YOUNGSTOWN CITY SCHOOLS
MATH: PRECALCULUS
UNIT 1: GRAPHING RATIONAL FUNCTIONS (3 WEEKS) 2013-14
Synopsis: This unit will begin with a review of linear, absolute value, piecewise, polynomial irrational, exponential, log, and trig functions. After reviewing these functions, rational functions will be taught in depth.STANDARDS
F.IF7d Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. (+) graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
MATH PRACTICES
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning
LITERACY STANDARDS
L-2 Communicate using correct mathematical terminology
L-7 Research mathematics topics or related problems
L-9 Apply [details of mathematical] readings/use information found in texts to support reasoning, and develop a “works cited document” for research done to solve a problem.
MOTIVATION / TEACHER NOTES1. Students will read the article “Why Graphing Is So Important” in the link below and also attached to the unit on pages 8 & 9, then discuss the article with the class. They will then search for occupations that deal with math and present them to the class or research how graphs can be used in their potential field and turn in a “works cited document” showing their sources. (F.IF.7a-e, L-2, L-7, L-9)
http://www.mathworksheetscenter.com/mathtips/whygraphingisimportant.html
2. Students establish personal and academic goals for the unit
3. Teacher previews the Authentic Assessment so students know what to expect at end of unit.
TEACHING-LEARNING / TEACHER NOTES /
Vocabulary:
Linear / Slope / x-intercepts / y-intercepts
Continuous / Vertical shift / Horizontal shift / End behavior
Maximum / Minimum / Vertex / Piece-wise
Polynomial / Zeros / Exponential / Period
Midline / Amplitude / holes / Slant asymptotes
Horizontal asymptotes / Vertical asymptotes / extrapolate
Note: Numbers 1 through 10 are review items. You want to keep the graphs fairly simple.
1. A pretest on factoring will be given to check students’ skills. If they are weak, use the weak areas for review, insuring that they are comfortable factoring trinomials, difference of two squares, sum and difference of two cubes, expressions with common factors and factoring by grouping. (MP.4, MP.8, L-2)
2. Review graphing linear functions: f(x) = mx + b – discuss slope, x and y intercepts, and continuity. (F.IF.7a, MP.2, MP.4, MP.8, L-2)
3. Review graphing exponential functions: f(x) = ax, examine when a>1 and when 0<a<1, end behavior, continuity, x and y intercepts, if any. (F.IF.7e, MP.2, MP.4, MP.8, L-2)
4. Review the procedure for regressions in the calculator, shown on page 10 of this unit. Reinforce with examples from the textbook on pages 742 & 743. (F.IF.7.e, MP.1, MP.2, MP.4, MP.5, MP.7)
5. Review graphing absolute value functions – x and y intercepts, discuss vertical and horizontal shift of f(x) = |x|, maximum, minimum, continuity, and end-behavior. (F.IF.7b, MP.2, MP.4, MP.8, L-2)
6. Review graphing piecewise functions: f(x) = (F.IF.7b)
7. Look at elevator function: (F.IF.7b, MP.2, MP.4, MP.8, L-2)
http://education.ti.com/calculators/timathnspired/US/Activities/Detail?sa=5025&t=5070&id=16902
8. Review graphing polynomial functions and relations: (F.IF.7c, MP.2, MP.4, MP.8, L-2)
a. Parabolas – f(x) = a(x-h)2+k, f(x) = ax2+bx+c, f(x) = a(x-b)(x-c),
x = a(y-h)2+k, x = ay2+by+c, x = a(y-b)(y-c): discuss vertex, sign of “a”, maximum, minimum, x and y intercepts, zeros, end behavior, continuity, vertical and horizontal shift of f(x) = ax2. For real life examples to reinforce these concepts refer to section 10-5 in the textbook.
b. F(x) = x3, f(x) = x4, f(x) = x5, f(x) = x6, look at similarities and differences and vertical and horizontal shifts and end behavior. (F.IF.7a, MP.2, MP.4, MP.5, MP.8, L-2)
9. Review graphing irrational functions: f(x) = , f(x) = , discuss horizontal and vertical shifts. (F.IF.7b, MP.2, MP.4, MP.8, L-2)
10. Review graphing log functions: f(x) = log x, f(x) = lnx, examine end behavior and continuity. (F.IF.7e, MP.2, MP.4, MP.8, L-2)
11. Review graphing trig functions: examine period, midline, amplitude for sine and cosine. Discuss period for tangent. Discuss horizontal and vertical shift for sine cosine and tangent. (F.IF.7e, MP.2, MP.4, MP.8, L-2)
12. Teach graphing rational functions. (F.IF.7d, MP.1, MP.2, MP.4, MP.5, MP.7, MP.8, L-2)
good web site for asymptotes: http://www.purplemath.com/modules/asymtote2.htm
a) Have students try to graph F(x) = and discuss their results. Then proceed to these steps:
i. Find vertical asymptotes by setting denominator equal to 0 which is the point at which the function is undefined, so vertical asymptote is x = 0
ii. Find horizontal asymptotes – when x approaches infinity or negative infinity, what value does f(x) or y approach. In this case y approaches 0 because as x gets very large, gets very small. So the horizontal asymptote is y = 0. If the greatest power of x in the denominator is larger than the greatest power of x in the numerator, the horizontal asymptote will be y = 0. If the greatest powers of x are equal in the numerator and denominator, then the horizontal asymptote is found by dividing each term in the numerator and denominator by x to that power. Using the fact that approaches zero as x approaches infinity, all terms with will become 0 and what is left is the horizontal asymptote.
