Pre-Cal

1. Quadratics
   1. Quadratics
      1. Parabola

         Annotations:

         • [Image: http://mathforum.org/cgraph/history/pictures/glossary/parabola.gif] (http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=u74A2-_Poxn2sM&tbnid=jfLcrr3p1zs6jM:&ved=0CAUQjRw&url=http%3A%2F%2Fmathforum.org%2Fcgraph%2Fhistory%2Fglossary.html&ei=BRtnUfXqG5CC9QSzioHAAg&bvm=bv.45107431,d.eWU&psig=AFQjCNHFtH8Qzh3OAYoXdRuXe02Vx2fsSg&ust=1365798006146604)
         • When asked to write the standard form of the equation of the parabola whose vertex is (1,2) and that passes through the point (3,-6) you can find the equation by using the vertex. The vertex of a parabola is (h,k). Because we know that our vertex is (1,2), we also know that our h is qual to 1 and our k is equal to 2. The next step is to substitue h and k in standard form. ( stand form: f(x)=a(x-h)^2+k ) So, our equation now looks like the following: f(x)=a(x-1)^2+2. Because the parabola passes through the point (3,-6), it follows that f(3)=-6. So, the next step would look like....-6=a(3-1)^2+2                                     -6=4a+2                                      -2=a So, the equation in standform is f(x)= -2(x-1)^2+2  
      2. Minimum/Maximum

         Annotations:

         • By writing the quadratic function f(x)=ax^2+ bx+c in standard from, f(x)=a(x+(b/2a))^2+(c-(b^2/4a)) you can see that the vertex occurs at x=-b/2a, which implies the following... 1) If a > 0, f has a minimum at x=-b/2a 2) If a
2. Rational Function Graphing
   1. Zeros

      Annotations:

      • If f is a polynomial function and a is a real number, the following statements are equivalent. 1. x=a is a zero of the function f 2. x=a is a solution of the polynomial equation f(x)=0 3. (x-a) is a factor of the polynomial f(x) 4. (a,0) is an x-intercept of the graph of f
      • [Image: http://www2.norwalk-city.k12.oh.us/wordpress/precalc0910/files/2009/09/Graph-1.JPG] (http://www2.norwalk-city.k12.oh.us/wordpress/precalc0910/2009/09/21/2-3-real-zeroes-of-polynomial-functions/)
   2. Asymptotes
      1. Oblique
      2. Horizontal and Vertical

         Annotations:

         • 1. The line x=a is a vertical asymptote of the graph of f if f(x) increases to infinity or f(x) increases toward negative infinity as x increases toward a, either from the right or from the left. 2. The line y=b is a horizontal asymptote of the graph of f if f(x) increases toward b as x increases toward infinity or x increases toward negative infinity.
         • [Image: http://www.mathnstuff.com/math/spoken/here/1words/a/a28ex1.gif] (http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=5tg7pJOMSAMH8M&tbnid=j7zp-IjQUSa-sM:&ved=0CAUQjRw&url=http%3A%2F%2Fwww.mathnstuff.com%2Fmath%2Fspoken%2Fhere%2F1words%2Fa%2Fa28.htm&ei=SixnUeLGAoSc9QTX2YHgBw&bvm=bv.45107431,d.eWU&psig=AFQjCNFOuBHSYJ620K0F5DSQKuzl6DdXYA&ust=1365802435306782) [Image: http://www.nku.edu/~longa/classes/mat119/days/images/03p278a.jpg] (http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=W3-Fs4kq-lqsvM&tbnid=JkZwlyUn8gaA8M:&ved=0CAUQjRw&url=http%3A%2F%2Fwww.nku.edu%2F~longa%2Fclasses%2Fmat119%2Fdays%2F3.7.rationals.html&ei=TC5nUay9IoWQ8wSbmYDgDQ&bvm=bv.45107431,d.eWU&psig=AFQjCNGdUu_TZSinMhHESOk23QqKp9Vf7Q&ust=1365802939311169)
      3. Slant

         Annotations:

