4x 2-5x+3. 7x-1 HOMEWORK 1-1
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1 HOMEWORK 1-1 As it is always the case that correct answers without sufficient mathematical justification may not receive full credit, make sure that you show all your work. Please circle, draw a box around, highlight, or otherwise clearly indicate your final answer for each question. By signing your name above, you attest to the fact that the work you are presenting is wholly your own. This work is due in class as assigned by the instructor and noted in the course syllabus. Attach all work to Perform the operations and simplify the expression: 1. 12a 5 b 1 4 a2 b 4 2. ( 4x 2 y 3 ) 2 3. (2x 3 y) 2 (2xy 2 ) 4 4. (4x 2 y 3 ) 2 ( 2y x 2)3 5. 5(3x 7y) 4(2y x) 6. 2y(3x 2 y) + 7y( x + 2y) 7. (2x 5y)(x + 4y) 8. 5(3x 4y) (2x) 2 5(2x) Find the perimeter of the following rectangle: 4x 2-5x+3 7x-1
2 HOMEWORK 1-2 As it is always the case that correct answers without sufficient mathematical justification may not receive full credit, make sure that you show all your work. Please circle, draw a box around, highlight, or otherwise clearly indicate your final answer for each question. By signing your name above, you attest to the fact that the work you are presenting is wholly your own. This work is due in class as assigned by the instructor and noted in the course syllabus. Attach all work to DO not use a calculator to get the answer!! 1. Simplify the expression: 1) ( 6) 2 2) ( 3) 2 3) ) ) (4 3) 2 6) (5 10)(5 + 10) 7) ) Rationalize the denominator: 1) )
3 HOMEWORK 1-3 As it is always the case that correct answers without sufficient mathematical justification may not receive full credit, make sure that you show all your work. Please circle, draw a box around, highlight, or otherwise clearly indicate your final answer for each question. By signing your name above, you attest to the fact that the work you are presenting is wholly your own. This work is due in class as assigned by the instructor and noted in the course syllabus. Attach all work to DO not use a calculator to get the answer!! Simplify and Write it in the standard form. 1) 169 2) 27 3) 108 4) 44 5) (3 + 2i) + (4 3i) 6) (6 3i) (4 4i) 7) (2 3i)(2 + 3i) 8) i 50 9) 5 3 2i 10) 3 i 4+3i
4 HOMEWORK 1-4 As it is always the case that correct answers without sufficient mathematical justification may not receive full credit, make sure that you show all your work. Please circle, draw a box around, highlight, or otherwise clearly indicate your final answer for each question. By signing your name above, you attest to the fact that the work you are presenting is wholly your own. This work is due in class as assigned by the instructor and noted in the course syllabus. Attach all work to Factor completely the following expression 1) 8x 3 y 2 4x 2 y x 2 y 3 2) 36a 2 49b 2 3) 2x 3 3x 2 18x ) x 2 x 30 5) x 2 15x ) 3x x 8 7) 6x 2 13x 5 8) 2x 2 7x + 3 9) 8x ) 3x 3 27x