iii. Draw vertical and horizontal asymptotes with dashed lines. These are the lines the function approaches gradually. The function will never cross the vertical asymptotes but can cross the horizontal asymptotes.
iv. Find x intercepts (zeros of the function) and y intercepts, if any.
v. Set up an xy chart and plot a few points in each region of the graph. Connect to form the graph and discuss end behavior.
x y
1 1
2 ½
-1 -1
-2 -½
b. Graph: f(x) = . Have students proceed through steps i through v in the previous example.
c. Graph: f(x) =
i. Find vertical asymptotes, x = 0
ii. Find slant asymptotes. There are slant asymptotes because the greatest power of x in the denominator is exactly one degree smaller than the numerator. To find them, use long division to divide the denominator into the numerator. The result is x – 2 + . As x approaches infinity, approaches 0 so the slant asymptote is y = x – 2.
iii. Draw the vertical and slant asymptotes.
iv. Find the x and/or y intercepts and plot them.
v. Set up an xy chart using values in each region of the graph. Connect to form a graph and discuss end behavior.
d. Graph: f(x) =
i. This function can be factored and reduced which means it has a hole. The x value of the hole appears when the common factor is set equal to zero and solved. The y value of the hole is found by replacing the x value into the reduced form of the function. f(x) = = = x – 5, so this is a graph of a line: f(x) = x–5 with a hole at (2, 7).
e. Have students work on the following worksheet: http://kutasoftware.com/FreeWorksheets/Alg2Worksheets/Graphing%20Rational%20Functions.pdf
f. Have students practice graphs with slant asymptotes:
i. f(x) =
ii. f(x) =
13. Worksheet on identifying functions is attached to the unit on pages 11-14.
Note: There are excellent videos showing the procedure used to graph each of these functions. Search the net and use those that are most appropriate for your class. (F.IF.7d, MP.2, MP.4, MP.5, MP.7, MP.8, L-2)
TRADITIONAL ASSESSMENT / TEACHER NOTES
1. Paper-pencil test with M-C questions.
TEACHER ASSESSMENT / TEACHER NOTES
1. 2- and 4-point questions
2. other teacher assessments
AUTHENTIC ASSESSMENT / TEACHER NOTES
1. Given a set of data, students are to plot the points by hand on graph paper and choose a function of best fit. Then, using technology, they are to arrive at a regression equation that best fits their data. Then they are to extrapolate and answer a question about the future data not included in this set. (F.IF.7d, MP.1, MP.2, MP.4. MP.5, MP.6, MP.7, MP.8, L-2)
2. Students evaluate goals for the Unit.
AUTHENTIC ASSESSMENT UNIT 3 (F.IF.7.d, MP.1, MP.2, MP.4, MP.5, MP.8)
1. A Tomahawk Cruise ship in the South Pacific miss fires a missile. The missile goes over the side of the ship and hits the water. In the data, x is the number of seconds after the missile is launched and y is the number of feet above water for the missile.
X 0 0.5 1 1.5 2 2.5 3
Y 128 140 144 140 128 108 80
Plot the data on graph paper, then using your TI calculator plot the points and use the regression capabilities to determine a regression equation for this data. After how many seconds will missile hit the water using your regression equation? What does the model predict the height will be after 3.5 seconds?
______
2. The table below show the amount of money, in billions of dollars, spent on pollution control in the U.S. for the years 1983 – 1989.
Year 1983 1984 1985 1986 1987 1988 1989
Amt. 61.8 68.9 74.6 78.7 81.5 86.1 91.3
Spent
Plot the data on graph paper (use 1, 2, 3, etc for the years), then using your TI calculator plot the points and use the regression capabilities to determine a regression equation for this data. What does the model predict the spending would be in 2000? Do you feel this is good model to predict spending on pollution control? Why or why not?
______
3. The concentration (in milligrams per liter) of a medication in a patient’s blood as time passes is given by the data in the table below. Plot the data on graph paper, then using your TI calculator plot the points and use the regression capabilities to determine a regression equation for this data. How long will it take for the medication of wear off?
