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REFINED Resources. The resources on this page have been updated and revised to align with the refined K-12 mathematics TEKS. These refined TEKS were adopted by the Texas State Board of Education in 2005–06 and implemented in 2006–07.

(2A.1) Foundations for functions. The student uses properties and attributes of functions and applies functions to problem situations.

(2A.1.A) Foundations for functions. The student uses properties and attributes of functions and applies functions to problem situations. The student is expected to identify the mathematical domains and ranges of functions and determine reasonable domain and range values for continuous and discrete situations.

Clarifying Activity

Students describe the domain and range for the following example:

The diagram shows a door and its position as it swings from closed to open. The width of the door is 120 cm. The distance WC is 150 cm.

Click here for a larger version of this graphic.

The distance d can be regarded as a function of the angle POW. Students state the domain and range of this function and describe values for each that would make sense.

The distance s can be regarded as a function of the angle POW. Students state the domain and range of this function and describe values for each that would make sense.

(2A.1.B) Foundations for functions. The student uses properties and attributes of functions and applies functions to problem situations. The student is expected to collect and organize data, make and interpret scatterplots, fit the graph of a function to the data, interpret the results, and proceed to model, predict, and make decisions and critical judgments.

Clarifying Activity

Students make pendulums by tying a small weight to one end of a 36-inch string. To measure the period of the pendulum, students hold the other end of the string, extend the string, raise the weight, and release.

Students record the time necessary for 10 complete swings back and forth, then divide by ten to obtain the period (the time necessary to complete one swing). Students repeat this process using 6 to 8 different lengths and record the results in a table. For example,

Students make a scatter plot and determine the curve of best fit.

Students use their findings to predict the period of the pendulum for any given string length. Then they make the pendulum to test their prediction.

Note: These data generally produce an exponential curve.

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(2A.2) Foundations for functions. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills to simplify algebraic expressions and solve equations and inequalities in problem situations.

(2A.2.A) Foundations for functions. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills to simplify algebraic expressions and solve equations and inequalities in problem situations. The student is expected to use tools including factoring, and properties of exponents to simplify expressions and to transform and solve equations.

Clarifying Activity

Students use matrices to transform and solve equations.

To predict fuel usage for new delivery routes, Express Delivery conducted a study of the fuel usage of one typical truck. Use the information shown in the table below to find the rates of fuel usage for rush hour driving, city street driving, and freeway driving.

Students create a system of equations to represent the problem situation.

2x + 9y + 3z = 15
7x + 8y + 3z = 24
6x + 18y + 6z = 34

Students use a matrix equation to solve the system.

The truck uses 2 gallons of fuel per hour during rush hour driving, 1 gallon per hour for city street driving, and 2/3 gallon per hour for freeway driving.

Additional Clarifying Activity

Students use properties of exponents to transform and solve equations and inequalities.

(NOTE: This CA is meant to illustrate properties of exponents. See step 2 in the Sample Response.)

A particular type of account earns interest at a rate r = 0.07 = (7%) per year. In other words, if you deposit $1,000, after n years your account will be worth $1,000 (1.07)ⁿ.

In general, at an interest rate of r, after n years your $1,000 will be worth $1,000 (1 + r)ⁿ.

What interest rate r would be required if you want your $1,000 to double after 5 years?

Sample student solution:

The condition in the problem can be expressed in this equation: $1,000 (1 + r)⁵ = $2,000.

Solve for r:

• (1 + r)⁵ = 2 (Dividing both sides by $1,000.)
• (1 + r) = 2^1/5 (Raising both sides to the 1/5 power.)
• r = 2^1/5 - 1 (Subtracting 1 from both sides.)
• r ≈ 0.15 = 15% (Doing the arithmetic.)

(2A.2.B) Foundations for functions. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills to simplify algebraic expressions and solve equations and inequalities in problem situations. The student is expected to use complex numbers to describe the solutions of quadratic equations.

Clarifying Activity

Students calculate the discriminant of quadratic equations, such as:

x² - 2x + 5 = 0

To determine whether there are real or complex roots.

Students then determine the root(s) using the quadratic equation.

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(2A.3) Foundations for functions. 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.

(2A.3.A) Foundations for functions. 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. The student is expected to analyze situations and formulate systems of equations in two or more unknowns or inequalities in two unknowns to solve problems.

