Quadrant IV whose coordinates

PLEASE READ THE QUESTIONS/PROBLEMS CAREFULLY; SHOW ALL YOUR WORK AND REASONING, Just the answer, without supporting work, will receive no credit. 1. [12 points] In the Cartesian Coordinate Plane (the x-y plane), using careful and correct scaling, (a) Choose/graph a point P4 (an ordered pair) in Quadrant IV whose coordinates (distances from the x-axis and the y-axis) are not the same; graph the point carefully and show its coordinates as an ordered pair, noting that in the fourth quadrant the coordinates are with different signs. (b) Draw the horizontal line y = 2 and label it. (c) Draw the vertical line x = −1 and label it. (d) Find point P1 in Quadrant I such that points P1 and P4 are symmetric about the horizontal line y = 2 (P1 and P4 are reflections of each other about the horizontal line y = 2); graph point P1 carefully and show its coordinates as ordered pair. You may carefully apply the concept of reflection in this and the following part(s); pay attention to where the “mirror” would be, as discussed in class. (e) Find point P3 in Quadrant III such that points P3 and P4 are symmetric about the vertical line x = −1 (P3 and P4 are reflections of each other about the vertical line x = 1); graph point P3 carefully and show its coordinates as an ordered pair. (f) Find the distance between Points P1 and P3. 2. [6 points] (a) Write the coordinates of a point of your choice (point Q) in Quadrant III of the x-y plane. (b) Using “the formula for the distance between two points,” find the distance between point Q and the origin. (c)) Noting “the formula for the distance between two points” and your experience with Part (b) above, what would be a “clean-cut” formula for the distance between point P(x,y) and the origin? (d) [For Extra Credit (1 point)] How can we come up with the answer to Part (c) above without using the distance formula? 3. [6 points] The midpoint between two points is M(1, -3). (a) Find the endpoints of the line segment AB with the midpoint M such that Point A would be in Quadrant II and Point B would be in Quadrant IV. (b) Find the endpoints of the line segment CD with the midpoint M such that Point C would be in Quadrant III and Point D would be in Quadrant I. (c) How many answers do you think the problem would have for each part above? Briefly explain. 4. [6 points] Suppose a solution set to an inequality is (−, −3)  (−3, 3)  (3, ). (a) Show the solution set graphically on a real number line. (b) Show the solution set using the set-builder method, involving inequality symbol(s). 5. [6 points] (a) Determine whether the ordered pair (5, -1/3) is a solution of the equation -3y + 4 = 5. (b) Solve the equation for y. (c) Find an ordered pair, different from the given one above, that would be a solution to the equation. 6. [8 points] Briefly support your answers to the following questions: (a) Determine whether the relation {(5, 1), (4, 3), (-4, 1), (0, -4)} is a function. Support your answer by explanation or plotting. (b) Find the domain of the relation. (c) Find the range of the relation. (d) If the given relation is a function, add a point to the set to change it to a relation that would not be a function. 7. [5 points] 8. [4 points] 9. [9 points] If f(x) is a piecewise-defined function as follows, compute the value(s) of the function at x = −1/2, x = −1, x = 1, x = −3, and x = 5/4 NOTE: Keep working with integers or fractions and present your answers as integers or fractions. 10. [9 points] The graph of function y = f(x) is provided below. (a) (b) Examining the graph carefully, what would be the value of f(3)? (c) What is (are) the x-intercept(s) of the function? (d) What is (are) the y-intercept(s) of the function? (e) What is the domain of the function f(x)? (f) What is the range of the function f(x)? 11. [5 Points] Find the domain of the following function and show it in interval format: f(x) = √2𝑥−2 𝑥−2 12. [9 points] If h(z) = 3 – 2z and g(z) = 1– z, provide answers to Parts (a) through (e) below: (a) (b) (c) (d) (e) (f) Find (h – g)(z) and simplify. Find (h – g)(1/4) Find (hg)(z) Find (hg)(0) Find (h/g)(z) Find (h/g)(1). 13. [6 points] Briefly support your answer to each part of the question asked below. 14. [9 points] Hideaway Vacation Cabins charges $150 plus $25 per person for a weekly rental fee. Each cabin can accommodate no more than 4 people. (a) Write an equation (a mathematical model) that can be used to determine the total cost, C(p), of a weekly rental for p persons. (b) Find the total rental cost for 3 persons. (c) Determine the domain and the range of the function C(p) within the context of the problem.
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