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EDITED Resources. The resources on this page have been aligned with the revised K-12 mathematics TEKS. Necessary updates to the resources are in progress and will be completed Fall 2006. These revised TEKS were adopted by the Texas State Board of Education in 2005–06, with full implementation scheduled for 2006–07.

(P.1) The student defines functions, describes characteristics of functions, and translates among verbal, numerical, graphical, and symbolic representations of functions, including polynomial, rational, power (including radical), exponential, logarithmic, trigonometric and piece-wise defined functions.

(P.1.A) The student defines functions, describes characteristics of functions, and translates among verbal, numerical, graphical, and symbolic representations of functions, including polynomial, rational, power (including radical), exponential, logarithmic, trigonometric and piece-wise defined functions. The student is expected to describe parent functions symbolically and graphically, including
f(x) = xⁿ,
f(x) = ln x,
f(x) = log_ax,
f(x) = 1/x,
f(x) = e^x,
f(x) = |x|,
f(x) = a^x
f(x) = sin x
f(x) = arcsin x, etc.

Clarifying Activity

Students match the graphs below with possible functions from the list given.

1. y = 1/x²
2. y = x³
3. y = 3^x
4. y = sin x
5. y = log x
6. y = 1/x
7. y =
8. y = (1/3)x
9. y = |x|
10. y = cos x
11. . y = x⁴
12. 

After students match the graphs to the functions above, they discuss their reasoning. Students discuss the following questions to clarify their thinking.

• How would you distinguish between the graphs of the following pairs of functions?
  1. y = x⁴ and y = x³
  2. y = 3^x and y = (1/3)^x
  3. y = 1/x and y = (1/3)^x
  4. y = x⁴ and y = | x |
  5. y = log x and y =
  6. y = sin x and y = cos x
  7. y = 1/x and y = (1/x)²
• Describe general characteristics of each parent function in the list, especially the characteristics that help you distinguish among the graphs of the functions.

Sample Student Response: The exponential functions are always positive and have the x-axis as an asymptote.

(P.1.B) The student defines functions, describes characteristics of functions, and translates among verbal, numerical, graphical, and symbolic representations of functions, including polynomial, rational, power (including radical), exponential, logarithmic, trigonometric and piece-wise defined functions. The student is expected to determine the domain and range of functions using graphs, tables, and symbols.

Clarifying Activity

Students create a table of values and a graph for a rational function such as:

Students find the domain and range of the function by examining the table, graph and symbolic form of the function. They discuss the benefits of each representation for finding the domain and range.

Additional Clarifying Activity

Students consider the following parametric equations for a curve, C:

1. x = t; y = t²
2. x= t²; y = t⁴
3. x = (1 - t²)^1/2; y = 1 - t²
4. x = t^-1; y = t^-2
5. x = ; y = t

1. For each curve, describe restrictions, if any, on the parameter, t.
2. For each curve, describe its domain and range.
3. Use a graphing calculator in parametric mode to graph each curve in the xy-plane. Consider your responses to Questions 1 and 2 in setting your viewing window. Make a sketch of each graph on paper.
4. Compare the different curves.
5. For which curves is y a function of x? Represent these functions symbolically. Compare the domains and ranges of these functions with your results in Question 3.

For best results, students should use as tstep the default value (e.g., π/24). Suggested windows are:

1. -10 ≤ t ≤ 10, -10 ≤ x ≤ 10, -1 ≤ y ≤ 100
2. Same as (a)
3. -1 ≤ t ≤ 1, 0 ≤ x ≤ 2, -1 ≤ y ≤ 2
4. Same as (a)
5. Same as (a)

(P.1.C) The student defines functions, describes characteristics of functions, and translates among verbal, numerical, graphical, and symbolic representations of functions, including polynomial, rational, power (including radical), exponential, logarithmic, trigonometric and piece-wise defined functions. The student is expected to describe symmetry of graphs of even and odd functions.

Clarifying Activity

After a discussion of the definitions of symmetry and even and odd functions, students are given the partial graph of a function—for example,

—and engage in the following tasks:

• Complete the graph so that it is the graph of an even function. Explain why it is the graph of an even function.
• Complete the graph so that it is the graph of an odd function. Explain why it is the graph of an odd function.
• How would you describe an even function algebraically? How would you describe an odd function algebraically?

Expected Responses: Students examine the even function graphs and show how a function is even on the graph with f(x) = f(-x). Students look at the odd function graph, mark a point (x,f(x)), find the point which they know is on the graph because the function is odd, and name the coordinates of this point. For an odd function, f(-x) = -f(x).

Additional Clarifying Activity

Students examine the graphs of the functions given in Clarifying Activity P.1.A to determine if the functions are even, odd, or neither. Students verify their conclusions using a table and algebraically. They describe the connections between symmetry of the graph of the function, entries in the table of values for the function, and the function's symbolic form.