5 HOMEWORK 1-5 As it is always the case that correct answers without sufficient mathematical justification may not receive full credit, make sure that you show all your work. Please circle, draw a box around, highlight, or otherwise clearly indicate your final answer for each question. By signing your name above, you attest to the fact that the work you are presenting is wholly your own. This work is due in class as assigned by the instructor and noted in the course syllabus. Attach all work to Find the real and complex solutions of the following each equation. 1) 2x 3 4 = 5 2) 3(4x 2) = 5x 9 3) 4x = 6x 4) 6x 2 x 2 = 0 5) x 2 + 8x = 3 6) 0 = 16x ) 4x 2 12x 3 = 0 8) 2x 3 7x 2 4x = 0 9) 3x 3 2x 2 12x + 8 = 0 10) 3x 4 = 12x 2
6 HOMEWORK 1-6 As it is always the case that correct answers without sufficient mathematical justification may not receive full credit, make sure that you show all your work. Please circle, draw a box around, highlight, or otherwise clearly indicate your final answer for each question. By signing your name above, you attest to the fact that the work you are presenting is wholly your own. This work is due in class as assigned by the instructor and noted in the course syllabus. Attach all work to Find the real and complex solutions of the following each equation. 1) (x + 2) (x + 2) 12 = 0 2) x 4 5x 2 36 = 0 3) (x + 7) 2 2(x + 7) 24 = 0 4) x 4 + 3x 2 4 = 0 5) 2x + 3 = x 6) 8 2x = x 7) 7x 10 = x 8) x 4 = 81 9) (x 2 2x) 2 11(x 2 2x) 24 = 0 10) 3 x+4 = x 1+2x
7 HOMEWORK 1-7 As it is always the case that correct answers without sufficient mathematical justification may not receive full credit, make sure that you show all your work. Please circle, draw a box around, highlight, or otherwise clearly indicate your final answer for each question. By signing your name above, you attest to the fact that the work you are presenting is wholly your own. This work is due in class as assigned by the instructor and noted in the course syllabus. Attach all work to 1. Find the midpoint(m) and the distance(d) between two points ( 4,2) and (6, 4). 2. Find the midpoint(m) and the distance(d) between two points ( 1,4) and (3, 2). 3. Find the midpoint(m) and the distance(d) between two points ( 3,2) and ( 1, 2). 4. Find the center and the radius of the circle x 2 + y 2 4x + 10y = 8 5. Find the center and the radius of the circle x 2 + y 2 + 2x 16y + 4 = 0 6. Find the center and the radius of the circle 2x 2 + 2y 2 12x + 8y 10 = 0 7. Find the equation of a circle which its radius is 3 and its center is (2, 3). 8. Find the equation of a circle which its radius is 4 and it passes through a point (0,2). 9. Find the equation of a circle whose two diameter end points are ( 6,2) and (4, 8). 10. Find the equation of a circle whose two diameter end points are ( 4,2) and (2,8).
8 HOMEWORK Find the intercepts of 3x 4y = 9 2. Find the intercepts of y = 3 2x x 3 3. Determine whether each the relation is a function or not 1) y = x 3 4 x 2) x 2 + 5y 2 = 7 3) y = x + 5 4) y = 6x 4 3x Determine whether the relation is symmetric about x-axis/ y-axis/origin/ or neither. 1) y = 5x 2 3 2) y = 4x x 3 3) y = 1 2 x3 3
9 HOMEWORK Find the domain of each the following function. Write your answer in the interval notation. 1) y = x 2 5x + 9 2) y = x 5 x+3 3) y = x+2 x 2 x 12 4) y = 4x 12 5) y = 3 5x 6) y = x+2 x 3 2. The graph of a function f(x) = a(x 2) 3 (x 4) 2 contain a point (3, 2). Find the value of a. 3. If a function f(x) = 3x 3 kx 2 + 2x 5 has a zero 1, find the value of k. 4. If a function f(x) = 2x 3 + kx 2 8x + 12 has a zero 2, find two other zeros of f(x).