Clarifying Activity

Students write a system of equations and/or inequalities to represent a problem situation such as the following:

The U.I.L. academic team is going to the state meet. There are 32 people going on the trip. There are 5 people who can drive and 2 types of vehicles, vans and cars. A van seats 8 people, and a car seats 4 people, including drivers. How many vans and cars does the team need for the trip?

System of equations/inequalities:

Let v = number of vans and c = number of cars

v + c ≤ 5

8v + 4c = 32

(2A.3.B) Foundations for functions. 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. The student is expected to use algebraic methods, graphs, tables, or matrices to solve systems of equations or inequalities.

Clarifying Activity

Students write a system of equations and/or inequalities to represent a problem situation such as the following:

The U.I.L. academic team is going to the state meet. There are 32 people going on the trip. There are 5 people who can drive and 2 types of vehicles, vans and cars. A van seats 8 people, and a car seats 4 people, including drivers. How many vans and cars does the team need for the trip?

System of equations/inequalities:

Let v = number of vans and c = number of cars

v + c ≤ 5

8v + 4c = 32

Students represent the system of equations by putting the equations into slope-intercept form and using graphing technology. Then they solve the system using table, graph, intersection, and tracing features of the technology.

Students solve the system algebraically using a method of their choice: substitution or linear combination.

(2A.3.C) Foundations for functions. 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. The student is expected to interpret and determine the reasonableness of solutions to systems of equations or inequalities for given contexts.

Clarifying Activity

Students discuss when answers to linear systems would not be reasonable.

For example, what answers would not be reasonable for the problem posed in 2A.3.B?

Possible student solution: Negative numbers or fractional answers would not be reasonable answers for the number of cars or vans.

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(2A.4) Algebra and geometry. The student connects algebraic and geometric representations of functions.

(2A.4.A) Algebra and geometry. The student connects algebraic and geometric representations of functions. The student is expected to identify and sketch graphs of parent functions, including linear (f(x) = x), quadratic (f(x) = x²), exponential (f(x) = a^x), and logarithmic (f(x) = log_ax) functions, absolute value of x (f(x) = |x|), square root of x (f(x) = √x), and reciprocal of x (f(x) = 1/x).

Clarifying Activity

Students identify and sketch the parent functions listed below as they learn them. (By the end of the course students should be able to complete the chart below.)

Students also connect each parent function with its inverse function in the chart. The inverses of two of the functions (y = x and y = 1/x) are the functions themselves.

Note: It is not clear in the case of y = a^x what the parent function is. We have chosen to use y = 2^x but one might want to investigate y = 10^x or y = (1/2)^x.

(2A.4.B) Algebra and geometry. The student connects algebraic and geometric representations of functions. The student is expected to 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.

Clarifying Activity

Students use a graphing calculator to graph the parent function y = x. Then, for each equation related to the parent function, students predict what they think the new graph will look like, use the graphing calculator to check their prediction, sketch the graph, and describe the change. Students repeat the process with the parent function y = x². After they investigate examples related to both parent functions, students write general statements about:

• the effects of changing the coefficient of the x² term;
• the effects of changing the coefficient of the x term; and
• the effects of adding or subtracting a constant.

Additional Clarifying Activity

Students use a graphing calculator to identify parameter changes of the graphs of the parent function y = x². Each of the functions shown below is graphed in a calculator standard decimal window (-4.7, 4.7, 1, -3.1, 3.1, 1). Students determine each function.

Answers:

(1) y = x² + 1

(2) y = (x - 2)²

(3) y = (x + 1)² - 2

(4) y = (1/2)x² - 3

(2A.4.C) Algebra and geometry. The student connects algebraic and geometric representations of functions. The student is expected to describe and analyze the relationship between a function and its inverse.

Clarifying Activity

Students recognize an inverse relationship between a pair of functions using tables, graphs, and algebraic representations.

Using a table:

Students note that x and y values are interchanged.

Using a graph:

Students notice that the graphs of two functions that are inverses of each other are reflections across the line y = x.

Using algebraic representation:

Students show that f(g(x)) = x and g(f x)) = x or using different functional notation f(f^-1(x)) = x and f^-1(f x)) = x to prove that the functions are inverses of each other.

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(2A.5) Algebra and geometry. The student knows the relationship between the geometric and algebraic descriptions of conic sections.

(2A.5.A) Algebra and geometry. The student knows the relationship between the geometric and algebraic descriptions of conic sections. The student is expected to describe a conic section as the intersection of a plane and a cone.

