Algebra / Trigonometry Review (Notes for MAT0) NOTE: For more review on any of these topics just navigate to my MAT187 Precalculus page and check in the Help section for the topic(s) you wish to review! Try the math problems on THIS first page while I visit with each student and make sure they are properly enrolled in the course! We will go over these problems as soon as I am done visiting with each of you. You will NOT have time to attempt all of these BUT do what you can! 1. Find the equation of the line that passes through the two points,7 and 5, slope-intercept form.. Write your final answer in Factor each of the following! (See the net page for some notes on factoring) 5 48 7 8 5 6 5 6 4 4 4 1 6 6 8 15 6 8 4 e e 1 1 1 1 DAY # 1 HOMEWORK: 1. Read through the class syllabus (several times) that your instructor gives you during the first class period!. Visit the class website and eplore all links. Be sure to read the instructors philosophy on Attendance, Homework and Grades.. Fill out the student information sheet given to you during the first class period. The instructor will collect your student information sheet at the beginning of the second class period! Be sure to fill out each blank on the student information site and initial and sign where asked to! 4. Start looking at the homework posted on the class web page for section 1.1!
Here is a general Factoring Strategy that you should use to factor polynomials. 1. Always factor out the GCF(Greatest Common Factor) first.. Net check the number of terms in your polynomial. A. Two terms (KNOW all three of these formulas!) i. Factor the difference of two squares a b a ba b ii. Factor the difference of two cubes a b a b a ab b iii. Factor the sum of two cubes a b a b a ab b B. Three terms ---- try reverse foil (or some other method that YOU are good at) (Note: sometimes a three term polynomial will factor into the product of two trinomials) C. Four terms ---- try factor by grouping D. For five or more terms OR if none of the above work you could try to use the rational roots theorem(from Precalculus/College Algebra) or Newton s Method (which we will learn later in this course)!. Repeat step until all factors are prime.
II. Completing the square (adding zero) Rewrite the following quadratic function in graphing form f a h k f 1 III. Working with fractions (be sure that you know what the various properties of fractions are) PROPERTIES we will discuss. 1. a c ac b d bd.. a a c ac b b c bc a c a d b d b c c 0 Do YOU know why this is? 4. a c ad bc b d bd 5. a b a b c c c 6. a b a b b a c c c DO NOT INVENT YOUR OWN PROPERTIES! (I will discuss some common mistakes shortly) A. Multiplying by 1 - something you will do A LOT in Calculus. Here are some eamples of where YOU have previously multiplied by 1 (Property ) E. Adding or subtracting fractions 5 6 4 E. Rewriting comple fractions in standard form i 4i E. Rationalizing the denominator 5 9
B. Multiplying or Dividing fractions (Properties 1 and ) 9 7 1 1 C. Adding and Subtracting fractions 5 11 0 9 0 D. Simplifying fractions. (Cancel FACTORS NOT TERMS) i.e using property backwards E. 6 4 6 4 6 or 4
E. 4 4 We will sometimes use properties 5 and 6 to split up fractions to help us with a calculus problem. E. sin 1 cos sin 1 cos or sin 1 cos IV. Evaluating polynomials. 4 f 5 5 9 11 Find 4 f 5 5 9 11 = = = OR you can use synthetic division and the remainder theorem! 5 5 9 11 OR you can use your calculator (TI.Table Set.Indep Var.Ask)
6 5 4 g 4 5. Use synthetic division to find g 1. Note that g is missing the term. Shortcut for evaluating a polynomial at = 1. Since g 1 0 (from the remainder theorem ) we say that 1 is a zero and that 1 is a factor of factor theorem ) Synthetic division can often be helpful in factoring when traditional methods fail. g (from the 54 4 7 4 In College Algebra / Precalculus you should have studied The Rational Roots Theorem and used it to help factor or find zeros for polynomials. (See the last page of this packet for more review on this) 4 f 11 10 4
V. Graphs of relations 1. Be able to interpret a graph. Consider the graph of y f given below. 1 A. f B. f C. Y-intercept = D. How many zeros? E. Is y f a function? Eplain F. State the Domain and Range using interval notation. Domain: Range: y f. You need to KNOW what the graphs of some basic functions look like from your College Algebra + Trig classes (or Precalculus class). In particular you should know WHEN (if ever) these functions are zero. Look these up yourself if you do not already know them and draw them in your notes! y y y y y y ln y e y sin y cos y tan 1. Be sure that you understand the definition of Absolute value Official Mathematical Definition: Eamples: w if w 0 w w if w 0 5 0 1 Write f 4 as a piecewise defined function and graph the result.
