Lighthouse Initiative for Texas Classrooms

TEKS: Precalculus

Read an introduction to Texas Essential Knowledge and Skills charts.

TEKS Examples Commentary

111.35 PRECALCULUS (ONE HALF TO ONE CREDIT)

(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 piecewise-defined functions.

(A) describe parent functions symbolically and graphically, including

  • • f(x) = xn
  • • f(x) = ln x
  • • f(x) = logax
  • • f(x) = 1/x
  • • f(x) = ex
  • • f(x) = |x|
  • • f(x) = ax
  • • f(x) = sin x
  • • f(x) = arcsin x

etc.;

(B) determine the domain and range of functions using graphs, tables, and symbols;

(C) describe symmetry of graphs of even and odd functions;

(D) recognize and use connections among significant values of a function (zeros, maximum values, and minimum values, etc.), points on the graph of a function, and the symbolic representation of a function; and

 

 

(E) investigate the concepts of continuity, end behavior, asymptotes, and limits and connect these characteristics to functions represented graphically and numerically.

f(x)= 6x3+x2+1
           2x-6

a) Determine any and all points of discontinuity in f(x). Find algebraically and verify graphically. Use a window of -10 ≤ x ≤ 10 and -50 ≤ y ≤ 300.

b) Find the end behavior model and graph it and f(x) on the same axes using windows of -20 ≤ x ≤ 20, -200 ≤ y ≤ 1000 and -30 ≤ x ≤ 30 - 500 ≤ y ≤ 3000.

c) Confirm the asymptotes and end behavior
i) graphically
ii) with tables

x

2

2.5

2.9

2.99

2.999

3

3.001

3.01

3.1

3.5

4

f(x)

 

x

-1000

-100

-10

10

100

1000

f(x)


d) As x approaches infinity, what does f(x) approach? As x approaches negative infinity, what does f(x) approach? As x approaches 3 from the right, what does f(x) approach? As x approaches 3 from the left, what does f(x) approach?

AP* Calculus Concept: Limits

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

(A) apply basic transformations, including

  • a · f(x)
  • f(x) + d
  • f(x - c)
  • f(b · x), and
  • compositions with absolute value functions, including |f(x)| and
  • f(|x|)

to the parent functions;

(B) perform operations including composition on functions, find inverses, and describe these procedures and results verbally, numerically, symbolically, and graphically, and

 

 

(C) investigate identities graphically and verify them symbolically, including logarithmic properties, trigonometric identities, and exponential properties.

f(x) = sin 2x
g(x) = 2sin x cos x

a) Put values for x, sin 2x, and 2sinxcosx, -π ≤ x ≤ π, in a table.

b) Graph f(x) and g(x) on the same axes.

c) Verify that f(x) = g(x) at x = π / 6.

d) Use the properties of circular functions to prove that sin 2x = 2sin x cos x.

 

 

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

 

 

 

 

(A) investigate properties of trigonometric and polynomial functions;

(B) use functions such as logarithmic, exponential, trigonometric, polynomial, etc. to model real-life data;

(C) use regression to determine the appropriateness of a linear function to model real-life data (including technology to determine the correlation coefficient); and

(D) use properties of functions to analyze and solve problems and make predictions.

Year Fed Debt
1980 0.91
1981 0.99
1982 1.1
1983 1.4
1984 1.6
1986 2.1
1987 2.3
1988 2.6
1989 2.9
1990 3.2
1991 3.6

(Debt values are in trillions)

a) Draw a scatter plot of the data.

b) Calculate the equation of the line of best fit.

c) Use the equation from part (b) to predict the debt in 1985 as well as 1994.

d) The scatter plot appears to be a curve, suggesting exponential growth. Linearize the curved data by graphing a scatter plot of (year, ln debt).

e) Calculate the regression line for the re-expressed data.

f ) Use properties of exponents/logarithms to rewrite the equation in part (e) into its exponential form.

g) Use the exponential equation in part (f) to predict the debt in 1985 and 1994.

h) Which of the two sets of predictions seem to be more accurate? Explain.

This includes the AP* Statistics concepts of transformations to achieve linearity, specifically logarithmic and power transformations.

(E) 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.