Clarifying Activity

Students use models of cones (for example, plastic models with colored water, snow cone cups, or modeling clay and fishing line) to show cross-sections of the cone: circles, ellipses, parabolas, and hyperbolas. Students describe the cuts and relate them to the representations.

(2A.5.B) Algebra and geometry. The student knows the relationship between the geometric and algebraic descriptions of conic sections. The student is expected to sketch graphs of conic sections to relate simple parameter changes in the equation to corresponding changes in the graphs.

Clarifying Activity

Students graph the following equations of circles on graph paper, or using a graphing utility, and complete the table.

Students fill in a similar chart for other conics and make generalization statements about how parameter changes in the equations change the graphs.

(2A.5.C) Algebra and geometry. The student knows the relationship between the geometric and algebraic descriptions of conic sections. The student is expected to identify symmetries from graphs of conic sections.

Clarifying Activity

Students use the symmetry of a conic to help them make its graph. By knowing that the point (1, 2.9) is on the graph of an ellipse centered at the origin, students use their knowledge of symmetry of an ellipse to find three other points on the ellipse.

Solutions: (1, 2.9); (-1, 2.9); (1, -2.9); (-1, -2.9)

(2A.5.D) Algebra and geometry. The student knows the relationship between the geometric and algebraic descriptions of conic sections. The student is expected to identify the conic section from a given equation.

Clarifying Activity

From several examples of various conics in both graph and equation form, students look for patterns and create rules to describe types of equations using characteristics of the general form of a quadratic (conic): Ax² + Bxy + Cy² + Dx + Ey + F = 0. (NOTE: In Algebra II, the quadratics used always have B= 0, so there is no xy term.) For example:

• For a circle: A = C
• For a parabola: Either A or C equals zero, but not both.
• For a non-circular ellipse: A ≠ C, but they have the same sign.
• For a hyperbola: A and C have opposite signs.

After creating a rule, students test their rule with other examples.

(2A.5.E) Algebra and geometry. The student knows the relationship between the geometric and algebraic descriptions of conic sections. The student is expected to use the method of completing the square.

Clarifying Activity

For a simple situation, students model the process of completing the square. For example, to complete the square for the expression x² + ax , students create a model like the one below to show that a term of (a/2)² must be added.

Click here for a larger version of this graphic.

Additional Clarifying Activities

• Students model the process of completing the square for the expression x² + 4x. Students first model the process using algebra blocks.

  

  Students use blocks to form a square.

  

  To complete the square add 4 units.

  

  Then, students model the process of completing the square symbolically for the function x² + 4x.

  

  (The process of completing the square allows us to convert equations in general form to standard form. The standard form provides information that assists us in graphing.)

  Next, students describe the graph of y = (x + 2)² - 4 in comparison to the graph of y = x².

  Possible student solution: The graph of y = x² is shifted 2 units to the left and 4 units down.

• Students use the method of completing the square to translate equations into standard form in order to sketch the conic.

  For example:

  General Form:

  

  Standard Form:

  

  

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(2A.6) Quadratic and square root functions. The student understands that quadratic functions can be represented in different ways and translates among their various representations.

(2A.6.A) Quadratic and square root functions. The student understands that quadratic functions can be represented in different ways and translates among their various representations. The student is expected to 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.

Clarifying Activity

Students describe the domain and range for the following example.

If a ball is thrown upwards from a tower that is 31 meters tall at a velocity of 24 meters per second, the height in meters of the ball above the ground is given by the function:

h(t) = 31 + 24t - 4.9t²

Here, t is the time in seconds since the ball was thrown. The graph of h(t) is given below:

For the interpretation given, what is a reasonable domain and range for the function h(t)?

(From the graph, approximately: domain 0 ≤ t ≤ 6, range 0 ≤ h(t) ≤ 60.)

(2A.6.B) Quadratic and square root functions. The student understands that quadratic functions can be represented in different ways and translates among their various representations. The student is expected to relate representations of quadratic functions, such as algebraic, tabular, graphical, and verbal descriptions.

Clarifying Activity

Students extend the problem in (2A.6.A) to answer questions using multiple representations.

If a ball is thrown upwards from a tower that is 31 meters tall at a velocity of 24 meters per second, the height in meters of the ball above the ground is given by the function:

h(t) = 31 + 24t - 4.9t²

• What is the maximum height achieved by the ball, and at approximately what time does the ball achieve that height?

  (From the graph: h ≈ 60 m and t ≈ 2.4 sec.)

