Math 175 MT#1 Additional Material Study Sheet
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1 Math 175 MT#1 Additional Material Study Sheet Use the following functions for this worksheet : w( x) = ; f ( x) = 3x 11x 4 ; p( x) = 2x x 17x + 12 ; 2 + x ( ) 3 ; ( ) ; ( ) 2 k x = x + c x = x x + x x + t x = x 1) Give the domain and range for the function wx ( ). Use the function wx ( ) defined at the beginning of the test. Use interval notation, and display the domain and the range to the right of the Dw and Rw below. D w R w : : f ( x+ h) f( x) 2) Remembering that the difference quotient for any function, f ( x ), is defined to be, h find the difference quotient for f ( x ). Again use the function f ( x ) defined at the beginning of the test. (10 points) f ( x+ h) f( x) h =
2 3) Consider p( x ) defined at the beginning of the test Use your graphing calculator with the following window settings X : [-5,5,1] & Y : [-20,40,10] to get an idea of all the real zeros (one is rational, and two are irrational conjugates). a) Using the Rational Root Theorem, list all the possible (zeros) rational roots of px ( ) = 0. (2 points) b) Use the Lower & Upper Bound Theorem and Synthetic Division to show that x = 4 is a lower bound and x = 4 is an upper bound, for all the real zeros of p( x ). (2 points) Is x = 4 a lower bound? Is x = 4 an upper bound? x = 4 is a lower bound x = 4 is an upper bound c) Using either long or synthetic division and the Remainder Theorem, divide p( x ) by kx ( ) to show that x = 3 is a rational zero of p( x ). Use the functions p( x) and k( x ) defined at the beginning of the test. (2 points) d) Use the Factor Theorem and the results of part (c) to state p( x ) in factored form (that is, with integer coefficients). (2 points) px ( ) =
3 e) Use the Quadratic Formula to find the two irrational conjugate zeros of p( x ). [Round to the closest hundredth.] (2 points) 4) Consider #3 further a) Use the Location Theorem and Bisection method, as well as the table given and started, to find the irrational zero of p( x ) that is closest to the origin [the point (0,0)], accurate to one decimal (the tenths) place. You should be filling in the correct sign + or -, in the blank cells of the table (2.5 points) (a,b) Midpoint (m) P(a) P(m) P(b) (0,1) (0.5,1) (0.5,0.75) (0.625,0.75) (0.6875,0.75) The irrational zero that is closest to the origin [the point (0,0)], accurate to one decimal (the tenths) place, is (~ 0.7, 0) b) Use the Zero command on your graphing calculator to find all three of the real zeros of p( x ). Round to the closest thousandth, if necessary. Insert the zeros in the given ordered pairs. (1.5 points) Zero # 1 : (,0) Zero # 2 : (,0) Zero # 3 : (,0)
4 c) Use the minimum and maximum commands on your graphing calculator, to find the two relative extrema of p( x ). Display the extrema as ordered pairs, and round both the x and y coordinates to the closest hundredth. (2 points) Relative maximum: Relative minimum: d) Using the zeros and relative extrema of p( x ) found in parts (b) & (c) above, sketch the graph of p( x ), on the graph paper provided below. [Hint: Use a scale of one unit per tick mark on the x-axis, and ten units per tick mark on the y-axis.] (2 points) y x e) Give all the intervals (using interval notation), where px ( ) > 0 (where p( x ) is greater than zero). (2 points) 5) Consider cx ( ) defined at the beginning of the test Use your graphing calculator with the following window settings X : [-2,6,1] & Y : [-50,200,50] to get an idea of all the zeros (one is rational and of multiplicity 2, and two are complex conjugates). a) Using either long or synthetic division and the Remainder Theorem, divide cx ( ) by tx ( ) and then the resultant quotient, by tx ( ) again, to show that x = 2 is a rational zero (of multiplicity 2) of cx ( ). Use the functions c( x) and t( x) defined at the beginning of the test. (2 points)
5 b) Use the Factor Theorem and the results of part (a) to state cx ( ) in factored form (that is, with integer coefficients). (2 points) cx ( ) = c) Use the Quadratic Formula to find the two complex conjugate zeros of cx ( ). (2 points) d) Use the minimum command on your graphing calculator, to verify that, x = 2, the root of multiplicity 2, is a turning point of cx ( ). Display the minimum exactly as it does on your calculator screen. (1 point) Minimum X = Y = e) Use the value command on your graphing calculator, by setting x = 0, to find the y-intercept of cx ( ). Display the y-intercept as an ordered pair. (1 point) Y-intercept : ( 0, ) f) Using the zero/turning point and the y-intercept of cx ( ) found in parts (a),(d) & (e) above, sketch the graph of cx ( ), on the graph paper provided below. Make sure it doesn t look exactly like a parabola, as you should see it does not have symmetry. You may wish to use the TABLE command on your calculator or value command to in order to fill out the T-table below for graphing purposes. [Hint: Use a scale of one unit per tick mark on the x-axis, and fifty units per tick mark on the y-axis.] (2 points) y x c(x) x
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