Math 150 Study Guide

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$Math 151 Syllabus Summer 2013$

Math 150 Study Guide• Basic vertical asymptote of multiplicity two: The Reciprocal SquareFunction y = 1x 2• Infinite discontinuity• May NOT be touched or crossed• Finite (or “defined”) x values that make y undefined• Occur whenever the denominator has real roots• If lim f xx →m( ) = ∞ or - ∞ or ± ∞ (check the sign chart for one-sided and two-( ) has a vertical asymptote at x = m .sided limits), then f x• VA multiplicity of 1 or 2; multiplicity affects the sign chartk) Jump Discontinuity• Basic Jump: y = x x = x ⎧⎪−1 if x < 0= ⎨x ⎩⎪ 1 if x > 0• Occurs when an x –value makes y indeterminate AND the function involvesan absolute valuel) End Behaviour/Non-Vertical Asymptotes• Zoomed out behaviour, what the graph looks like when you zoom out farenough• May or may not be touched or crossed• Found as the quotient of a rational function ( y = q x( ) from the properfraction form); the degree of the quotient polynomial prescribes the “type”of non-VA: horizontal, slant, parabolic, etc.. Furthermore, the type dependsof the difference in the degrees of the numerator and denominatorpolynomialsm) Special Case of Non-VA: Horizontal (constant quotient) Asymptote• Non-VA of degree 0• Infinite x produces finite y (compare with vertical asymptotes, for whichfinite x produces infinite y)( ) = k means that y = k is a horizontal asymptote of f ( x )• lim f xx →±∞n) About the different equation forms: Recognize various forms for equations ofrational function graphs, be able to extract properties from various equationforms, be able to graph from such forms, be able to determine the mostappropriate form in given situations, and be able to rearrange from one formto another42. Solving Equations in One Variable:a) get LCD© Raelene Dufresne 2013 16 of 22
Math 150 Study Guideb) consider domain restrictions to eliminate possible extraneous solutionsc) clear fractions by multiplying both sides of the equation by the LCDd) solve the resulting polynomial using factoring, rational root theorem, etc.e) eliminate any extraneous solutions43. Solving Inequalities in One Variable:a) never solve as an equation! (Well, there are a few instances when you can, butin general, and usually, you cannot.)b) solve graphically by comparing heights of two different functions (the leftside and right side functions)c) be clever in rearranging terms in the inequality to make solving by graphingeasier. For example, rearrange x 2 + 2x + 5 > 0 as x 2 + 2x > −5 , both of whichare easy to sketch; you should quickly note that y = x 2 + 2x is ALWAYShigher than y = −5 and therefore the solution is x ∈° .d) Solve by sign chart (which is a numerical and partial graphical method), usefulwhen the complete graph is not required. VERY IMPORTANT NOTES:• All terms must be on one side (let’s say the left) and ZERO on the other• The expression on the left must be FULLY FACTORED; all factors(expressions in parentheses) then must be joined either by multiplication ordivision, NOT by addition and subtraction. (Why? Do you know what (-)times (-) is? YES .. positive. But do you know what (-) + (-) is? No ... itdepends on HOW BIG each negative quantity is.44. Graph: Know all steps and required informationa) absolute value, polynomial, power/root, greatest integer, rational, exponential,logarithmic, logistic and piecewise functions and some relations (circles andhorizontal parabolas) by handb) use sign charts to aid in graphing functions45. Modeling and Optimization:a) Regression curves; scatter plots; interpolation and extrapolationb) Real world domain restrictions and extraneous solutionsc) Writing word problems algebraicallyd) Geometry formulas• Circle: C = 2πr and A = πr 2• Surface Area: sum of all surfaces’ areas• Volume: V = 1 n A ⋅ h , where n = 1,2 or 3basee) Optimization (max/min word problems)• Define all quantities using meaningful variables different from x and y• Define independent quantities using lower case variables and dependentquantities using upper case variables© Raelene Dufresne 2013 17 of 22
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Math 150 Study Guide

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