• At what time(s) does the ball reach a height of h = 50 meters? Answer in two ways, directly from the graph, and by solving an equation.

  (From the graph, the times are about t = 1 sec and t = 4 sec. Solving the equation 50 = 31 + 24t - 4.9t² leads first to 4.9t² - 24t + 19 = 0, and then, by the quadratic formula, to ≈ t 0.99 and t ≈ 3.90.)

• Solve the inequality h(t) > 40 graphically, and interpret in terms of the situation.

  (From the graph, h(t) > 40 for, approximately, the values 0.5 ≤ t ≤ 4.5. The meaning is that the ball was at a height greater than 40 meters for about a 4 second interval from about 0.5 sec to about 4.5 sec.)

(2A.6.C) Quadratic and square root functions. The student understands that quadratic functions can be represented in different ways and translates among their various representations. The student is expected to determine a quadratic function from its roots or a graph.

Clarifying Activity

Students determine a quadratic function given the following information:

The roots of a quadratic function are -2/3 and 6. What is the function?

(Answer: y = 3x² - 16x - 12)

Below is the graph of a quadratic function. What is the function?

(Answer: y = -2x² + x + 3)

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(2A.7) Quadratic and square root functions. The student interprets and describes the effects of changes in the parameters of quadratic functions in applied and mathematical situations.

(2A.7.A) Quadratic and square root functions. The student interprets and describes the effects of changes in the parameters of quadratic functions in applied and mathematical situations. The student is expected to 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.

Clarifying Activity

Students use their knowledge of parent functions to answer questions such as the following about the functions represented by:

y₁ = 2(x - 3)² + 1

y₂ = 2x² - 12x + 19

• Justify that the functions are the same using algebraic methods.
• What information about the sketch can you determine from y₁ = 2(x - 3)² + 1?

  (Possible answers: The parabola opens up with a minimum point at (3, 1). The parabola is stretched out compared to the parent function.)

• What information about the sketch can you determine from y₂ = 2x² - 12x + 19?

  (Possible answers: The parabola opens up, is stretched out, and crosses the y-axis at 19.)

• Using information from Question 1 and Question 2, hand sketch a graph of the function, then check using graphing technology.

(2A.7.B) Quadratic and square root functions. The student interprets and describes the effects of changes in the parameters of quadratic functions in applied and mathematical situations. The student is expected to 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.

Clarifying Activity

Students investigate and describe the effects of changes in a, h, and k on the graphs of y = a(x - h)² + k.

For example, students graph the following functions, then complete the chart and discuss the effects of changes in "k" in the graph of y = (x - 2)² + k.

Students do similar investigations to determine the effects of changing "h" and "a".

Given an equation, students predict the effects of changes on the parent function. For example, students predict the changes to the parent function that occur in the following functions:

y = (x + 4)² + 1 (The parent function is shifted left 4 units and up 1 unit.)

y = 5(x - 6)² - 2 (The parent function is stretched by a factor of 5, shifted right 6 units and down 2 units.)

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(2A.8) Quadratic and square root functions. 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.

(2A.8.A) Quadratic and square root functions. 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. The student is expected to analyze situations involving quadratic functions and formulate quadratic equations or inequalities to solve problems.

Clarifying Activity

Students formulate a quadratic equation to solve problems similar to the following:

A box with a square base and no lid is to be made from a square piece of metal by cutting squares from the corners and folding up the sides. The cut-off squares are 5 cm on a side. If the volume of the box is 100 cm³, find the dimensions of the original piece of metal.

(The equation for finding the dimensions is 5(x - 10)(x - 10) = 100.)

(2A.8.B) Quadratic and square root functions. 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. The student is expected to analyze and interpret the solutions of quadratic equations using discriminants and solve quadratic equations using the quadratic formula.

Clarifying Activity

Students use the discriminant to predict the type of roots for a given quadratic equation.

Example: Determine if the following equations will have two real roots, one real root, or no real roots (two imaginary roots).

• 2x² + 8x + 12 = 0 (two imaginary roots since the discriminant is negative, d = -32)
• x² - 4x + 3 = 0 (two real roots since the discriminant is positive, d = 4)
• 4x² + 20x + 25 = 0 (one real root since the discriminant is zero)

Students then solve the equations above using the quadratic formula

and verify the types of roots by examining the graphs.

(2A.8.C) Quadratic and square root functions. 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. The student is expected to compare and translate between algebraic and graphical solutions of quadratic equations.

