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MATH127 (1)

Final Exam Review great review; covers both midterms aswell!

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Clinton Loo

MATH 127- Exam Review Number systems- integers (Z), real numbers (R), rational numbers (Q) Set Notation- if S is a set; xeS means x is an element in the set xeS means x is not an element in the set if T is also a set; SUT (Union) consists of elements in both sets (OR) SNT (Intersection) consists of elements common to both sets (AND) Intervals (a,b) endpoints not contained NOTE: + infinity must have a round bracket [a,b] end points contained Inequalities 1. If a < b then a+c < b+c 2. If a < b and c< d then a+c < b+d 3. If a < b and c>0 then acbc (negative sign switches inequality) 5. If 01/b 2 2 Quadratic Formula- if ax +bx+c = 0 then x = -b+ sqrt(b -4ac)/2a 1. b -4ac > 0 then 2 solutions 2 2. b -4ac < 0 then no solutions 3. b -4ac = 0 then 1 solution Absolute Value-the absolute value of a number a, |a|, is the distance from the number a to 0 on the real number line Defn: |a|= a if a >= 0; -a if a <0 2 1. Sqrt (a ) = |a| 2. |a| >= 0, for any aeR 3. Sqrt(a) means positive root; -sqrt(a) means the negative root 4. |ab|=|a||b| 5. |a/b|=|a|/|b|; b cannot equal 0 n n 6. |a |=|a| Triangle Inequality |a+b| <= |a| + |b| Coordination Geometry and Lines 2 2 |P 1 2=sqrt((x -2 )1+(y -2 )1) Lines- to find the equation of a line two things are needed. 1. Slope (m) 2. A point of the line (x,y) Defn- the slope of a non-vertical line passing through the points (x ,y ) and (1 ,1 ) is giv2n 2y m=rise/run = (y -y 2/(1 -x )2 1 1. Point slope form: the equation of a line with slope m and passing through the point (x ,y ) is o o y-y om(x-x ) or y=m(x-x ) +y o o 2. Slope intercept form: the equation when slope is m and the y-intercept is b is y=mx+b Conics- an equation of a circle with radius r and centered at (h,k) is given by (x-h) +(y-k) =r 2 2 2 Parabolas- a parabola is the graph of the equation of the form y=ax +bx+c 2 2 2 2 2 Ellipses-the curve with the equation x /a +y /b =1 where a,b > 0 is an ellipse in standard position (centered at the origin) 2 2 2 2 Hyperbola-the curve with the equation x /a - y /b =1 where a,b >0 is a hyperbola Functions- a function is a relation between sets D(domain) and R(range) that assigns to each element xeD precisely one element f(x)eR (not 2 different ones).The domain consists of all the elements where f is defined. The range consists of all possible values y=f(x) as x varies throughout the domain. Y is determined by x (independent variable). Y (dependent variable) depends on x. Vertical Line Test- a curve is the graph of a function of x, if and only if, no vertical line intersects the curve more than once Piecewise functions- these are functions that are defined by different formulas on different intervals Symmetry 1. A function f is an even function if f(x)=f(-x) for all x in its domain; symmetric about the y-axis 2. A function f is an odd function if f(-x)=-f(x) for all x in its domain; symmetric about the x and y-axis Increasing Function- a function is increasing on an interval, I, if f(x ) < f(x 1 for x2,x 1 e 1, 2here x f(x )1for x ,2 1 e I,1wh2re x 0 and even it will look like a parabola and if a is odd it will ook like a cubic function 4. Rational Functions y=p(x)/q(x), where p(x) and q(x) are polynomials Transformations of Functions Translations(suppose c>0) 1. Y=f(x) +c is a vertical shift up c units 2. Y=f(x) c is a vertical shift down c units 3. Y=f(x-c) is a horizontal shift right c units 4. Y=f(x+c) is a horizontal shift left c unitsStretching (suppose c>1) 1. Y=cf(x) is a vertical stretch of factor c 2. Y=1/cf(x) is a vertical compression of factor c 3. Y=f(cx) is a horizontal compression of factor c 4. Y=f(1/cx) is a horizontal shift of factor c Reflections (suppose y=f(x)) 1. Y=-f(x) is a reflection of the graph in the x-axis 2. Y=f(-x) is a reflection of the graph in the y-axis Combination of Functions 1. Addition (f+g)(x) = f(x)+g(x); domain is the intersection of the domain of f and g 2. Subtraction (f-g)(x) = f(x)-g(x); domain is the intersection of the domain of f and g 3. Multiplication (fg)(x)=f(x)g(x); domain is the intersection of the domain of f and g 4. Division (f/g)(x) = f(x)/g(x); domain is the intersection of the domain of f and g, except where g(x) =0 5. Composition (fog)(x) = f(g(x)); everywhere you see an x it becomes replaced with g(x); domain is all x in the domain of g, such that, g is in the domain of f x Exponential Functions-an exponential function is a function of the form f(x) = a where a>0 1. When xeZ, x> 0 a = a(x times) 2. When xeZ, x>0 a =1/a (reciprocal) x (m/n) m 3. When xeQ, x=m/n then a =a =nth root (a ) 4. If a>0 it will never touch the x-axis (ho
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