10 HOMEWORK Let f(x) = 3x 2 5x ) f( 2) 2) f(5x) 3) f(x 1) 2. Let f(x) = 2x 2 + 7x 5 1) f( 3) 2) f(3x) 3) f(x 2) 3. Let f(x) = x 3 x+5 1) f( 3) 2) f(x 1) 2x 5, if x < 1 4. Find the values of f( 3) and f(2) if f(x) = { 3, if x = 1 4 x 3, if x > 1 5. If f(x) = { 1 2x if x < 1 3 if x = 1, find f(1). 3x 2 if x > 1
11 HOMEWORK Let f(x) = 2x 3 and g(x) = 4x + 5. Find the following 1) (f g)(x) 2) (f g)(x) 3) (f + g)( 2) 4) Find the domain of ( f ) (x) g 2. Let f(x) = 4x + 3 and g(x) = 3x 2. Find the following 1) (f g)( 2) 2) (f g)(x) 3) (f + g)(x) 4) Find the domain of ( f ) (2) g 3. For the graph to the right, 1) Is it a function? 2) If so, find the domain and range. 3) f( 4) =? 4) Find the intercepts. 5) Find the local maximum 6) Find the decreasing interval(s)
12 HOMEWORK The graph of y = x 5 is shifted left by 3 units, reflected across the x-axis, and shifted up 4 units. Write the resulting equation. 2. The graph of y = x 3 is shifted right by 2 units, shifted down 5 units, and reflected across the x-axis. Write the resulting equation The graph of y = x resulting equation. is shifted right by 4 units, vertically stretch by factor 4, and shifted down 5 units. Write the 4. The graph of y = x is shifted down by 3 units, reflected across the x-axis, and shifted left 6 units. Write the resulting equation. 5. Describe the transformations that have been applied to obtain the function from the given base function : y = 4 x 3 + 3; y = x
13 HOMEWORK Find the slope and y-intercept of a line 5x 7y = Find the slope and y-intercept of a line x 2y = Find the slope of a line which contains points ( 3,4) and (1, 2). 4. Find the equation of a line such that its slope is 3 and it contains a point (2, 1). 5. Find the equation of a line which contains two points ( 2,3) and (4, 2). 6. Find the equation of a line which has x-intercept 2 and y-intercept A line l passes through two points ( 1,1) and (7, n) and its slope is 1. What is the value of n? 4 8. Find the constant a if a line x + ay = 3 contains a point (1,2) 9. A car rental agency charges a weekly rate of $150 for a car and an additional charge of 5 cents for each mile driven. How many miles can you travel in a week for $500? 10. Find the positive number a if the area of a right triangle bounded by ax + 3y = 6, x-axis, and y-axis is 32.
14 HOMEWORK Find the equation of a line which is parallel to y = 1 x 7 and contains a point (3,1) Find the equation of a line which is parallel to 2x + 3y = 7 and contains a point (5, 2). 3. Find the equation of a line which is parallel to y = 5 and contains a point (5, 2). 4. Find the equation of a line which is perpendicular to y = 2x + 5 and contains a point (4, 1). 5. Find the equation of a line which is perpendicular to 3x y = 5 and contains a point (3,1). 6. Find the equation of a line which is perpendicular to y = 5 and contains a point (5, 2). 7. The slope of line l is 2 and its y-intercept is 1. What is the equation of a line perpendicular to the line l that contains a point ( 1,3)? 8. (extra) In the xy-plane below, ABCD is a square and point E is the center of the square. The coordinates of points C and E are (7, 2) and (1, 0), respectively. Find an equation of the line that passes through points B and D.
15 HOMEWORK 3-3 x + 2y = 1 1. Solve the system of linear equations: { 2x + 3y = 5 3x + 4y = 3 2. Solve the system of linear equations: { x 2y = 4 x = y 3 3. Solve the system of linear equations: { x + 2y = 6 2 3x + 4y = Solve the system of linear equations: { 2y x = 19 x y = 5 5. Solve the system of linear equations: { 3x + 3y = 7 4x 6y = 8 6. Solve the system of linear equations: { 6x + 9y = You inherit $10, 000 and will invest the money in two stocks paying 6% and 11% annual interest, respectively. How much should be invested at 6% interest stock option if the total interest earned for the year is to be $900? 8. A movie theater sells tickets for $9.00 each, with seniors for $7.00. One evening the theater sold 600 tickets and took in $4760 in revenue. How many of seniors ticket were sold? 9. At a lunch stand, each hamburger has 50 more calories than each order of fries. If 2 hamburgers and 3 orders of fries have a total of 1700 calories, how many calories does a hamburger have? 10. Solve the system of linear equations: { x 4 y = 1 x 5 y = 3 (hint: first find the value of 1 x and 1 y )
16 HOMEWORK Find the vertex of the quadratic function f(x) = 3x 2 + 2x Find the maximum or minimum of the quadratic function f(x) = 2x 2 6x Find the increasing interval and the decreasing interval of f(x) = 5x x Find the domain and range of the function f(x) = 3x 2 12x Find the intercepts of the quadratic function f(x) = x 2 + 3x Find the equation of a quadratic function such that its vertex is ( 3,1) and it contains a point (1,9). 7. Tom has 200 feet of fencing available to enclose a rectangular field. One side of the field lies along the highway, so only three sides require fencing. Find the largest area of the rectangular field. 8. The length of a rectangle is 3 cm longer than its width. The area of the rectangle is 154 cm square centimeters. Find the length of the rectangle. 9. A person standing close to the edge on the top of a 512-meter building throws a ball vertically upward. The quadratic function h(t) = 16t models the ball s height above the ground, h(t), in meter, t seconds after it was thrown. How many seconds does it take until the ball finally hits the ground? 10. An object is thrown vertically upward and its distance s(t) in feet above the ground after t seconds is given by the formula: s(t) = 4t t 1) Find its maximum distance above the ground. 2) When will it hit the ground again?
