve problems.

Mathetmatics, Algebra II, Figure 1

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.

This question is based on the 1996 AB6 AP Calculus question.

(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 graphs, tables, and algebraic methods.

Consider the curve defined by x2 + xy + y2 = 27. Solve for y in terms of x and use your calculator to graph the curve. Determine the intercepts algebraically and verify graphically.

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.

Graph the region given by the quadratic system 25x2 - 9y2 ≥ 225, x≥0 and x≤8 on a coordinate plane. Rotate the region about the x-axis and sketch the resulting three-dimensional solid.

This introduces the calculus concept of solids of revolution.

Also addresses TEKS 2A.8B and 2A.8C.

(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.

 

 

 

 

 

 

(F) analyze situations modeled by square root functions, formulate equations or inequalities, select a method, and solve problems.

To start this activity you will need a stopwatch, a piece of string at least 1.7 meters long, a ruler, and a weight.

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.

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.

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.

(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 / √(x2 - 4)

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 a park. The city plans to build a brick fence along 3 sides of the park that will cost $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.

This relates to the calculus concept of optimization.

(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, and predict the effects of parameter changes on the graphs of exponential and logarithmic functions, describe limitations on the domains and ranges, and examine asymptotic behavior.

Let f be the function given by f(x) = 2xe2x

Use your graphing calculator to do the following:

a) find all horizontal asymptotes of f(x)

b) locate the absolute minimum value of f

c) determine the domain and range of f

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) = Cekt, where S is measured in billions of gallons per year and t is measured in years from the 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?

This question is based on the 1996 AB/BC3 AP Calculus exam.

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