CBC MATHEMATICS DIVISION MATH 1324-Exam MATHEMATICS DIVISION MATH 1324-Exam Formula Sheets Equations and Functions Linear Equations • = + ...

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<ul><li><p> CBC MATHEMATICS DIVISION MATH 1324-Exam Formula Sheets </p><p> Equations and Functions Linear Equations </p><p> = + 1 = ( 1) = = </p><p> Linear Function: () = + </p><p> =21</p><p>21 ; 2 1 0 </p><p> General Form of Quadratic Function: () = 2 + + , ( 0) </p><p> Quadratic Formula </p><p> =24 </p><p>2 </p><p> Formulas to find Vertex (, ) </p><p> = </p><p>2 = ()2 + () + , </p><p> or (</p><p>2, (</p><p>2)), or (</p><p>2,</p><p>42</p><p>4) </p><p> Axis of symmetry: = </p><p> Vertex Form of Quadratic Function: () = ( )2 + vertex (, ) </p><p> Polynomial function: () = + 1</p><p>1 + + 11 + 0 </p><p> Rational function: () =()</p><p>() , () and () are polynomials, but () 0. </p><p> Vertical Asymptotes </p><p> For () in simplified form, if () = 0, then = is a Vertical Asymptote. </p><p> Horizontal Asymptote </p><p> = 0 is the Horizontal Asymptote if degree of () &lt; degree of (). </p><p> =</p><p> is the Horizontal Asymptote if degree of () = degree of (), </p><p>where is leading coefficient of () and is leading coefficient of (). </p><p> If degree of () &gt; degree of (), then there is no Horizontal Asymptote. </p><p> Exponential Function: () = , where &gt; 0, 1. </p><p> Properties of Exponential Functions: &gt; 0, &gt; 0, 1, 1, and , real.</p><p> = + , () = , () = </p><p> . </p><p>= , (</p><p>)</p><p>=</p><p> = , if and only if = . For 0, = , if and only if = . </p></li><li><p> CBC MATHEMATICS DIVISION MATH 1324-Exam Formula Sheets </p><p>Auth:C.Villarreal-Professor 2015Falll </p><p> Logarithmic Function: () = log() </p><p> log(1) = 0 , log() = 1 , log() = , log(</p><p> ) = </p><p> log( ) = log() + log() </p><p> log ( </p><p> ) = log() log() </p><p> log() = log() </p><p> If log() = log(), then = . </p><p> If = , then log() = log(). </p><p> Change of Base formula log() = log()</p><p>log() or log() = </p><p>ln()</p><p>ln() </p><p> Function Transformations </p><p> Reflections </p><p> = () reflect () about -axis = () reflect () about -axis </p><p> Stretch and Compress </p><p> = (), &gt; 0 vertical: stretch () if &gt; 1 : compress () if 0 &lt; &lt; 1 </p><p> = (), &gt; 0 horizontal: stretch () if 0 &lt; &lt; 1 : compress () if &gt; 1 </p><p> Shifts </p><p> = () + , &gt; 0 vertical: shift () up = () , &gt; 0 : shift () down </p><p> = ( + ) &gt; 0 horizontal: shift () left </p><p> = ( ), &gt; 0 : shift () right </p><p> System of Equations and Matrices </p><p> 3 Matrix Row Operations: Switch any two rows. Multiply any row by a nonzero constant. Add any constant-multiple row to another. </p><p> Solve a system of equations(Gaussian Elimination) Rewrite system of equations as augmented matrix. Apply row operations to obtain Row Echelon form. Write row equations and solve for solutions.(back-substitute if necessary) </p></li><li><p> CBC MATHEMATICS DIVISION MATH 1324-Exam Formula Sheets </p><p>Auth:C.Villarreal-Professor 2015Falll </p><p> Basic Properties of Matrices </p><p> ( + ) + = + ( + ) + = + + 0 = 0 + = </p><p> + () = () + = 0 () = () = = </p><p> If is square matrix and 1 exists, then 1 = 1 = . ( + ) = + , ( + ) = + If = , then + = + If = , then = and = </p><p> Matrix Equation: = = 1, provided is square and 1 exists. </p><p> Simplex Method: Summary of Problem Types </p><p>Problem Type </p><p>Constraints Constants Objective Function </p><p>Coefficients Method </p><p>Maximization Nonnegative Real Simplex Minimization Real Nonnegative Dual Maximization Mixed Nonnegative Real Big Minimization Mixed Nonnegative Real Big </p><p> Exponential Models Formulas </p><p> Simple Interest: = </p><p> Compound Interest: = (1 + ) = (1 +</p><p>)</p><p> Continuous Compounding: = = </p><p> Annual Percentage Yield(Effective Rate of Interest): </p><p> Compounding times per year = (1 +</p><p>)</p><p> 1 </p><p> Compounding continuously per year = 1 </p><p> Annuity(Future Value): = ((1+</p><p>)</p><p> 1 </p><p>(</p><p>)</p><p>) </p><p> Annuity(Present Value): = ( 1 (1+</p><p>)</p><p>() </p><p>(</p><p>)</p><p>) </p><p> Logistic Model: () =</p><p>1+() </p></li><li><p> CBC MATHEMATICS DIVISION MATH 1324-Exam Formula Sheets </p><p>Auth:C.Villarreal-Professor 2015Falll </p><p> Counting Principles </p><p> ( ) = () + () ( ) </p><p> ! = ( 1)( 2) 2 1 </p><p> 0! = 1 , ! = ( 1)! </p><p> =!</p><p>()! for 0 </p><p> =!</p><p>!()! for 0 </p><p> Probability </p><p> () =()</p><p>() </p><p> ( ) = () + () ( ) </p><p> Odds for =()</p><p>()=</p><p>()</p><p>1() where () 1. </p><p> Odds against =()</p><p>() where () 0. </p><p> Conditional Probability: (|) =()</p><p>() where () 0. </p><p> (|) =()</p><p>() where () 0. </p><p> Product Rule: ( ) = ()(|) = ()(|), where () 0, () 0. </p><p> If ( ) = ()(), then and independent. </p><p> If () 0 and () 0, and either (|) = () or (|) = (), then and are independent. </p><p> If 1, 2, , are independent, then (1 2 ) = (1) (2) () </p><p> Bayes Formula: (1|) =(1)</p><p>()=</p><p>(1)</p><p>(1)+(2)++() </p><p> =(|1)(1)</p><p>(|1)(1)+(|2)(2)++(|)() </p></li></ul>

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