         • If the degree of the numerator is exactly one more than the degree of the denominater, the graph of the function has a slant asymptote. For instance, by dividing x+1 into x^2-x you have.. f(x)= (x^2-x)/(x+1)= (x-2)+(2/x+1) As x increases or decreases without bound, the remainder term 2/(x+1) approaches 0, so the graph of f approaches the line y=x-2.
         • [Image: http://www.nabla.hr/SlantAsymptFig.gif] (http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=6xLmyoLjSb_HhM&tbnid=b0zPGjHVm9ILUM:&ved=0CAUQjRw&url=http%3A%2F%2Fwww.nabla.hr%2FZ_CalculusLimitOfFunction_2.htm&ei=ri5nUaqUEoL28gTu4oC4Cg&bvm=bv.45107431,d.eWU&psig=AFQjCNGdUu_TZSinMhHESOk23QqKp9Vf7Q&ust=1365802939311169)
   3. Higher Degree
      1. Zeros

         Annotations:

         • Find the polynomial function with the following zeros.. -1/2, 3, 3 Note that the zero x= -1/2 corresponds to either (x+ 1/2) or (2x+1). To avoid fractions, choose the second factor and write f(x)=(2x+1)(x-3)^2 f(x)=(2x+1)(x^2-6x+9)=2x^3-11x^2+12x+9
         • To use the Rational Zero Test, first list all rational numbers whose numerators are factors of the constant term and whose denominators are factors of the leading coefficient. possible rational zeros=factors of constant term/factors of leading coefficient Now that you have formed this list of possible rational zeros, use the trial-and-error method to determine which, if any, are actual zeros of the polynomial. Note that when the leading coefficient is 1, the possible rational zeros are simply the factors of the constant term. 
3. Linear

   Annotations:

   • [Image: http://upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Linear_Function_Graph.svg/300px-Linear_Function_Graph.svg.png] (http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=EPVJyaSLaQ2VuM&tbnid=XXxRt9BThY1YFM:&ved=0CAUQjRw&url=http%3A%2F%2Fen.wikipedia.org%2Fwiki%2FLinear_equation&ei=NzdnUZv5Doj68QTn64CwCg&bvm=bv.45107431,d.eWU&psig=AFQjCNHI_AibpRLvh8lDBiOa0qPe20Sj_w&ust=1365805231427505)
   1. Slope

      Annotations:

      • The slope m of the nonvertical line through (x1,y1) and (x2,y2) is... m=(y2-y1)/(x2-x1)=change in y/change inx where x1 cannot equal x2. ****When the formula for slope is used, the order of subtraction is important!
      • 1. A line with positive slope (m>0) rises from left to right 2. A line with negative slope (m
      • Point-Slope Form: y-y1=m(x-x1) Slope-Intercept Form y=mx+b
   2. Summary of Equations

      Annotations:

      • 1. General Form: Ax+By+C=0 2. Vertical Line: x=a 3. Horizontal Line: y=b 4. Slope-intercept Form: y=mx+b 5. Point-slope form: y-y1=m(x-x1)

1. The Unit Circle

   Annotations:

   • [Image: http://instaedu.com/media/tutor_session_docs/questions/3a4/3a4b7202888228e3dffdd3b794c3f655/UnitCircle.gif] (http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=1T2uFJlJGRL6cM&tbnid=FyuDfKO6GiIszM:&ved=0CAUQjRw&url=http%3A%2F%2Finstaedu.com%2FPre-Calculus-questions%2FCan-you-show-me-the-steps-to-the-following-problem-sin45-%2B-cos60%2F&ei=zTdnUZmJPIXm8QTB54HgAg&bvm=bv.45107431,d.eWU&psig=AFQjCNE_zKLA_PAAHwtpKMf2tk7Uld0J7Q&ust=1365805379950125)
   • The unite circle is used to determine the value of sine and cosine of differnt angles. it is also used to solve trig equations. One common theory used by the unit circle is the Pythagorean theorem.If (x, y) is a point on the unit circle, then |x| and |y| are the lengths of the legs of a right triangle whose hypotenuse has length 1, leaving x and y to satisfy the equation.. x^2 + y^2 = 1
2. Basic Trig Functions
   1. Right Triangle Definitions