(P.1.D) The student defines functions, describes characteristics of functions, and translates among verbal, numerical, graphical, and symbolic representations of functions, including polynomial, rational, power (including radical), exponential, logarithmic, trigonometric and piece-wise defined functions. The student is expected to recognize and use connections among significant values of a function (zeros, maximum values, minimum values, etc.), points on the graph of a function, and the symbolic representation of a function.

Clarifying Activity

Students algebraically determine the roots of the following function

f(x) = 0.5x(x + 2)(x - 4)

and verify them with a graphing calculator, using a "friendly window." Then students mentally estimate the coordinates of the maximum and minimum points that occur between consecutive roots and refine their estimates by using the trace feature of the calculator. Students use their understanding of the continuity of this function to explain why the function must have a minimum or a maximum point between consecutive roots.

Additional Clarifying Activity

Students:

• write a cubic function with roots at x = -1/2, 4, 7.

  f(x) = _____________________

• define three other cubic functions with these same roots.
• for each of the four cubic functions, estimate the coordinates of the maximum and minimum points occurring between consecutive roots.
• discuss the coordinates of the maximum or minimum point occurring between two consecutive roots and what parameter in (attribute of) the cubic function causes the maximum (or minimum) point between two consecutive roots to vary.

Expected Response: Students notice that other cubic functions with roots at x = -1/2, 4, and 7 are found by modifying the basic cubic function y = (x + 1/2)(x - 4)(x - 7) with a scalar, k, to form the family of functions f(x) = k(x + 1/2)(x - 4)(x - 7). Students should reach the conclusion that, as the cubic is modified by varying the value of the scalar, k, the x-coordinate of a maximum or minimum point does not change but the value of the y-coordinate of that point varies. They also should see the connection between the value of the scalar, k, and the way in which the y-coordinate varies.

(P.1.E) The student defines functions, describes characteristics of functions, and translates among verbal, numerical, graphical, and symbolic representations of functions, including polynomial, rational, power (including radical), exponential, logarithmic, trigonometric and piece-wise defined functions. The student is expected to investigate the concepts of continuity, end behavior, asymptotes, and limits and connect these characteristics to functions represented graphically and numerically.

Clarifying Activity

Students discuss the continuity and/or discontinuity of the following graphs:

For each graph, students give two intervals on which the graph is continuous and two intervals on which the graph is discontinuous (if applicable).

Students then investigate the continuity/discontinuity of the following functions:

1. 
2. 
3. 
4. f(x) = x³ - 2x² + 3x - 4
5. 
6. 
7. f(x) = sin()

through engagement in the following tasks:

• For each function, predict from its symbolic form where it is discontinuous.
• Use a calculator to generate a table of values for and a graph of each function. What do the table and the graph tell you about the continuity of the function?
• For each function give two intervals on which the function is continuous.
• For each function which has discontinuities give an interval on which that function is discontinuous.

Additional Clarifying Activity

Students use calculators to generate a table of values and the graph of the following rational function

and answer the following questions:

1. Describe the domain and range of the function.
2. Describe the continuity of the function.
3. As x approaches -2, what happens to f(x)?
4. What is the end-behavior of the function, i.e., as x approaches ±∞, what happens to f(x)?
5. What asymptotes does the graph of f have? How are the asymptotes related to your responses to Questions 1–4 above?

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(P.2) The student interprets the meaning of the symbolic representations of functions and operations on functions to solve meaningful problems.

(P.2.A) The student interprets the meaning of the symbolic representations of functions and operations on functions within a context. The student is expected to apply basic transformations, including a f(x), f(x)+ d, f(x - c), f(bx), and compositions with absolute value functions, including |f(x)|, and f(|x|), to the parent functions.

Clarifying Activity

Students are given six copies of the following "parent function" graph:

Students graph one of each of the following functions on each of the copies of the "parent function" graph.

1. y = 3 f(x)
2. y = -3 f(x)
3. y = f(x - 3) and y = f(x + 3)
4. y = f(x) + 3 and y = f(x) - 3
5. y = f(|x|)
6. y = |f(x)|

For each of the six graphs comparing the "parent function" to one of the other functions, students summarize how the parent function has been transformed, using correct mathematical descriptors such as translation, reflection, stretch/shrink.

Additional Clarifying Activity

Students analyze the following graph of a sinusoidal function with parent function y = sin x:

• Determine a function, y = a sin (b(x - c)) + d, that this graph represents. Describe what effect the parameters a, b, c and d have on the parent function, y = sin x.
• If the given graph is shifted p units to the left, what function would describe it? Justify your response.