Clarifying Activity

Students solve the equation x² - 4x + 3 = 0 and find the roots (a) using the graphing calculator and (b) algebraically. Students should note that the solutions they find algebraically are the places where the graph crosses the x axis. (Students should be given several opportunities to investigate this relationship.)

(2A.8.D) Quadratic and square root functions. 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. The student is expected to solve quadratic equations and inequalities using graphs, tables, and algebraic methods.

Clarifying Activities

Students use a variety of ways (and make connections between the methods) to solve a quadratic equation. For example, solve x² - x - 6 = 0 using a variety of methods.

Graphing:

Students use a graphing calculator to graph y = x² - x - 6.

Students trace to find the roots (zeros) of the function. (Some calculators allow students to calculate the roots, use a table feature to locate the roots, or solve by other methods.)

Algebraic:

Students solve the equation x² - x - 6 = 0 by

Factoring:

Completing the square:

Using the quadratic formula:

Additional Clarifying Activity

Students use a variety of ways to solve a quadratic inequality. For example, solve x² - x - 6 < 0 using a variety of methods.

Graphing:

Students use a graphing calculator to graph y = x² - x - 6.

Students trace to find the roots of the function and solve the inequality by inspection of the graph. (Some calculators allow students to calculate the roots, use a table feature to locate the roots, or solve by other methods.) (The solution of the inequality will be the interval consisting of x values for which the graph of the function lies below the x-axis: -2 < x < 3.)

Algebraic:

Students solve the inequality x² - x - 6 < 0 by

Factoring:

(x + 2)(x - 3) < 0

Then setting up the following cases and solving:

CASE 1: (x + 2) < 0 and (x - 3) > 0

CASE 2: (x + 2) > 0 and (x - 3) < 0

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(2A.9) Quadratic and square root functions. 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.

(2A.9.A) Quadratic and square root functions. 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. The student is expected to 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.

Clarifying Activity

Students investigate and describe the effects of changes in "h" and "k" in the equation .

For example:

Students use a graphing calculator to investigate the following functions. Students then complete the chart and discuss how changes in "h" affect the graph of .

Students answer the following question: How does changing the value of "h" in affect the parent function ?

Students use a graphing calculator to investigate the following functions. Students then complete the following chart and discuss how changes in "k" affect the graph of .

Students answer the following question: How does changing the value of "k" in affect the parent function ?

Given a function, students predict the effects of changes on the parent function. For example, students predict the changes to the parent function that occur in the following functions:

(The parent function is shifted left 4 units and up 1 unit.)

(The parent function is stretched by a factor of 5, shifted right 6 units and down 2 units.)

(2A.9.B) Quadratic and square root functions. 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. The student is expected to relate representations of square root functions, such as algebraic, tabular, graphical, and verbal descriptions.

Clarifying Activity

Students investigate with several representations. Responding to the problem, students

• describe the function based on the parent function (verbal description);

  (The parent function is the square root function, shifted 4 to the right and 2 up.)

• find the domain and range from the graph (graphical); and

  (Domain: {x: x ≥ 4}; Range: {y: y ≥ 2})

• explain what "error" means for certain values in a graphing calculator table (tabular).

  

  (Students can check values algebraically and make connections with domain and range from the graph.)

(2A.9.C) Quadratic and square root functions. 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. The student is expected to 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.

Clarifying Activity

Students formulate a square root function to solve problems similar to the following:

If an object is dropped from a height "c" in meters, the height of the object after "t" seconds is given by the function h(t) = -4.9t² + c. Formulate a function to determine how long it will take the object to reach the ground from a given height "c". Describe the domain and range for the new function.

Determining reasonable values for domain:

For h(t) = 0 , . The domain represents the height from which the object will be dropped. Therefore, the values of the domain must be positive. Domain: c > 0

Determining reasonable values for range:

Since the range represents time, the values of the range must be positive. Range: t > 0

(2A.9.D) Quadratic and square root functions. 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. The student is expected to determine solutions of square root equations using graphs, tables, and algebraic methods.

Clarifying Activity

Students use a variety of ways to solve a square-root equation such as

Graphing:

Students use a graphing calculator to graph simultaneously the functions and Y₂ = 3. Students can trace to find the point of intersection of the two functions. (Some calculators allow students to calculate a point of intersection, use a table feature to locate a point of intersection, or solve by other methods.)