      Annotations:

      • Let x be an acute angle of a right triangle. Then the six trigonometric functions of the angle x are defined as follows... sin x= opp/hyp cos x=adj/hyp tan x=opp/adj csc x=hyp/opp sec x=hyp/adj cot x=adj/opp
   2. Definitions of Any Angle

      Annotations:

      • Let K be an angle in standard position with (x,y) a point with (x,y) on the terminal side of K and r= the square root of x^2+y^2 and it cannot equl 0. sin K=y/r cos K=x/r tan K=y/x, x cannot equal 0 cot K=x/y, y cannot equal 0 sec K=r/x, x cannot equal 0 csc K=r/y, y cannot equal 0
   3. Sin, Cos, and Tan Functions

      Annotations:

      • [Image: http://www.snow.edu/jonathanb/images/y_sinx_blue.GIF] (http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=7IwFQ3Bsbs72bM&tbnid=8qprFral_KnSsM:&ved=0CAUQjRw&url=http%3A%2F%2Fwww.snow.edu%2Fjonathanb%2FCourses%2FMath1210%2Farcsine.html&ei=WzpnUc2UC4Hc8wTnlIHQDw&bvm=bv.45107431,d.eWU&psig=AFQjCNETS1nrh7l9Djw-1Y7qu3GKpqRW1g&ust=1365806027687816)
      • [Image: http://www.math.ubc.ca/~cass/courses/m309-03a/a1/endersby/images/cosnew.gif] (http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=SWN-xv4Y9hg-pM&tbnid=au2R-TWN-rM1KM:&ved=0CAUQjRw&url=http%3A%2F%2Fwww.math.ubc.ca%2F~cass%2Fcourses%2Fm309-03a%2Fa1%2Fendersby%2Fwebpage.html&ei=mz1nUZfKGY229gTvr4DYBw&bvm=bv.45107431,d.eWU&psig=AFQjCNGxWW9bwcsWkfOx_vRGztwVZ6_6sw&ust=1365806864225414)
      • [Image: http://pages.physics.cornell.edu/courses/p101-102/math/images/tan1.gif] (http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=i2oufN7wi4HxZM&tbnid=LGannpiGa8F5vM:&ved=0CAUQjRw&url=http%3A%2F%2Fpages.physics.cornell.edu%2Fcourses%2Fp101-102%2Fmath%2Ftrigonometric.html&ei=1T1nUdP_JJKO8wS1pIDICg&bvm=bv.45107431,d.eWU&psig=AFQjCNExh5axwTByX4z4pnGkPzLXbgTsXg&ust=1365806914583536)
      • The graphs of y= a sin(bx-c) and y= a cos(bx-c) have the following characteristics. (assume b >0) Amplitude= the absolute value of a Period= 2(pi)/b The left and right end points of a one-cycle interval can be determined by solving the equations bx-c=0 abd bx-c=2(pi)
3. Analytic Trig
   1. Trig Identities

      Annotations:

      • sin u=1/csc u cos u=1/sec u tan u=1/cot u csc u=1/cos u sec u =1/cos u cot u=1/tan u
      • tan u=sin u/cos u cot u=cos u/sin u
      • sin^2u+cos^2u=1 1+tan^2u=sec^2u 1+cot^2u=csc^2u
      • sin(-u)=-sin u cos(-u)=cos u tan(-u)=-tan u csc(-u)=-csc u sec(-u)=sec u cot(-u)=-cot u
      1. Verifying

         Annotations:

         • Verify the identity (sin x/1+cos x)+(cos x/sin x)=csc x (sin x/1+cos x)+(cos x/sin x)=(sin x)(sin x)+(cos x)(1+ cos x)/(1 + cos x)(sin x)                                              =sin^2x + cos^2x + cos x/(1 + cos x)(sin x)  multiply                                               =1+cos x/(1 + cos x)(sin x)   pythagorean identity                                               =csc x   divide out common factor and use reciprocal identity                                          
      2. Sum and Difference