(P.2.B) The student interprets the meaning of the symbolic representations of functions to solve meaningful problems. The student is expected to perform operations including composition on functions, find inverses, and describe these procedures and results verbally, numerically, symbolically, and graphically.

Clarifying Activity

Students answer the following questions:

1. Given f(x) = sin x. Describe its domain and range.

   Choose one of the following expressions for g(x):

   • 2/x
   • x²
   • 

   Describe the domain and range of g(x), based on your choice.

2. Determine the composite function f(g(x)). Describe its domain and range.
3. Determine the composite function g(f(x)). Describe its domain and range.
4. Verify your results to Questions 2 and 3 using the table and/or graphing features of your calculator.
5. Compare the composite functions f(g(x)) and g(f(x)) for the functions you chose for f and g.

Additional Clarifying Activity

Students consider the function f(x) = . Then they graph the function and its inverse using parametric equations. (Possible settings: [-4.5, 4.5] by [-3,3], Tmin = -3, Tmax = 3, tstep = 0.1; x_1T = t , y_1T = , x_2T = y_1T , y_2T = x_1T , x_3T = t , y_3T = t.)

Students use the graph and tables to determine the domain and range of a function and its inverse; trace along the graphs; move from one graph to the other and note the coordinates; and describe how they know that the functions are inverses.

Students may also graph y = x on the same grid, describe the symmetry, and describe the restricted domain and range.

Additional Clarifying Activity

Students determine symbolically, the inverse, f^-1, of the following rational function:

Students explore the behavior of f and f^-1 using the table and graphing features of their calculators; compare the domains, ranges, and intercepts of these functions; and compare the continuity, asymptotes, and end-behavior of these functions.

(P.2.C) The student interprets the meaning of the symbolic representations of functions to solve meaningful problems. The student is expected to investigate identities graphically and verify them symbolically, including logarithmic properties, trigonometric identities, and exponential properties.

Clarifying Activity

Students use a graphing calculator to complete the table and graphs of the functions in each given pair. Students use both the table and graph as evidence to decide if y₁ = y₂. If y₁ and y₂ appear to be equal, they verify the equality symbolically.

1. y₁ = ln (3x), y₂ = 3 ln(x)
2. y₁ = ln(3x), y₂ = ln(3) + ln(x)
3. y₁ = sin(2x), y₂ = 2 sin(x)
4. y₁ = sin(2x), y₂ = 2sin(x)cos(x)
5. y₁ = 1 + x, y₂ = e^x
6. y₁ = 5x, y₂ = 10^{(log (5x))}

Students should identify the intervals for which y₁ = y₂. Students also should recognize that, in Question 6, y₁ is undefined for x ≤ 0, but y₁ = y₂ for x > 0.

As an extension, students should graph y₃ = y₁ - y₂. Also, students should use their table to check values of y₃ to verify whether y₁ = y₂.

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(P.3) The student uses functions and their properties, tools and technology, to model and solve real-life problems.

(P.3.A) The student uses functions and their properties, tools and technology, to model and solve meaningful problems. The student is expected to investigate properties of trigonometric and polynomial functions.

(P.3.B) The student uses functions and their properties, tools and technology, to model and solve meaningful problems. The student is expected to use functions such as logarithmic, exponential, trigonometric, polynomial, etc. to model real-life data.

Clarifying Activity

The students derive, interpret, and apply the sinusoidal model

f(x) = a sin (b(x - c)) + d

in the following problem (Source: Foerster, Paul. Precalculus with Trigonometry. Addison-Wesley, 1986, pg. 62)

Suppose that you are on the beach at Port Aransas, Texas. At 2:00 PM on March 19, the tide is in (i.e., the water is at its deepest). At that time you find that depth of the water at the end of the pier is 1.5 m. At 8:00 PM the same day when the tide is out, you find that the depth of the water is 1.1 m. Assume the depth of the water varies sinusoidally with time.

1. Derive the sinusoidal model for this problem expressing the depth of the water in terms of the number of hours since 12:00 noon on March 19. Explain the meaning of the parameters a, b, c, and d in the model.
2. Use the model to predict the depth of the water at 4:00 PM on March 19, 7:00 AM on March 20, and 5:00 PM on March 20.
3. At what time will the first low tide occur on March 20?
4. What is the earliest time on March 20 that the water will be 1.27 m deep?

As an extension, students can research the connection between tidal flow and models for predictions of global warming.

Additional Clarifying Activity

Students determine an exponential model to fit data on space debris that is predicted to grow at a fixed percent per year:

A = A₀ (1 + r)^t

where A₀ = the initial amount, r = the annual growth rate, t = the number of years, and A = the accumulated amount of debris.