Algebraic:

Students solve the equation by

• squaring both sides: 2x - 1 = 9
• solving for x: x = 5
• checking:

Students use the same methods described with equalities to solve square-root inequalities such as:

(2A.9.E) Quadratic and square root functions. 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. The student is expected to determine solutions of square root inequalities using graphs and tables.

Clarifying Activity

Tsunami waves are often created by earthquakes deep within the ocean floor. The speed of a tsunami wave depends on the depth of the ocean. The square root function s = 3.1√d represents this relationship, where d is the depth in meters of the ocean and s is the speed in meters per second of the wave.

Using a graph and table, students determine the possible ocean depths that would produce a tsunami with speed greater than 200 m/s. Students explain how they can use their table to support the findings in the graph and how they can use their graph to support the findings in the table.

(2A.9.F) Quadratic and square root functions. 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. The student is expected to analyze situations modeled by square root functions, formulate equations or inequalities, select a method, and solve problems.

Clarifying Activity

Students formulate a square-root equation to solve problems similar to the following:

For a pendulum, the period (T) in seconds is given by where L is the length in meters and g is the acceleration due to gravity (approximately 9.8 m/sec².) Glenn is swinging back and forth on the end of a rope swing. He sees the neighbor's dog leaping up at him from the ground below. He quickly scrambles halfway up the rope to safety. Relaxing, he finds that the period of his swing is one second shorter now that his "pendulum" is half its original length. How long is the rope swing?

While the algebra is extremely tedious in this problem, a solution can be easily found using graphing technology as the intersection of two functions. Let and

Answer: about 2.9 meters

(2A.9.G) Quadratic and square root functions. 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. The student is expected to connect inverses of square root functions with quadratic functions.

Clarifying Activities

Students determine the inverse of the following quadratic function, f(x) = x² + 2. Students discuss with the teacher and each other that in the original function, f(x) = x² + 2, the value of x is squared and then 2 units are added. The inverse of those operations would involve subtracting 2 from x and then taking the square root, thus forming the inverse,

Algebraically:

f(x) = x² + 2

y = x² + 2

x = y² + 2

x - 2 = y²

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(2A.10) Rational functions. 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.

(2A.10.A) Rational functions. 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. The student is expected to 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.

Clarifying Activity

Given a rational function , students investigate the function using graph and algebra to answer the following:

• For what value of x is the denominator equal to zero? (x = 2)
• State the domain of the function. {x: x ≠ 2}
• State the range of the function. (All real numbers except 0)
• Complete the tables below to investigate asymptotic behavior.

  

  

  As x approaches 2 from the left , f(x) approaches ______________.

  As x approaches 2 from the right , f(x) approaches _______________.

(2A.10.B) Rational functions. 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. The student is expected to analyze various representations of rational functions with respect to problem situations.

Clarifying Activity

Students are given the following situation:

Liz plays basketball on the high school basketball team. She has made 18 of her last 30 free throws.

Algebraic:

Students write a function to model Liz's percentage if she makes "x" more consecutive free throws.

Answer:

Tabular:

Students make a table showing the relationship between the number of free throws made and the number shot.

Graphical:

Students use a graphing calculator to make a scatter plot of the data on a graph and compare the graph of the data to the graph of the original function,

(2A.10.C) Rational functions. 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. The student is expected to 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.

Clarifying Activity

Students formulate a rational function for a problem similar to the following one and determine a reasonable domain and range.

Liz plays basketball on the high school basketball team. She has made 18 of her last 30 free throws. Write a function to model her percentage if she makes "x" more consecutive free throws. Describe the domain and range for your new function.

Formulating a rational function:

Determining reasonable values for domain:

Since the domain represents the number of free throws, the values of the domain must be positive integers.

Determining reasonable values for range:

Since the range represents the percentage of her free throws after completing 18 out of 30, the values of the range must be positive and must be greater than .

Determining reasonableness of solutions:

Will Liz's free throw percentage ever be less than 50%? Will it ever be equal to 100%? Why or why not?

(2A.10.D) Rational functions. 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. The student is expected to determine the solutions of rational equations using graphs, tables, and algebraic methods.

Clarifying Activity

Partway through a course, S has earned 130 points out of a possible 325.

• If there are going to be 175 more points that can be earned, what is S's highest possible percentage?

  If S earns all 175 points, the percentage will be (100) () = 61%.

• If there are going to be 275 more points that can be earned, what is S's highest possible percentage?