         Annotations:

         • cos(A+B) = cos A cos B − sin A sin B sin(A+B) = sin A cos B + cos A sin B cos(A−B) = cos A cos B + sin A sin B sin(A−B) = sin A cos B − cos A sin B tan(A+B) = (tan A + tan B) / (1 − tan A tan B) tan(A−B) = (tan A − tan B) / (1 + tan A tan B)

Annotations:

• [Image: http://people.richland.edu/james/lecture/m116/logs/log2.gif] (http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=w6KYNUz8YLp20M&tbnid=U5kYdBmihVLWHM:&ved=0CAUQjRw&url=http%3A%2F%2Fwww.richland.edu%2Fjames%2Flecture%2Fm116%2Flogs%2Flogs.html&ei=AkdnUd--CYy29gS064DIBg&bvm=bv.45107431,d.eWU&psig=AFQjCNH4nU5SGt7APhoajgV3aDEncb5giA&ust=1365809006921596)

1. Exponential

   Annotations:

   • The expontential function f with base a is denoted by.. f(x)=a^x where a > 0, a cannot equal 1, and x is any real number.
   • [Image: http://tutorial.math.lamar.edu/Classes/CalcI/ExpFunctions_files/image001.gif] (http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=YHOmnU-u8U55DM&tbnid=Hh1Rj7FTp7ZYdM:&ved=0CAUQjRw&url=http%3A%2F%2Ftutorial.math.lamar.edu%2FClasses%2FCalcI%2FExpFunctions.aspx&ei=9URnUY6ZAY3a9QTQ6YDwCg&bvm=bv.45107431,d.eWU&psig=AFQjCNHt88U4f_ePwEZ6x7In-7KdMARhfg&ust=1365808736979848)
2. Logarithmic

   Annotations:

   • For x > 0, a > 0, and a cannot equal 1.  y=log(base a)x if and only if x=a^y The function given by f(x)=log(base a)x is called the logarithmic function with base a.
   • [Image: http://www.sosmath.com/algebra/logs/log4/log004.gif] (http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=PEjzGVtf4-524M&tbnid=G7HFklqH0BtVvM:&ved=0CAUQjRw&url=http%3A%2F%2Fwww.sosmath.com%2Falgebra%2Flogs%2Flog4%2Flog4.html&ei=9EVnUf7DFom49gSKuoHoDw&bvm=bv.45107431,d.eWU&psig=AFQjCNH4nU5SGt7APhoajgV3aDEncb5giA&ust=1365809006921596)
   1. Properties

      Annotations:

      • 1. log(base a)1= 0 because a^0=1 2. log(base a)a=1 because a^1=a 3. log(base a)a^x=x and a^log(base a)x=x     (inverse properties) 4. If log(base a)x=log(base a)y, the x=y          (one-to-one property)
      1. Base Properties

         Annotations:

         • 1. loga (uv) = loga u + loga v 2. ln (uv) = ln u + ln v 3. loga (u / v) = loga u - loga v 4. ln (u / v) = ln u - ln v 5. loga un = n loga u 6. ln un = n ln u
   2. Natural Log

      Annotations:

      • For x > 0, y=ln x if and only if x=e^y The function given by f(x)=log(base e)x=ln x is called the natural logarithmic function. ***Note that the natural logarithm ln x is written without a base because the base is understood to be e.
      1. Properties

         Annotations:

         • 1. ln 1=0 because e^0=1 2. ln e=1 because e^1=e 3. ln e^x=x and e^ln x=x   (inverse properties) 4. If ln x = ln y, then x=y    (one-to-one property)
3. Solving Strategies

   Annotations:

   • 1. Rewrite the original equation in a from that allows the use of the one-to-one properties of exponential or logarithmic functions. 2. Rewrite an exponential equation in logarithmic form and apply the Inverse Property of logarithmic functions. 3. Rewrite a logarithmic equation in exponential form and apply the Inverse Property of exponential functions.