Space Debris Accumulation

The following table gives the accumulation of space debris in millions of pounds, with the amount of debris increasing at a fixed percent (10% or 20%) per year. In 1990, scientists estimated that a total of 4 million pounds of debris was in Earth orbit. (Source: NCTM 1997. Mission Mathematics, Grades 9-12, pg. 24)

Students determine the models for 10% annual growth and 20% annual growth symbolically and verify their models by using the STAT plot and Reg. features of their graphing calculators.

Students use their models to compare the amount of accumulated space debris in the years 2000, 2010, 2020, etc.

Students investigate doubling time for each of the models.

As an extension, students could investigate what would happen if space debris stopped accumulating in a certain year (e.g., 2005) and began to deplete by a certain amount (e.g., 15% per year).

(P.3.C) The student uses functions and their properties, tools and technology, to model and solve meaningful problems. The student is expected to use regression to determine the appropriateness of a linear function to model real-life data (including using technology to determine the correlation coefficient).

Clarifying Activity

Students use technology to fit regression models to collected data (or data provided) that has been generated in a real-life situation. For example:

National Health Expenditures

(Source: Larson, Roland. Precalculus with Limits, 2nd Ed. Houghton Mifflin, 1997. Pg. 300)

The table below gives the national health expenditures, y (in billions of dollars) for the years 1982 through 1991. The years are given by x where x = 2 corresponds to 1982.

• Use a graphing calculator to plot the data.
• Use the regression feature of your calculator to fit an exponential model to the data.
• Use the regression feature of your calculator to fit a logarithmic model to the data.
• Which model, exponential or logarithmic, appears to best describe the data?
• For both models, predict health expenditures for 2000, 2010, 2025.
• If the rate of growth of health care costs could be slowed, which model may be better for the future? Explain.

Chemical Reaction

(Based on Problem 102, pg. 306, in Larson's Precalculus with Limits, 2nd Ed. Houghton Mifflin, 1997.)

The data in the table gives the yield, y (in milligrams), of a chemical reaction after t minutes.

• Use a graphing calculator to plot the data.
• Use the calculator to fit a linear model to the data.
• Use the calculator to fit a logarithmic model to the data.
• Which is a better model of the data? Explain your reasoning.

(P.3.D) The student uses functions and their properties, tools and technology, to model and solve meaningful problems. The student is expected to use properties of functions to analyze and solve problems and make predictions.

Clarifying Activity

Students use functions to analyze the following set of data and answer questions:

The San Antonio City Public Service provided data on monthly use of electricity from January 1996 to September 1997.

The data was plotted and the following sinusoidal model of the data was determined. In the model, x represents the number of months since August '96, and y represents monthly electricity use in megawatt hours.

y = 1403963.5 + 420897 cos [(π/6)x]

• Explain the meaning of the constants in this sinusoidal model of monthly electricity use.
• Use the model to predict when the electricity use is least and when it is greatest.
• If the population of the city increased or decreased, how might the model change?
• If energy conservation measures restricted summer use of air conditioning, how might the model change?

(P.3.E) The student uses functions and their properties, tools and technology, to model and solve meaningful problems. The student is expected to solve problems from physical situations using trigonometry, including the use of Law of Sines, Law of Cosines, and area formulas and incorporate radian measure where needed.

Clarifying Activity

Students apply the Law of Sines to solve the following problem:

A forest ranger at an observation point A along a straight road sights a fire in the direction 32ƒ east of north. Another ranger at a second observation point B 10 miles due east of point A on the road sights the same fire 48ƒ west of north.

• Sketch the situation, including the given information on the sketch.
• Find the distance from each observation point to the fire.
• Fire fighters must proceed by foot from the road to the fire. Find the shortest distance from the road to the fire.

(Source: Demana, F., Waits, B. and Clemens, S. Precalculus Mathematics. Addison-Wesley, 1994, p. 433.)

Additional Clarifying Activity

Students apply the Law of Cosines to solve the following problem:

A 100-foot vertical tower is to be erected on the side of a hill that makes a 6ƒ angle with the horizontal. The tower will be secured with guy wires from the top of the tower to the ground 75 feet uphill from the base of the tower and 75 feet downhill from the base of the tower.

• Sketch the situation, including the given information on the sketch.
• Find the length of the guy wires that secure the tower.

(Source: Larson, Roland. Precalculus with Limits, 2nd Ed. Houghton Mifflin, 1997, p. 475.)

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(P.4) The student uses sequences and series as well as tools and technology to represent, analyze, and solve real-life problems.

(P.4.A) The student uses sequences and series as well as tools and technology to represent, analyze, and solve real-life problems. The student is expected to represent patterns using arithmetic and geometric sequences and series.