4. Be able to graph (and read) a piecewise defined function. Graph: 1 1 g 1 1 Note: Sometimes the pieces matchup and the function is continuous at that value and other times the pieces do NOT matchup and the function is discontinuous at that value of. We will study the concept of continuity in Calculus. VI. Trigonometry 1. You should know some basic trigonometric identities and be able to use them to simplify trigonometric epressions. (see a later page in these notes for a list of trigonometric identities). Eamples: Simplify A. cos 1 sin B. cos sin C. 4 1 tan sec. You should know your unit circle and how to use it to find trig function values AND inverse trig function values. See a page later in these notes for more information on this. Eamples: Draw partial unit circles to show where the value for each of the following comes from 1 1 1. sin. tan. arcsin 4. tan
Here are 8 trigonometric identities that you studied in your Trigonometry or Precalculus class. You do NOT need to memorize all 8 of them BUT there are several that you should know because they come up frequently in Calculus. Trigonometric Identities 1. Sin 1. Csc 1 Cos. Sec 1 Tan Cot 4. Sin Sin 5. Cos Cos 6. Tan Tan 7. Sin Tan 8. Cos Cot Cos Sin 9. Cos Sin 1 10. Cot 1 Csc 11. 1 Tan Sec 1. CosA B CosACosB SinASinB 1. CosA B CosACosB SinASinB 14. SinA B SinACosB CosASinB 15. SinA B SinACosB CosASinB 16. TanA B TanA TanB 1 TanATanB 18. Sin Cos 17. TanA B TanA TanB 1 TanATanB 19. Cos Sin note: # s 18 and 19 also hold for the function pairs.tan, cot AND sec, csc 0. Sin SinCos 1. Cos Cos Sin Cos 1 =1 Sin Tan. Tan 1 Tan. 1 Cos Sin 4. 1 Cos Cos 5. 1 Cos Tan 1 Cos 6. Sin 1 Cos 7. Cos 1 Cos 8. Tan 1 Cos 1 Cos = Sin 1 Cos = 1 Cos Sin The identities that come up often in calculus are # s 1, 7, 9 11, 1 15, 0, 1. Note: # s 4 6 state that our trigonometric functions are either odd or even functions! A function is even if f f for all in the domain of the function. A function is odd if f f for all in the domain of the function. Those of you going on to take Calc II will also need to know the power reducing identities # s and 4
Here is a copy of The Unit Circle YOU should know this completely!!!!! As YOU know from your Trigonometry background, the coordinate of a point on the unit circle is the cosine of the given angle and the y coordinate of a point on the unit circle is the sine of the given angle. In terms of the coordinates on the unit circle we know. cos sin y tan y 1 1 y sec csc cot y Note: the equation of the unit circle is y 1 and as is the case with every graph, if a point lies on the graph then the coordinates of the point must make the equation true. So, if you take any of the coordinates shown on the graph and substitute them into the equation y 1 you will get a true statement.
When you studied Trigonometry you restricted the domain on your trigonometric functions so they became 1-to-1 and consequently would have an inverse that was also a function. 1 1 f sin f sin 1 1 1 y1 y f f 1 1 cos 0 cos 1 1 1 y 1 0 y y y 1 1 f tan f tan y Knowing the RANGE for the inverse trigonometric functions will be very important later in this class. REMEMBER THAT IF YOU NEED TO REVIEW MORE TRIGONOMETRY THEN YOU CAN ALWAYS GO TO MY MAT187 PRECACLULUS PAGE AND CHECK IN THE HELP SECTION TOWARRDS THE BOTTOM FOR VARIOUS TRIGONOMETRY TOPICS.
Rational Roots Theorem: f a a a... a a a be a polynomial with integer coefficients. n n1 n Let n n1 n 1 0 If the polynomial has any rational zeros (roots), p/q, then p must be an integer factor of a 0 and q must be a factor of a n. Eample: List the possible rational zeros for 4 p : 1,, 4 q : 1, p 1 4 : 1,,,, 4, q f 11 10 4. Other important polynomial theorems for College Algebra / Precalculus. Conjugate Pairs Theorems. i. If your polynomial has rational coefficients and a b c is a zero then so is it s conjugate a b c ii. If your polynomial has real coefficients and a bi is a comple zero then so is it s conjugate a bi A) The Remainder Theorem. If you wish to evaluate a polynomial at a number c just do synthetic division using c and whatever remainder you get will be f (c). Note: This works for ANY number, integer, irrational or imaginary. B) The Factor Theorem. If doing synthetic division with c yields a remainder of zero then we say that c is a zero (or root) of f () AND it means that ( c ) is a factor of f (). C) The Upper Bound Theorem If doing synthetic division with a positive number yields a whole row of non-negatives then there is no zero greater than the one that you just tried. D) The Lower Bound Theorem If doing synthetic division with a negative number yields a whole row of alternating signs then there is no zero smaller than the one that you just tried. E) The Intermediate Value Theorem. For any polynomial P(), with real coefficients, if a is not equal to b and if P(a) and P(b) have opposite sings (one negative and one positive) then P() MUST have at least one zero in the interval (a, b). Note: The Intermediate Value Theorem holds for any CONTINUOUS function. We will study the idea of continuity in MAT0.