1. Increasing and Decreasing

   Annotations:

   • A function f is increasing on an interval if, for and x1 and x2 in the interval, x1 f(x2) A function f is constant on an interval if, for any x1 and x2 in the interval, f(x1)=f(x2)
2. Relative Min/Max

   Annotations:

   • [Image: http://www.ltcconline.net/greenl/courses/103a/keys/mid1PracticeExam/img1D.jpg] (http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=0FzDCcewQ6-WSM&tbnid=fOnVOjDuZ03_cM:&ved=0CAUQjRw&url=http%3A%2F%2Fwww.ltcconline.net%2Fgreenl%2Fcourses%2F103a%2Fkeys%2Fmid1PracticeExam%2FExam1PracticeKey.htm&ei=gk1nUYaAK4XA8ATS5YFI&bvm=bv.45107431,d.eWU&psig=AFQjCNGmkfKjrt57W6QlLaVtdXsyYjlvzw&ust=1365810926005997)
3. Even and Odd

   Annotations:

   • How to Tell: When a graph is symmetric to the y-axis its an even function. When a graph is symmetric to the origin its an odd function. When a graph is symmetric to the x-axis its not a function. Test for Even and Odd: A function f is even if, for each x in the domain of f, f(-x)=f(x) A function is odd if, for each x in the domain of f, f(-x)=-f(x)
   • [Image: http://image.tutorvista.com/Qimages/QD/53120.gif] (http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=nJrtDvZo4WFnnM&tbnid=nA4bLC5LQJPo-M:&ved=0CAUQjRw&url=http%3A%2F%2Fworksheets.tutorvista.com%2Feven-and-odd-function-worksheet.html&ei=8E5nUY3uK4nY9QTX4oC4Aw&bvm=bv.45107431,d.eWU&psig=AFQjCNHcyTBDw01ehBgBU-MMpoBp5BcjOQ&ust=1365811300529091) Graph A is an odd function, graph B is niether, and graph C is an even function.
4. Shifts

   Annotations:

   • Let c be a positive real number. Vertical and horizontal shifts in the graph of y=f(x) are represented as follows. 1. vertical shift c units upward: h(x)=f(x)+c 2. vertical shift c units downward:  h(x)=f(x)-c 3. horizontal shift c units to the right:  h(x)=f(x-c) 4. horizontal shift c units to the left:   h(x)=f(x+c)  
   • [Image: http://dl.uncw.edu/digilib/mathematics/algebra/mat111hb/pandr/quadratic/shift.gif] (http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=1ioMiLmnzTavMM&tbnid=q-MOZfcDEHdD8M:&ved=0CAUQjRw&url=http%3A%2F%2Fdl.uncw.edu%2Fdigilib%2Fmathematics%2Falgebra%2Fmat111hb%2Fpandr%2Fquadratic%2Fquadratic.html&ei=7lFnUcv6EIus8ATNq4C4Bg&bvm=bv.45107431,d.eWU&psig=AFQjCNHcWz45Q3Qv9N73Rj7cf142OrHbIg&ust=1365812066485655)
5. Inverse Functions

   Annotations:

   • Let f and g be two functions such that.. f(g(x))=x  for every x in the domain of g and g(f(x))=x  for every x in the domin of f Under these conditions, the function g is the inverse function of the function f. The function g is denoted by f^-1. So, f(f^-1(x))=x and f^-1(f(x))=x The domain of f must be equal to the range of f^-1, and the range of f must be equal to the domain of f^-1.
   • [Image: http://library.thinkquest.org/2647/media/parainv.gif] (http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=31bYvuvs4OsN7M&tbnid=zWH3_PkN7MwuXM:&ved=0CAUQjRw&url=http%3A%2F%2Flibrary.thinkquest.org%2F2647%2Falgebra%2Fftinvers.htm&ei=xVFnUYuKHZCA9gT6woCACQ&bvm=bv.45107431,d.eWU&psig=AFQjCNEsRzgYBIW1hPVhVG06x23pDyiZpg&ust=1365812022134884)