Clarifying Activity

Students create the Sierpinski carpet using the following procedure:

Begin with a sheet of graph paper and draw a large square (where the length of the sides is easily divisible by three). Subdivide the square into nine equal subsquares. Shade in the "middle" square as shown above. Shading represents removing this area from the original square. Next, subdivide the remaining non-shaded squares into nine equal subsquares and shade in the "middle" square of each of these. This process continues to form the Sierpinski Carpet.

(Reference: Peitgen, Heinz-Otto et al. Fractals for the Classroom, Strategic Activities, Vol. I. Springer-Verlag, 1991, pp. 15-16.)

Students analyze and represent the patterns in the Sierpinski carpet by working on the following tasks:

1. Write as a sequence the area that remains at each stage of removal. Assume that the area of the original square (removal stage n = 0) is 1. Give a symbolic representation of the general term, a_n.

   

2. Write as a sequence the area removed at each stage. Give a symbolic representation of the general term b_n, the area removed to get to stage n.

   

3. Describe the type of sequence in Question 1 and in Question 2. How do they compare? Write an equation that shows the relationship between a_n and b_n and interpret this equation in terms of the situation.
4. Write a series that gives the total area removed from the original square in the first n stages. What does the sum of this series approach as n becomes large? Interpret this in terms of the situation.

Solutions:

1. At every stage there are a certain number of white squares remaining. To get to the next stage, of each of these squares is removed. This leaves of the area of the previous stage. The sequence of areas remaining is thus:

   

2. By question 1 the total area remaining at stage n-1 is (8/9)^n-1. To get to stage n we need to remove 1/9 of this. So, the amount removed at stage n is (1/9)(8/9)^n-1.

   

3. The sequence {a_n} is a geometric sequence with a₀ = 1 and ratio 8/9. The sequence {b_n} is a geometric sequence with b₁ = 1/9 and ratio 8/9. The relationship between the sequences is b_n = (1/9)a_n-1. The interpretation is that we need to remove 1/9 of the (n-1)st stage to get to the nth stage.
4. The series is the sum of the first n stages of {b_n}. This is given by

   

   From the formula for the sum of a geometric series, the sum of the first n stages is

   

   As n becomes large, the term (8/9)ⁿ becomes very small. Hence the total area removed approaches 1, the area of the original square.

   (However, not every point is removed! See Peitgen, listed above, for more information.)

(P.4.B) The student uses sequences and series as well as tools and technology to represent, analyze, and solve real-life problems. The student is expected to use arithmetic, geometric and other sequences and series to solve real-life problems.

Clarifying Activity

Students are given the following situation.

An auditorium has 16 seats in the first row, 18 seats in the second row, 20 seats in the third row and so on. How many seats are in the first 5 rows?

Solution:

Method 1

S = 16 + 18 + 20 + 22 + 24
S = 24 + 22 + 20 + 18 + 16
2S = 40 + 40 + 40 + 40 + 40
2S = 5 (40)
S = 1/2 (5)(40)
S = 100

Method 2

Connect the process used in Method 1 to a formula for the sum of an arithmetic series.

Students use the formula to find the number of rows needed for a seating capacity of 450.

(P.4.C) The student uses sequences and series as well as tools and technology to represent, analyze, and solve real-life problems. The student is expected to describe limits of sequences and apply their properties to investigate convergent and divergent series.

Clarifying Activity

Students work in groups of three. They are given a piece of paper (to represent a large brownie or piece of cake), and they are asked to divide the paper into 4 equal parts with each person taking a piece. The remaining piece is then divided into 4 equal pieces, and one of these pieces is distributed to the each member of the group, leaving a piece left over. The group continues this process as long as possible. (The diagram below shows what each student gets.)

Note, the sum of the darker shaded portions (1/4 + 1/16 + 1/64 + . . . ) represents the amount received by each person.

Students consider the limit of this sequence and the behavior of the resulting series by discussing the following questions:

1. How much of the original piece of paper would each student have?
2. What is the limit of the sequence?
3. How can you write the total amount of paper you have as a series? Is it convergent? If so, what does it converge to?

Additional Clarifying Activity

In Clarifying Activity P.4.A with the Sierpinski Carpet, students:

• explain in words and show analytically (by summing the geometric series) why the series that represents the area of the removed (shaded) squares converges or diverges. If convergent, what does the series converge to? What does this tell you about the sequence of remaining areas?
• explain in words and show analytically why the series of perimeters of the shaded squares converges or diverges.

(P.4.D) The student uses sequences and series as well as tools and technology to represent, analyze, and solve real-life problems. The student is expected to apply sequences and series to solve problems including sums and binomial expansion.

Clarifying Activity

Students use sequences and series to solve the following problem:

Leaking Oil Tanker Problem

Yesterday, an oil tanker collided with a Coast Guard cutter off the California coast at 9:00 AM, and the disabled tanker began to spill oil from its damaged hull. The rate of flow of oil into the Pacific Ocean was measured over a ten hour period. It was found that the data could be modeled by the function below:

f(x) = 0.03 (x³ - 27x² + 110 x + 600)

Below is the graph of this function over the ten hour period of the spill.

