Precalculus Review Lecture Notes PDF

Lecture 1: Precalculus Review Part 1 Recognizing linear equations Solving equations Solving Inequalities Defining the Absolute Value Function A linear equation in x has the form ax + b = c where a, b, and c are constants....

Lecture 1: Precalculus Review Part 1 Recognizing linear equations Solving equations Solving Inequalities Defining the Absolute Value Function A linear equation in x has the form ax + b = c where a, b, and c are constants. x+m ex. Solve = 1 for x. mx + 5 mats ex = myts-m AmeX +m : mxts S m S meX( -m) = - GX = - - my ↳  An inequality compares two values. a → b means that a is greater than or equal to b. a < b means that a is less than b. & 2 ex. ↓2 x↑3 210 2e 2 X - 3 - 2x + 6 2 ] - - ↓ T 3 9 3 - X 3 * - X - -Lxtz ⑫ - - 8eX 24 O & X - 3 20 -Ly 2 - >07 X439XL3 x - X = (-0 3) vS , I (4, , 8)  ex. Solve the following equations: > Xi(x 1) a. x3 ↑ x2 = 0 - - = 0 E- X = 0 * (m- 1) -2 (m-1) Cm-1' 1 2 b. 8m2(m ↑ 1) 3 + 2m(m ↑ 1)↑ 3 = 0 - 2m(m-1) 2 - mm + = 0 - 1) 1] 2m - m = O-4m2-4m + - - 43 22m-1/2 (m- ) 2m-12p = Y M = 0 ' m & ↔ c. 2 ↑ x2 ↑ x = 0 Ex X + = 2 - xh = xh 0 - ((x-1)20 check 2x" -2 = > - > 8 0 of s x2-1 ; V I 2 ; x1 Exit 1 0 1fX 5 - X = -1 - : - St 0 - = > - > - E -1 = 0 X= 1- Et - 1= 0 Definition of Absolute Value ! a a→0 If a is a real number, |a| =. ↑a a < 0 ex. | ↑ 4| = Sy, 4 ! X-1 XI0 , ex. |x ↑ 1| = - (x-1, XLO !X X7 O 1 x 70 x S - code X I I ex. = = |x| I finda - Y I XL8 x X ! 1 $ " -D # Lecture 2: Precalculus Review Part 2 Di!erent Types of Functions Domain/Range Unit Circle Solving Trigonometric Equations Translations and Transformations A function describes a relationship where every in- put has exactly one output. x↑2 ex. f (x) = , is a rational function. x↑1 The domain is XX1 or (-0 , 1) VLA , 4) ↔ ex. Find the domain of g(x) = x2 ↑ 5x. * 20 - xh-Sy : 0 x(x5/10 = 0 5 05 x , -( - 0 , 03 Is ,  Combinations of functions ex. Let f (x) = x2 ↑ 2 and g(x) = 2x. (f g)(x) = (2x)" - 2 All real numbers ; R Domain: " # f g (x) = L 2x Domain: ( - 5 07010 , 4) , (f + g)(x) = (f ↑ g)(x) = x2-2 2X x 27 24 - -  XX-1 Xy - 1 4 ↑ , ~ 1 x ex. If f (x) = and g(x) = 2 , find " # x+1 x ↑1 f ↓ C · (x) and its domain. g 7 X(x + 1) , 10 - , 0 v(0 , 1) ULL , a) Def. Consider the interval I where x1 and x2 are both in I. A function is increasing on an interval I if when x1 < x2, f (x1) < f (x2) A function is decreasing on an interval I if when x1 < x2, f (x1) > f (x2)  ~ Shif up ex. f (x) = 1 + ex (exponential) -F ! $ " # Transformations of the graph f (x): f (x) ± c vertical shifts f (x ± c) horizontal shifts vertical stretch 1 > 1 cf (x) vertical shrink 22 1 I honbutl ohnen f (cx) c) horemhl streckch Ch1 ↑f (x) reflect our X-axi) f (↑x) reflect our yaxis # ↔ ex. Graph y = 2 ↑ x↑1 NOTE: Perform horizontal shift first! Then reflect, and last do vertical shift. ! $ " # Trigonometry Review ↓ P(x,y) O θ hypotenus opposite θ adjacent A sin ω = & reciprocalscsc ω = H - # - O A A cos ω = sec ω = F - # An tan ω = o cot ω = J In general, let P (x, y) be any point $on the terminal side of ω (radians) and let- r = x2 + y 2 be the distance from the origin to point P.  3. Two basic triangles 30-60-90 4-12-90 /Tyu π /3 π /6 π /4 4. Unit circle (r = 1, so sin ω = y and cos ω = x) π/2 2π/3 π/3 3π/4 π/4 π/6 5π/6 π 0 7π/6 11π/6 5π/4 7π/4 4π/3 5π/3 3π/2  to E ↑ ω 0 ε/6 ε/4 ε/3 ε/2 sin ω O 112 Elz 5/2 A cos ω # 5/2 Fle 12 ⑨ tan ω O 5/3 A 5 undefined NOTE: - A ↓ sin x ↓ I , - I↓ cos x ↓ 1 5. Graphs y = sin x y = cos x y = tan x y = cot x y = sec x y = csc x In calculus, angles are measured in radians. Recall that radians measure the arc length counterclockwise along the unit circle. One rotation is 2εr radians AKA 360↗. One half rotation is ε radians AKA 180↗. Important Trigonometric Identities Pythagorean Identities sin2 x + cos2 x = 1 tan2 x + 1 = sec2 x 1 + cot2 x = csc2 x sin (2x) = 2 sin (x) cos (x) Even/Odd sin(↑x) = ↑ sin x (sin x is an odd function). cos(↑x) = cos x (cos x is an even function). Reciprocal Identities 1 1 1 csc x = sec x = cot x = sin x cos x tan x On the unit circle, we have x = cos ω and y = sin ω. ! $ " # ex. sin ε6 = t ε E ex. cos = - 4 2 Fahr -↔ ex. Solve 2 cos2 x ↑ - 3 cos x = 0, 0 < x < 2ε. - 5) - S Coox) /2 cosy 0 En = - 10x E 2205 X - 5 = E z = 10x & #