  If S earns all 275 points, the percentage will be (100) () = 67.5%.

• In general, suppose there are going to be x more points that can be earned. Express the highest possible percentage P(x) in terms of x and graph this relationship.

  If S earns all x points, the percentage will be P(x) = (100) (). See Question 4 for the graph.

• How many more points must be available if S is to be able to get a final score of 80%? Answer this question both using algebra and from a graph.

  If x more points are available, and if S earns x points, the percentage will be P(x) = (100) (). Setting this equal to 80% gives the equation (100) () = 80. Solving this equation for x gives x = 650.

  The graph of P(x) from Question 3 is below. The lines show that a percentage of 80% requires 650 more points, in accords with the solution just obtained.

  

Additional Clarifying Activity

Students use a variety of ways to solve rational equations such as:

Graphing:

Students use a graphing calculator to graph simultaneously the functions and . Students trace to find the point of intersection of the two functions. (Some calculators allow students to calculate a point of intersection, use a table feature to locate a point of intersection, or solve by other methods.)

Algebraic:

Students multiply both sides by the least common denominator to solve for x. {x = 4}.

(2A.10.E) Rational functions. 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. The student is expected to determine solutions of rational inequalities using graphs and tables.

Clarifying Activity

A manufacturer wants to design a cylindrical soda can that will hold 500 milliliters (mL) of soda. The relationship between the height of the can in centimeters and the radius of the can in centimeters is . The manufacturer's research has determined that an optimal can height is between 10 and 15 centimeters. Use a graph and a table to determine the possible range of radius measurements. Explain how the information in one supports the information in the other.

(2A.10.F) Rational functions. 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. The student is expected to analyze a situation modeled by a rational fuction, formulate an equation or inequality composed of a linear or quadratic function, and solve the problem.

Clarifying Activity

Students formulate a rational equation for a problem similar to the following one and determine a solution.

Liz plays basketball on the high school basketball team. She has made 18 of her last 30 free throws. How many consecutive free throws does she need to make to raise her percentage to 75%?

Formulating a rational equation:

(Using the method of their choice, students should find the solution to be 18 more free throws.)

(2A.10.G) Rational functions. 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. The student is expected to use functions to model and make predictions in problem situations involving direct and inverse variation.

Clarifying Activity

Students solve direct and inverse variation problems similar to the following ones:

Direct variation:

The amount of chlorine needed varies directly as the size of the pool. If 5 units of chlorine is the amount needed for 550 gallons of water, write a function representing the amount of chlorine needed for "x" gallons of water. Predict the amount of chlorine needed for 850 gallons of water.

Indirect variation:

The equation , models the amount of light "E" provided by a 50 watt light bulb, where E (measured in lux) depends on the distance "d" meters from the bulb (for d ≥ 1 meter). Find the amount of light on a desk 3 meters away from the light source. (about 6 lux)

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(2A.11) Exponential and logarithmic functions. 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.

(2A.11.A) Exponential and logarithmic functions. 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. The student is expected to develop the definition of logarithms by exploring and describing the relationship between exponential functions and their inverses.

Clarifying Activity

Students work through an activity like the following one to see the mathematical relationship between exponential and logarithmic functions.

Students complete Table 1 for f(x) = 2^x. Using blue ink, students graph the exponential function f(x) = 2^x by plotting the Table 1 values. Then students reverse the coordinates in each ordered pair and enter them in Table 2. They plot these points in the same coordinate plane using red ink. Students plot the line y = x with pencil and study the relationships between the graphs.

Students need to note that the two graphs above are inverse functions. One graph is a reflection of the other across the line y = x.

Students need to know that the inverse of, the exponential function with base 2, is called the logarithmic function with base 2.

Students find the value of each log:

Students recognize that the base can be any positive number except 1 (because 1^? = 1 --- any power of 1 is 1).

Students connect their exploration to the following definition:

Definition of Logarithm

If b and N are positive numbers (b ≠ 1), log_bN = k if and only if b^k = N.

(2A.11.B) Exponential and logarithmic functions. 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. The student is expected to 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.

Clarifying Activity

Students investigate and describe the effects of changes in "h" and "k" in the equation y = b^x-h + k. For example, students use graphing calculators to investigate the parent function in the table below. Students then complete a similar table, as is shown below, and discuss how changes in "h" affect the graph of y= 2^x-h.

Students answer the following question:

How does changing the value of "h" in y = 2^x-h affect the parent function y = 2^x?