1. Find the area of one of the rectangles shown above. What does the area represent in terms of the oil spill?
2. Write a series to estimate the total amount of oil that spilled during the ten hour period.
3. Use sigma notation to represent the sum of the series.

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(P.5) The student uses conic sections, their properties and parametric representations, as well as tools and technology, to model physical situations.

(P.5.A) The student uses conic sections, their properties and parametric representations, as well as tools and technology, to model physical situations. The student is expected to use conic sections to model motion, such as the graph of velocity vs. position of a pendulum and motions of planets.

Clarifying Activity

Students use conic sections to model motion in problems such as the following (from Larson, R. Precalculus with Limits, 2nd Ed. Houghton Mifflin, 1997, p. 771):

Halley's comet has an elliptical orbit with the sun as one focus. The eccentricity of the orbit is approximately 0.97. The length, a, of the semimajor axis of the orbit is about 36.23 astronomical units. (An astronomical unit is about 93 million miles.) To respond to the following, place the orbit center at the origin and place the major axis on the x axis.

1. Determine the distance, c, from orbit center to the foci and the length, b, of the minor axis.
2. Determine the rectangular equation of the elliptical orbit in standard form.
3. Using a reasonable scale, sketch on paper the orbit for Halley's comet, marking the focus that is the sun. Include on the sketch the lengths of the major and minor axes and the focal distance.
4. Find the least distance and the greatest distance between Halley's comet and the sun. (Use the graphing calculator to display a graph of the orbit and a table of values for distance between the comet and the sun.)
   • In Function mode,

     

   • In Parametric mode,

     

   • Compare the results from the Function and Parametric modes. Which method is preferable? Why?

Extension: Students can research orbits of other comets and orbits of planets. Students can research polar representation of conic sections and compare the three representations (rectangular, parametric, and polar).

Additional Clarifying Activity

In space science, orbital velocity of a celestial body is sometimes measured as a function of position. A similar problem is to find the relation between the velocity of a pendulum and its displacement. To investigate this, students conduct an experiment to measure the period (time in sec) of a pendulum as it relates to the pendulum's displacement from the vertical.

Suppose that an experiment was conducted on a 15 cm pendulum and the following model was found to best describe the data:

q = 0.2 cos(8t)

where q is the angular displacement of the pendulum from the vertical in radians and t is the time in seconds. Generate tables for t and q, using Tablestart = 0 and Dtbl = 0.05. To estimate velocity, build a table of values for Dq/Dt. In the following t, q, and Dq/Dt are given as L₁, L₂, and L₃ respectively.

Students use the STAT PLOT feature of the graphing calculator to explore the plot of velocity (L₃) as a function of displacement (L₂) and describe the results.

Suggested window:

Tmin = 0, Tmax = 8, Tstep = 0.13
Xmin = -1, Xmax = 1, Xscl = 0
Ymin = -2, Ymax = 2, Yscl = 0

Students should see that the plot appears to be an ellipse, centered at the origin, with a nearly vertical major axis. To write an equation of the ellipse that might fit the data they could estimate a and b from the plot.

(b ≈ (1.580 - (-1.566))/2 = 1.57 and a ≈ ((.2 - (-.1794))/2 = 0.1897)

To check their estimates, students graph the ellipse using parametric representation:

x_1T = a cos(T) and y_1T = b sin(T)

refining the values of a and b.

With the teacher's guidance, students note that, for this pendulum model, velocity, v, as a function of displacement, q, is given by

(P.5.B) The student uses conic sections, their properties and parametric representations, as well as tools and technology, to model physical situations. The student is expected to use properties of conic sections to describe physical phenomena such as the reflective properties of light and sound.

Clarifying Activity

Students use conic sections to analyze the following situation:

The world's largest solar-thermal complex, Luz International, is located in the Mojave Desert in California. It consists of 1.5 million parabolic mirrors that reflect the sun's rays onto tubes filled with oil. The heated oil produces boiled water to produce steam for a turbine.

A cross-section of the parabolic mirror is modeled by the equation 25y = x² where both x and y are measured in feet. How high above the vertex of the mirror is the heating tube? Assuming ground level corresponds to the directrix of the parabolic cross-section, how far above ground is the heating tube?

(Answer: The tube is 6.25 feet above the vertex of the mirror and 12.5 feet above ground.)

Extension: Students can research the actual size of one of the mirrors, its mounted height, its steam producing capacity, etc. and the productivity of this solar-thermal complex.