17 HOMEWORK Form a polynomial of degree 6 with zeros: 2 of multiplicity 3, 2 of multiplicity 2, and 3 of multiplicity Form a polynomial f(x) of degree 3 with zeros 2, 1, 3 and f(2) = Find zeros, the multiplicities of each zero, and decide whether it crosses/touches x-axis at each zero of the function f(x) = 5x 3 (3x 2) 2 (2x + 5) 5 4. Find zeros, the multiplicities of each zero, and decide whether it crosses/touches x-axis at each zero of the function f(x) = 2x 2 (x 2 4) 2 (x 2 2x 8) 3 5. Find a polynomial function with real coefficients that Degree 4; zeros: 2, 1, 3 2i 6. Find a polynomial function with real coefficients that Degree 4; zeros: 1, 1, 2 + i 7. Which of following is a factor of a polynomial 2x 3 5x 2 4x + 3? (A) (x + 3) (B) (x 1) (C) (2x 1) (D) (2x + 1) 8. Which of following is a factor of a polynomial 2x 3 7x 2 16x + 12? (A) x + 2 (B) 2x 1 (C) 2x + 3 (D) 3x 2
18 HOMEWORK Find quotient and remainder when we divide (using the synthetic method) (A) 4x 3 7x 2 11x + 5 by x 1 (B) 2x 4 3x 3 + 3x + 1 by x + 1 (C) 2x 3 + 5x 2 4 by x + 3 (D) x 3 7x + 9 by x 3 (E) 2x 3 + 3x 2 6x + 3 by x 1 2 (F) 3x 3 x 2 12x 7 by x Find quotient and remainder when we divide (using the long division) (A) 4x 3 + 4x 2 + x 5 by (2x 1) (B) 3x 3 4x x + 8 by (3x + 2) 3. List all the potential rational zeros of P(x) = 5x 4 + ax 3 + bx 2 18x + 6 where a, b are real numbers. 4. Find the quotient and remainder when 3x 4 4x 2 + 5x 2 is divided by x Find the remainder when f(x) = 3x x + 7 is divided by x Find the value of k such that f(x) = x 4 kx 3 + kx has x + 2 as a factor.