Time (hours) 0 0.5 1 1.5 2 2.5
Concentration 0 78.1 99.8 84.4 50.1 15.6
4. The data shows the average brain weight as a percentage of the body weight for different ages. Plot the data on graph paper, then using your TI calculator plot the points and use the regression capabilities to determine a regression equation for this data. Predict the brain weight at age 20 using the different models.
Age 0 2 4 6 8 10 12 14 16
brain weight 11 8 7 6 5 4.5 4 3.5 3.25
______
AUTHENTIC ASSESSMENT RUBRIC
Question # / ELEMENTS OF PROJECT / 0 / 1 / 21 / Plot points on graph paper / Did not attempt / Plotted some points / Plotted all points and connected them
Regression equation / Did not attempt / Regression equation does not fit data / Regression equation fits data closely
List seconds hit water / Did not attempt / Answer does not comply with regression equation / Answer complies with regression equation
Height after 3.5 sec. / Did not attempt / Answer does not comply with regression equation / Answer complies with regression equation
2 / Plot points on graph paper / Did not attempt / Plotted some points / Plotted all points and connected them
Regression equation / Did not attempt / Regression equation does not fit data / Regression equation fits data closely
Answer to spending amount in 2000 / Did not attempt / Answer does not comply with regression equation / Answer complies with regression equation
Good model with explanation / Did not attempt / Signified whether good model or not but no explanation / Signified whether good model and explained
3 / Plot points on graph paper / Did not attempt / Plotted some points / Plotted all points and connected them
Regression equation / Did not attempt / Regression equation does not fit data / Regression equation fits data closely
Time to wear off / Did not attempt / Answer does not comply with regression equation / Answer complies with regression equation
4 / Plot points on graph paper / Did not attempt / Plotted some points / Plotted all points and connected them
Regression equation / Did not attempt / Regression equation does not fit data / Regression equation fits data closely
Weight at 20 / Did not attempt / Answer does not comply with regression equation / Answer complies with regression equation
Possible answers to regression problems (students will have answers that vary)
1. f(x) = -16x2 + 32x + 128, missile hits water in 4 sec., after 3.5 sec. the height will be 39′.
2. f(x) = 4.6 x + 63.65, in 2000 the spending would be $147 billion, no, should not continue to increase.
3. f(x) = -56.2x2 + 139.3x + 9.35, it will take 2.55 seconds approximately
4. f(x) = 9.652*0.929x, at age 20 the brain weight would be 2.2 lbs.
Why is Graphing So Important in your Life Anyway?
A graph is a planned drawing, consisting of lines and relating numbers to one another. With the use of color and a little imagination you can quickly whip up a professional looking graph in no time at all. With technology at your fingertips you can make use of the computer.When doing calculations in everyday life we need the basic knowledge of making use of graphs. It is not just for those that excel in math, but for every student to use according to their needs.
When doing analysis of any kind, we need to make use of structure. This will be done by using a graph. Graphing is used daily. From stockbrokers to performance evaluation in companies. All use them to boost sales and meet deadlines.
Even simple calculations can be assessed better by using a graph. What about professional presentations? If you want to be taken seriously in the business world, you need a professional looking presentation. It will boost you and you will be the professional they hire.
What about annual sales figures. Whether you're a manager or a sales assistant, when the monthly targets aren't met, you need to know why and how to correct them.
Simplifying your life is the way of the future. For too long we as humans have taken to much work upon our shoulders, it's time to simplify our lives and to use the best tools for the job. Graphing is one of those tools that you just cannot be without.
Even planning your monthly budget can be benefited by drawing up a graph, do this for a period of 6 months and soon you will be able to see where you make mistakes and where you are prospering.
Accountants will benefit by using graphs to convey financial information to their clients. A graph can be very handy in collecting data and storing it in one place.
A graph can be a very effective tool in presenting visual information rather swiftly. Even students can use a graph as it is something simple to draw. Dot, lines and a little bit of knowledge can go a long way.
When measuring seismic waves a graph can identify any faulty areas and help to keep track of the situation.
When taking a survey a graph can be used to easily assess the information gathered into usable detail.
Remember that a good graph show clear facts and will be visually accurate. It will grab the attention of the reader and show data clearly. It will demonstrate and be simple.
The use of a graph will present data in a quick way, lift out the most important facts and will be easily remembered.
Define your target audience before you decide which type of graph to use. What are they expecting to see and what do they need to know.
For those in a computer field, as in networking, the use of graphs can be very useful to measure trafficking to a site.
Graphs are used in everyday life, from the local newspaper to the magazine stand. It is one of those skills that you simply cannot do without. Whatever your need or calculation, if used correctly, a graph can help you and make your life simpler.
Can you see the importance of using a graph? It is a very useful tool that can only be a benefit to you, no matter where you are in life. From students to professionals. A graph can help you keep track of things and to be on top of your game.
T/L # 4: Regression on the TI-83/84