Additional Clarifying Activity

Students investigate applications of conic sections, such as the following (from Demana, F., Waits, B., and Clemens, S. Precalculus Mathematics, A Graphing Approach. Addison-Wesley, 1994, p. 517):

Lenses in cameras, glasses, and telescopes use the reflective properties of both the parabola and the hyperbola. For example, in a telescope the main lens is parabolic with focus F₁ and vertex F₂. The secondary lens is hyperbolic with foci F₁ and F₂. The concave surface of the hyperbolic lens is coated with a reflective substance. Light rays reflect off the parabolic lens to the convex surface of the hyperbolic lens, which reflects the rays to the viewing source.

Suppose the equation for the parabolic lens is y² = 24x and the vertex for the hyperbolic lens is (4.5, 0).

1. What are the coordinates of the foci of the hyperbola?
2. Determine in standard form the equation of the hyperbola.
3. Find complete graphs of both conics in the same viewing window.
4. Explain how this lens arrangement works.

Answers:

1. (0,0) and (6,0)
2. (x - 3)² / 2.25 - y² / 6.75 = 1
3. Suggested window: [-3,10] by [-10,10]
4. Light is reflected from the parabolic surface toward the focus common to both conics. Then it is reflected from the hyperbolic surface toward the eyepiece at the second focus of the hyperbola.

(P.5.C) The student uses conic sections, their properties and parametric representations, as well as tools and technology, to model physical situations. The student is expected to convert between parametric and rectangular forms of functions and equations to graph them.

Clarifying Activity

Students convert from rectangular to parametric form in the following situation:

Some astronomers track incoming meteorites to find out whether or not they will strike Earth. Below are given equations of the Earth's surface and an incoming meteorite in rectangular coordinates (x and y are in thousands of kilometers).

Earth's Equation: x² + y² = 40
Meteorite's Equation: x = 0.25y² + 5

Will the meteorite strike the Earth's surface? If so, what are the coordinates of the point of impact?

1. Compare the procedure you would use in finding a symbolic solution to this system of equations without a calculator to exploring this problem with your graphing calculator in Function mode.

   [Sample response: Without the calculator, I would solve the second equation for y², and use substitution . . . .]

   What difficulties would you encounter in using the calculator to answer the questions?

   [Sample response: The equations are not in explicit form (y as a function of x) and also will require two function entries per equation to see a complete graph of each equation, etc.)]

2. Describe this situation with parametric equations.

   Parametric Equations for the Earth's Surface:

   Expected answer:

   x_1T = 40 cos(T)
   y_1T = 40 sin (T)

   Parametric Equations for Meteorite:

   Expected answer:

   x_2T = 0.25T² + 5
   y_2T = - (T)

   Graph the parametric equations in a window that shows complete graphs of both the Earth and the meteorite's path. If the meteorite impacts with Earth, give the coordinates of the point of impact.

   If the Earth's equations represented a small satellite in orbit close to the Earth's surface, would the meteorite and the satellite collide? Explain.

   Suggested window:

   Tmin = -8, Tmax = 8, Tstep = 0.13
   Xmin = -15.16, Xmax = 15.16, Xscl = 0
   Ymin = -10, Ymax = 10, Yscl = 0

   [Answer: The meteorite impacts the Earth's surface at about (6,2). If it were the orbit of the satellite, the meteorite would miss it because of the difference in time when the graphs pass through the point (6, 2).]

Additional Clarifying Activity

Students convert from parametric equations for a curve to a rectangular equation as part of the following problem (from Larson, R. Precalculus With Limits, 2nd ed., Houghton Mifflin, 1997, p. 797):

Consider the parametric equations x = 4 (cos q)² and y = 2 sin q

1. Use a graphing calculator to complete the following table.

   

2. Plot the points (x,y) generated in Question 1 and sketch a graph of the parametric equations.
3. Find the rectangular equation by eliminating the parameter. Use a graphing calculator to sketch its graph. How do the graphs differ?

Additional Clarifying Activity

See the Additional Clarifying Activity at c.1.b.

(P.5.D) The student uses conic sections, their properties and parametric representations, as well as tools and technology, to model physical situations. The student is expected to use parametric functions to simulate problems involving motion.

Clarifying Activity

Students use parametric representation to simulate motion in the following problems:

1. You must cross a river that is 4 miles wide. You set out to go straight across to the opposite shore at a speed of 8 miles per hour. However, the river has a current flowing at 6 miles per hour. How long will it take you to cross the river and how far downstream will you be?
   1. Simulate the motion in this problem using your graphing calculator in parametric mode.

