Problems on Quadratics, Straight Lines, Polynomials, Logs and Exponentials

1. (10 points) Express A and B as simplified, improper fraction:  For the calculation of A, cancel common factors in numerators and denominators before multiplication. For the calculation of B, use the least common denominator.

A =

25 27

.

81 100

B =

3 26

+

7 10

For A and B, there is 1 point for right answer and 4 points for following instruction on how to arrive at the answer. Show work!  If you use a calculator (not recommended), show the reasoning needed to arrive at your solution without its use.

2. (10 points)  Use algebra to find the intersection point of the two lines  

y = ¾x - ¼
y = ¼x + 2

3.  (4 points) (a) Find the slope of the straight line K if K is perpendicular to a line L with slope 2.
     (6 points) (b)  Find the slope and an equation for a straight line T which passes through [3,5] and [8, 20].        
                            Express the equation in slope intercept form y= m x + b

4.  (10 points) Compute product (x⁴+3x³+6x² + 3x + 1)(x² + 2x + 1) using  a column  method. 

5.  (10 points)  Use the polynomial long division to find polynomials a(x) (the quotient) and r(x) (the remainder)  such that 

x⁴+3x³+6x² + 4x + 2= a(x)(x² + 2x + 1) + r(x)

with degree (r(x)) < 2. 

6.   (2 points)  (i)  Use the ln(x) button on your calculator to fill in the table (or a copy of it) 

     (3 points)  (i) Use the e^x button on your calculator to complete in the table (or a copy of it). 

   (5 points)   (iii) Draw  the straight line y = x from [-2,-2] to [6,6] on graph paper.  Then on the same graph, use the above points to sketch graph of y = ln(x) for x in the interval [1/6, 6] and  y = exp(x) =  e^x  for x in the interval [-1.79, 1.79]. Label all curves.  

7. (3 points) (i) Solve x²-5x+ 4 =0 using factorization by inspection. Show all ways for 4 to equal  the product AB of two integers A and B.  
    (3 points) (ii) Solve x²-5x+4 = 0  with the quadratic formula. Show work.
    (4 points) (iii) Solve x² - 5x + 4 =0  starting with the method of completing the square. Show work.

8.  (4 points)  (i) Find the x- and y-intercepts of  the quadratic y =  x²-5x+4 and the straight line y =  -2x + 8 with the x- and y-axes. Hint: See 7(ii) or (iii).
     (3 points)  (ii) Sketch the curves  y =  x²-5x+4 and y =  -2x + 8. Identify the axis of symmetry for the quadratic. Label axes  and all intercepts.
     (3 points) (iii) Find the coordinates of the intersection points for the  quadratic y =  x²-5x+4 and  line y =  -2x + 8.  Show reasoning.  Hint:  x²-3x-4  = (x-4)(x+1). 

 

_______    |      |     |    //  _   _ \\ /\             /\   <|  (o)   (o)   |>   \     | |      /  \___ _/ ||  -/[]\-  ||    / \_  Professor Whyslopes: Site value lies in the difference between its ideas and yours.   If one site explanation is not to your liking, try another. Each one is different.

Two gaps The Old Algebra Gap:  Algebra  appears with too few words of explanation in high school and college mathematics.  Online Volumes 2 and 3 offer remedies.   Chapters 8 to 12 in Volume 2  put more words into the explanation and comprehension of algebra.  Chapter 14 in Volume 2 with its explicit discussion of the direct and indirect use a formulas identifies a unifying theme for mathematics and logic - all rules and patterns will be used forward and backwards. Chapters 2 to 6 and 12 to 18 in Volume 3 may further ease or avoid the very challenging use of algebra in the high level mathematics: calculus.    Calculus requires earlier high school mathematics at full strength: (i) This logically complete but long lesson on  complex numbers shows how to simplify the senior  high school exposition of circular trig functions upto to formulas in the plane  for vectors dot and cross-products.  The lesson provides the route that would have been taken in course design if the key element of the lesson, a December 2009 invention,  had been available in the 1950s.  For further algebra skill development. See the site coverage of fraction with units, proportionality,  ratios and rates, polynomials, quadratics functions  and straight line slopes and equations. The Arithmetic Gap: An exact and efficient mastery of arithmetic with decimals and fractions is best (required)  for the high level  study of mathematics alone and in science, technology and business.   Pages here on arithmetic with decimals and integers,  on  fractions and solving linear equations with fractional operations on stick diagrams may help fill the gap.  That exact and efficient command should be obtained in the last years of primary school and the first years of secondary school.     Skill mastery in mathematics has to be seen to believed.  To that end,  learn or teach how-to write and draw the steps in mathematical figuring or  reasoning  clearly. Do not try to save space by doing a sequence of step in one place. Instead, do or record the steps in sequence on a separate lines to make each step obvious and verifiable.

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