Students use graphing calculators to investigate the same parent function in a table similar to the one below. Students then complete the table and discuss how changes in "k" affect the graph of y = 2^x + k.

Students answer the following question:

How does changing the value of "k" in y = 2^x + k affect the parent function y = 2^x?

Given a function, students predict the effects of changes on the parent function. For example, students predict the changes to the parent function that occur in the following functions:

• y = 2^x+4 + 1 (The parent function is shifted left 4 units and up 1 unit.)
• y = 2^x-6 - 2 (The parent function is shifted right 6 units and down 2 units.)

Students complete tables like the one below to investigate asymptotic behavior.

As x approaches negative infinity, f(x) approaches ______________.

Students use a similar approach to investigate changes in the parameters and asymptotic behavior of a logarithmic function.

(2A.11.C) Exponential and logarithmic functions. 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. The student is expected to 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.

Clarifying Activity

Students formulate an exponential function for a problem similar to the following one and determine a reasonable domain and range.

The NCAA holds a championship basketball game each spring. The nation's top 64 teams in Division 1 are invited to play. When a team loses, it is out of the tournament. Describe the domain and range for a function that determines the number of teams left in the tournament f(x) after round "x":

Determining reasonable values for domain:

Since the domain represents the round, the values of the domain must be positive integers.

Determining reasonable values for range:

Since the range represents the number of teams left in the tournament, the values of the range must be positive even integers less than 64.

(2A.11.D) Exponential and logarithmic functions. 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. The student is expected to determine solutions of exponential and logarithmic equations using graphs, tables, and algebraic methods.

Clarifying Activity

Students use a variety of ways to solve exponential and logarithmic equations.

Example: 25^2x = 5^x+6

Graphing:

Students use a graphing calculator to graph simultaneously the functions Y₁ = 25^2x and Y₂ = 5^x+6. Students trace to find the point of intersection of the two functions. (Some calculators allow students to calculate a point of intersection, use a table feature to locate a point of intersection, or solve by other methods.)

Algebraic:

25^2x = 5^x+6

5^4x = 5^x+6

So, 4x = x + 6 and x = 2.

Example: 3^x = 30

Graphing:

Students use a graphing calculator to graph simultaneously the functions Y₁ = 3^x and Y₂ = 30. Students trace to find the point of intersection of the two functions. (Some calculators allow students to calculate a point of intersection, use a table feature to locate a point of intersection, or solve by other methods.

Algebraic:

3^x = 30

log 3^x = log 30

xlog3 = log 30

x ≈ 3.096

(2A.11.E) Exponential and logarithmic functions. 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. The student is expected to determine solutions of exponential and logarithmic inequalities using graphs and tables.

Clarifying Activity

Two friends, Toni and Maria, buy new cars on the same day. Toni purchases her car for $20,000. Her car's value will depreciate 8% per year. Maria buys a car with an original price of $16,000. Maria's car will depreciate only 7% per year. Use tables and graphs to determine how many years Toni's car will be more valuable than Maria's. Explain how you used the table and the graph to determine your solution.

(2A.11.F) Exponential and logarithmic functions. 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. The student is expected to analyze a situation modeled by an exponential function, formulate an equation or inequality, and solve the problem.

Clarifying Activity

Students formulate exponential equations for problems similar to the following and determine solutions.

Example 1: The NCAA holds a championship basketball tournament each spring. The nation's top 64 teams in Division 1 are invited to play. When a team loses, it is out of the tournament. Determine the number of teams left in the tournament after Round 5.

Students formulate an exponential equation and show how they used it to determine their answer. Using the equation below and a graphing calculator, student can determine a table of values or a graph and trace to answer the question.

Students may also use algebraic manipulation of the equation. Using the method of their choice, students determine that there will be 2 teams left in the tournament after Round 5.

Example 2: A biologist is observing a strain of bacteria growing in a petri dish. The population of the bacteria increases exponentially over time. If the first bacterium took 1 hour to divide into two bacteria, how long will it take for 30 bacteria in the dish to grow to at least 1,000?

Students formulate an exponential equation and show how they used it to determine their answer.

Using p(t) = 30(2)^t where t = hours and p = number of bacteria, students may use a graphing calculator to determine a table of values or a graph and trace to answer the question. Students may also use algebraic manipulation of the equation. Using the method of their choice, students determine that 1,000 ≤ 30(2)^t yields t ≥ 5.06 hours.)

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