      Sample Window:

      Tmin = 0, Tmax = 1, Tstep = 0.13
      Xmin = 0, Xmax = 4.7, Xscl = 1
      Ymin = -1, Ymax = 3.2, Yscl = 1

      Sample Parametric Equations:

      x_1T = 8T, y_1T = 6T

   2. Using your calculator simulation, how long will it take you to cross the river and how far will you actually travel?
   3. Verify your results to Part b using your knowledge about right triangles or vectors.
2. Two trains leave a common station on parallel tracks one hour apart. The first train travels at 50 mph, while the second train leaves one hour later at 55 mph. How many hours must pass before the two trains are even with each other on the tracks?
   1. Simulate the motion in this problem using your graphing calculator in parametric mode so that you see the motion of the trains on parallel tracks.

      Sample Window:

      Tmin = 0, Tmax = 17, Tstep = 0.13
      Xmin = 0, Xmax = 850, Xscl = 100
      Ymin = -2, Ymax = 4, Yscl = 1

      Sample Parametric Equations:

      Train A: x_1T = 50T, y_1T = 1
      Train B: x_2T = 55(T - 1), y_2T = 2

   2. Use the TRACE and TABLE features of the calculator to describe the relative times and positions of the trains as Train B overtakes Train A.
   3. Verify your results in Part b using your knowledge of relative rates.

Additional Clarifying Activity

Students explore and extend the following problem using parametric graphs and simultaneous mode (from Demana, F., Waits, B., and Clemens, S. Precalculus, a Graphing Approach, 3rd ed., Addison-Wesley, 1994, p. 499):

Eric is standing on the ground 75 feet from the base of a Ferris wheel that has a radius of 20 feet. Janice is on the Ferris wheel, which makes one revolution counterclockwise in 12 seconds. At the instant she is at point A (on the horizontal) Eric throws a ball to her. The release height of the ball is at the bottom of the Ferris wheel and he throws the ball at an angle of 60 degrees with the horizontal and velocity 60 feet per second. Assume that g = 32 ft/sec² and neglect air resistance. Will the ball reach Janice? If not, is the ball underthrown or overthrown?

1. Write parametric equations for the path of the Ferris wheel and parametric equations for the path of the ball. Tmin = 0 corresponds to both the release of the ball and the Ferris wheel moving from ground position (0,0).

   Answer:

   x_1T = 20cos(πT/6), y_1T = 20 sin (πT/6)

   Ball Path:

   x_2T = 75 - 30T, y_2T = - 16T² + 30()T)

2. Graph the sets of parametric equations in simultaneous mode. Describe what the graphs depict happening. Use the TRACE feature of your calculator to verify your description.
3. Graph the distance between the ball and Janice parametrically:

   x_3T = T, y_3T = ((x_1T - x_2T)2 + (y_1T - y_2T)²)^1/2 with 0 ≤ x ≤ 4 and 0 ≤ y ≤ 20.

   Describe what the distance graph tells you about the situation.

4. In the parametric equations for the ball, vary Eric's position from 75 feet to see how close he can get the ball to Janice. Try a number of different values in [65,80]. Keep all other values the same. Explore the graphs and the distance between Janice and the ball. Describe your results.
5. In the parametric equations for the Ferris wheel, vary Janice's position, A, on the Ferris wheel when the ball is thrown. Explore the graphs and the distance between Janice and the ball. Describe your results.

   Students should see that in order to do this they must add an appropriate angle value to πT/6 in the parametric equations for the Ferris wheel.

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(P.6) The student uses vectors to model physical situations.

(P.6.A) The student uses vectors to model physical situations. The student is expected to use the concept of vectors to model situations defined by magnitude and direction.

Clarifying Activity

Students draw vector diagrams to represent component and resultant forces in the following problem and determine component and resultant forces as vector magnitudes (from Foerster, P. Trigonometry: Functions and Applications. Addison-Wesley, 1977, p. 211, Problem 19):

The Canal Barge Problem

Freda Pulliam and Yank Hardy are on opposite sides of a canal, pulling a barge with tow ropes. Freda exerts a force of 50 pounds at 20 degrees to the canal, and Yank pulls at an angle of 15 degrees with just enough force so that the resultant force vector is directly along the canal.

Complete the diagram below, drawing in the force component and resultant vectors.

Find the number of pounds with which Yank must pull and the magnitude of the resultant vector.

(P.6.B) The student uses vectors to model physical situations. The student is expected to analyze and solve vector problems generated in real-life situations.

Clarifying Activity

Students use component and resultant vectors to represent and solve the following problem:

An airplane is headed 32 degrees north of west. Its speed with respect to the air is 900 kilometers per hour. The wind at the plane's altitude is from the southwest at 100 kilometers per hour. What is the true direction of the plane and what is its speed with respect to the ground?

Expected Answer:

38.3 degrees north of west at 882.9 kilometers per hour)

Extension: What adjustments would the pilot need to make to maintain the plane's original course and speed?

Students can do some research on the basic terminology of navigation and meteorology to help them better understand problems such as this one.

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