19 HOMEWORK Find all real zeros of the given polynomial function P, and use the real zeros to factor P(x) completely. 1) P(x) = 3x 4 + 4x 3 19x 2 8x ) P(x) = 5x 4 + x 3 35x x Find all real solutions of the equation. 1) 2x 3 + 3x 2 8x + 3 = 0 2) 6x 3 4x 2 + 3x 2 = 0 3) x 4 3x 3 + 6x 2 + 2x 60 = 0 3. Find all real/complex zeros of the polynomial function 1) P(x) = x 4 + 6x 3 + 6x 2 24x 40 2) P(x) = x 4 3x 3 + 7x x 26 3) P(x) = x 4 x 3 17x 2 + 5x + 60
20 HOMEWORK Find the domain, asymptotes, hole, x-intercepts, asymptotes (VA, HA, OA) of the rational function (A) f(x) = x 3 x 2 2x 8 (D) f(x) = 4x 1 2x+3 (B) f(x) = 2x2 2x 4 x 2 x 6 (E) f(x) = x 1 x 3 4x (C) f(x) = x2 4 x+3 (F) f(x) = x2 x 6 x+1 2. Find an equation of a ration function f that satisfies the given conditions. Vertical asymptote: x = 5 Horizontal asymptote: y = 1 x-intercept: 2
21 HOMEWORK Solve the inequality: x 2 + 3x Solve the inequality: x 2 x Solve the inequality: 2x 2 + 5x 3 < 0 4. Solve the inequality: 2x 3 18x 2 5. Solve the inequality: x 3 x 2 9x Solve the inequality: x 2 x Solve the inequality: x+3 x Solve the inequality: 9. Solve the inequality: x 1 x 2 +2x 8 < 0 2x 3 x 2 x Solve the inequality: 4x x+2 3
22 HOMEWORK 5-1, Decide whether the following function is one to one or not. 1) f(x) = x 3 3x 2) f(x) = (x 2) ) f(x) = 2 x 2. Find the inverse function of each function. 3 1) f(x) = x + 4 2) f(x) = x ) f(x) = (x 4) ) f(x) = 2x 3 4x+5 3 5) f(x) = x ) f(x) = 3x 1 3. Let f(x) = 3x 2 5x + 11, g(x) = 2x, and h(x) = x ) (f g)(x) 2) (f h)( 3) 3) (g f)(x) 4. Find the domain of f g when f(x) = 1 x 2 and g(x) = 1 x.
23 HOMEWORK Find the domain and range of y = 2 x Find the domain and range of y = log 3 (5 x) 3. Find the relation between the following two functions: y = 2 x, y = log 2 x 4. Write each equation in its equivalent exponent form: x = log b (64) 5. Write each equation in its equivalent logarithmic form: a x = b 6. Evaluate log 7 (6). Round your answer in the four decimals 7. Find the exact value of the following expression without using a calculator. 1) log 27 ( 1 ) 9 2) e2 ln(3) 3) log 2 (8) 8. Use logarithmic properties to expand the expression as much as possible. 9. Write the expression as one logarithm. ln ( y3 x + 1 x 5 w 2 ) 1) log(x 3 y 2 ) 2 log(x 2 y) 3 log ( x y ) 2) 2 ln ( y3 x ) 3 ln(y) ln(x4 y 2 )
24 HOMEWORK Solve the equation. (A) 2 x+5 = 64 (B) 9 5x 4 = 27 x+3 (C) 4 4 x = 2 x2 (D) 4 1 2x = 1 32 (E) ( 3 2 )2x+5 = 8 27 (F) ( 5 3 )x2 = ( 3 5 )2x 8 1 (G) e x2 = e 15x (H) e e0.3x = 400 (I) 4 x 9 2 x + 8 = 0 (J) 9 x 5 3 x 36 = 0 (K) 4x 3 e x + 3x 2 e x = 0
25 HOMEWORK Solve the equation. (A) log 6 (4x 5) = log 6 (2x + 1) (B) log 2 (2x 1) = 3 (C) log x 1 3 = 1 2 (E) log(x 7) = 2 (D) log 2 (x 3) = 2 (F) ln(x) + ln(x 1) = ln(12) (G) log 2 (x) + log 2 (x 6) = 4 (H) log 2 (5 x) + log 2 (5 + x) = 4 (I) log(2x 1) + log(x 9) = 2 (J) 2 ln(x + 1) = ln(3x + 7)
26 HOMEWORK The growth of a colony of bacteria is given by the equation, Q = 300e 0.15t. If there are initially 300 bacteria and t is given in minutes. After how many minutes will there be four times the initial number of bacteria? Round to 3 decimal places. 2. Find the accumulated amount on an investment of $6,500 at 2.5% annual interest, compounded monthly over 10 years. 3. Find the accumulated amount on an investment of $2,000 at 5% annual interest, compounded continuously over 5 years. 4. How many years will it take for an initial investment of $30,000 to grow to $70,000 in an account that compounded monthly at an interest rate 6%? 5. Ten-thousand dollars ($10,000) is invested in a savings account in which interest is compounded continuously at 11% per year. When will the account reach $35,000?
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