Precalculus Summer Packet
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1 Precalculus Summer Packet These problems are to be completed to the best of your ability by the first day of school You will be given the opportunity to ask questions about problems you found difficult during the first week of school You will be tested on this information at the beginning of the second week of school This packet will be graded as your first week s worth of homework DO NOT WAIT until the night before to do this!! YOU LL BE SORRY! These problems should be familiar to you They are review questions from Algebra I and Algebra II Below is a list of topics covered by these problems and formulas you may need Linear Equations: y = m + b, y y = m( ) Intersection of two lines: (graphing, substitution, elimination) Distance between two points: d = ( ) + ( y ) y Midpoint of a line segment: + y +, y y y Slopes of Lines:, parallel lines (have the same slope), perpendicular lines (have negative reciprocal slopes) Linear functions: Comple numbers: f ( ) = m + b a + bi, the number i =, comple conjugate a bi Solving quadratic equations: factoring, quadratic formula completing the square, the discriminant Quadratic functions: parabolas, verte, ais of symmetry, b ± b 4ac =, a y = a + b + c
2 For problems and, find the length of CD and the midpoint of CD 9 C 8,, D 7,5 C =,, D=, = ( ) = ( ) ( ) ( ) For problems and 4, solve the given system of equations + y = y = y = y = 6 5 Plot the points A = (,), B = ( 4, 6), C = ( 9,), and D = ( 7,7) bisect each other What kind of quadrilateral is ABCD? Show that AB and CD For problems 6 and 7, find the slope of the line joining the points whose coordinates are given 6 (,8) and (, ) 7 (, ) and (, 4 ) For problems 8, find the slope and the y-intercept of the line whose equation is given 8 y = 5 9 y= y= 7 Tell which of the given equations have parallel line graphs and which have perpendicular line graphs a y= 5 5 b y = + c y= Show that the line through (, ) and ( ) (,7) and (, ) 7, is perpendicular to the line through Find the value of k if the line joining ( 4, k ) and ( 6,8 ) and the line joining (,4) (,8 ) are a parallel b perpendicular For problems 4 7, write an equation of the line described 4 The line with y-intercept -6 and parallel to the line 5+ 4y= 5 The line through (,4) and ( 5,8 ) 6 The horizontal line through ( 5, 7) and
3 7 The vertical line through ( 5, 7) 8 a If f( ) =, find f ( ) and ( ) 4 f b Find the zero of f( ) 9 Let ht () 9 4t = Decide whether h( 45) h( 5) h( ) = For problems 8, simplify each epression ( 4 i) + ( 6+ 8i) 4 4( + 5i) ( 6i) 5 ( 6 i)( 6+ i) + 7 ( 8+ i)( 5i) 8 ( 4 5i) 6 ( 5 i 5)( 5 i 5) For problems 9 4, write each epression in the form a+ bi i + 5i 5 i 5 i 4 5 i+ i + i + i + i i 4 i 5 For problems 5 and 6, solve the equation by factoring = = 6 ( )( ) For problems 7 9, solve the equation by completing the square Give both real and imaginary roots 7 = z 6z 768= = For problems 4 4, solve the equation using the quadratic formula Give your answers in simplest radical form Give both real and imaginary roots = 4 t = t v 6 v = v 4 For problems 4 45, sketch each parabola Label the verte, the ais of symmetry, and the - and y-intercepts 4 = = y= ( + ) + 6
4 The problems that follow are a preview of your first test You probably want to try these! Let A = (, 6) and B = ( 4,) midpoint of AB Find the value of a if the point ( 4, ) Find the length of AB and the coordinates of the lies on the line + ay = 4 Solve the equations + y= and 6 y= 4 simultaneously Then sketch the graphs of the lines and label the intersection point 4 Find the slope and the y-intercept of the line 4 y= 7 5 Tell which of the following equations have parallel line graphs and which have perpendicular line graphs a + y= b y = + c 6 4y = 6 Write an equation of the line through ( 6, ) and (,) 7 Write an equation of the line through ( ) 5,5 and parallel to the line 4+ y= 8 Write an equation of the line with -intercept - and y-intercept -5 9 Write an equation of the vertical line through ( 4, ) a Describe, in a few sentences, the steps involved in finding an equation of the perpendicular bisector of a line segment b Write an equation of the perpendicular bisector of the segment joining the,8 points ( 7, ) and ( ) For problems 6, epress each comple number in the form a+ bi 5 8 ( + i) + i 4 + i i 5 4( i) 5 ( i) i
5 7 A teacher must grade eam papers Working for 6 minutes, she is able to grade eactly seven eams The function that models the number of eams graded is shown below in red Because some eams take longer to grade than others, the teacher does not work at a constant rate Nevertheless, her rate is close enough to being constant that a linear function (whose graph is the straight line shown in blue) can be used effectively to model her progress a Give an equation of the linear function that models the number of eams graded as a function of time b Use the equation to find approimately how long it will take to grade all the eams 8 Solve each equation for a 7 = 5 b 4= 9 c + + = 9 Sketch the graph of each parabola Label the verte, the ais of symmetry, and the intercepts y= 8 b a ( ) = Sketch the graphs of the line y= and the parabola coordinates of any point(s) of intersection = Find the
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