 

 

 

 

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

 

 

 

 

(A) represent patterns using arithmetic and geometric sequences and series;

Using (1,1), (2,5), (3,12), (4,22) ...

a) What is the 10th term in the sequence above?

b) Write an equation for the sequence of the ordinants.

c) Graph a scatter plot of the first 10 terms.

d) Graph the equation from part b on the same axes as the scatter plot.

e) Find the sum of the heights (above the independent axis) of each of the first 10 points.

f) Express the sum in part e using sigma notation.

g) Approximate the area between the curve (connecting the 10 points) and the x-axis using trapezoids.

h) If the ordered pairs represent (week, homework problems per week), what is the meaning of the area found in part g?

AP Calculus Concept: Accumulation 8

(B) use arithmetic, geometric, and other sequences and series to solve real-life problems;

A patient has a throat infection and is to take 300 mg of an antibiotic every 6 hours. At the end of 6 hours, about 3% of the medication is still in the body.

a) What quantity of medication is in the body immediately after the 4th dose?

b) What quantity of medication is in the body immediately after the 12th dose?

c) Assuming the patient continues taking the medication, what eventually happens to the drug level in the patient's body?

d) Show graphically and use a table of values to confirm the conclusion from part (c).

dose d

1

2

3

4

5

6

medication m

 

 

 

 

 

 


e) Use sigma notation to express the data as a series.

f) Complete the statement: "as the number of doses increases, ..."

 

AP Calculus Concept: Limits

 

 

(C) describe limits of sequences and apply their properties to investigate convergent and divergent series; and

Given that

 1  +  1   1 +...+ 1  
1·2   2·3   3·4     n(n+1)

=  n 
   n+1

a) Prove the given statement by induction.

b) Show that

Mathematics, Precalculus, Figure 1

c) Find

Mathematics, Precalculus, Figure 2 by finding the limit of the sequence of partial sums s1, s2, s3, s4, ...sn.

d) Confirm the answer to part "c" by graphing the partial sums as n increases.

e) Verbalize what you learned; "As n increased, ..."

These problems are particularly important for students who will be taking BC Calculus.

(D) apply sequences and series to solve problems including sums and binomial expansion.

Suppose Tim Duncan of the San Antonio Spurs is a 77% free throw shooter.

a) Use binomial expansion to determine the probability of Tim's making exactly 4 of 9 free throws in a game.

b) Determine the probability of Tim's making at least 4 of 9 free throws in a game.

c) Write (in the form of a chart) a probability distribution to determine the most likely outcome when Tim attempts 9 free throws.

d) What is the probability of the most likely outcome?

e) Draw a bar graph that represents the probability distribution of Tim's making 0 through 9 free throws.

AP Statistics concept: Discrete random variables and their probability distributions.

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

(A) use conic sections to model motion, such as the graph of velocity vs. position of a pendulum and motions of planets;

(B) use properties of conic sections to describe physical phenomena such as the reflective properties of light and sound;

 

 

 

(C) convert between parametric and rectangular forms of functions and equations to graph them; and

x = 2cos2t and y = sin 2t

a) Graph the conic section formed by x and y with your calculator set in parametric mode.

b) Using properties of circular functions, find a rectangular equation for the curve that contains no variables other than x and y.

c) Graph the equation from part b above.

d) Select values of t so that the graph of the parametric equations begins at (1,1) and ends at (2,0). Show a table of values and a graph that confirms your work.

e) Rewrite the parametric equations and choose values of t so that it graphs clockwise, starting and ending at (1,1). Show a table of values and a graph that confirms your work.

AP Calculus Concept: Parametric equations are used to describe motion. Students find parametric rates of change and determine velocity, acceleration, and arc length.

(D) use parametric functions to simulate problems involving motion.

A circle with equation (x+2)2 + y2 = 9 is intersected by a graph with parametric equations x = 2cos2t and y = sin2t.

a) Convert the circular equation above to parametric form. Convert the parametric equations to rectangular form.

b) Find the points of intersection between the circle and the other graph.

c) Let t represent the time, in seconds, of an object moving along the conic sections above. Will an object on the 1st conic section reach the 1st intersection point before, after, or at the same time as an object moving on the second conic section? Determine algebraically and confirm graphically.

d) Find the minimum distance between objects moving along the conic sections where 0 ≤ t ≤ 1. At what time does the minimum distance occur?

AP Calculus Concept: Optimization

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

(A) use the concept of vectors to model situations defined by magnitude and direction; and

(B) analyze and solve vector problems generated by real-life situations.

 

 

 

 

 

 

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