Chapter 3A Rectangular Coordinate System

Transcription

1 Fry Texas A&M University! Math 150! Spring 2015!!! Unit 4!!! 1 Chapter 3A Rectangular Coordinate System A is any set of ordered pairs of real numbers. The of the relation is the set of all first elements of the ordered pairs. The of the relation is the set of all second elements of the ordered pairs. Calculating Distance in the rectangular coordinate system: Find the distance d between the points A(1, 2) and B(-3, 4). Find the distance d between the points C(x 1, y 1 ) and D(x 2, y 2 ) Consider a general point B (x, y) that is 5 units from the origin. This equation is true for all points that are exactly 5 units from the origin. Sketch the graph of this equation.

2 Fry Texas A&M University! Math 150! Spring 2015!!! Unit 4!!! 2 Place the point B at (1, 2). List and graph 4 points that are 3 units from B. Is the origin 3 units from (1, 2)? Write an equation whose solution is all of the points that are 3 units from (1, 2) We could repeat this procedure starting at an arbitrary point (h, k) and find all the points that are a given distance called r from (h, k). In this case, we would find that This is the standard form of the equation of a centered at with radius. Write the equation of a circle centered at (-3, 0) with radius 2. What is the center of the circle (x + 4) 2 + (y 5) 2 = 6? What is the radius of this circle? Expand this equation and simplify.

3 Fry Texas A&M University! Math 150! Spring 2015!!! Unit 4!!! 3 The General Form of an Equation of a Circle is written Note that there is both an term and a term, but no term. The coefficients on the x 2 term and the y 2 are both Consider the equation 16x 2 +16y 2 +16x 56y + 5 = 0 Is this a circle? If so what is its center? What is its radius? Consider the equation 4x 2 + 4y 2 24x + 40y +135 = 0 Is this a circle? If so what is its center? What is its radius?

4 Fry Texas A&M University! Math 150! Spring 2015!!! Unit 4!!! 4 Remember if you are trying to calculate the midpoint of a line segment, you are trying to find a point that is halfway between the endpoints. The x-coordinate of the midpoint will be halfway between the x-coordinates of the endpoints. The y- coordinate of the midpoint will be halfway between the y-coordinates of the endpoints. Also remember that the average of two numbers is halfway between the two numbers. Suggested Problems for 3A: Text: 4, 5, 6a, 11-13, 14a, 15a, 16, 17a, My Previous Exams:!! Fall A: 3, 4! Spring A: 6,!!!!! Fall A: 2, 6,! Spring A: 1, 2,!!!!! Fall A: 3 Suggested Problems for 7B: Text: 2-6 My Previous Exams:!! Fall A: 10!!!!! Fall A: 10!! Spring A: 10!!!!! Fall A: 6

5 Fry Texas A&M University! Math 150! Spring 2015!!! Unit 4!!! 5 Chapter 3B Graphs of Equations Intercepts The y-intercept is where the graph crosses It occurs when. The x-intercept is where the graph crosses It occurs when. Find the intercepts of the graph for the following equation: x 2 + xy + 5y 2 = 25. To find the x-intercept To find the y-intercept Find the intercepts of the graph for the following equation: (x 4) 2 + (y + 2) 2 = 9.

6 Fry Texas A&M University! Math 150! Spring 2015!!! Unit 4!!! 6 Symmetry about the x-axis On the figure below, label the vertices in the first and second quadrants. Then reflect the image through the x-axis to create a polygon that is symmetric about the x-axis. Notice that if (a, b) is on the polygon, then so is A graph is symmetric about the x- axis iff for each (x, y) on the graph, is also on the graph. The test to see if the graph of an equation will be symmetric about the x-axis: Substitute -y for y in the equation. Simplify the equation. If the resulting equation is to the original equation, then the graph will be 1. Show that x + y 2 = 4 is symmetric about the x-axis.

7 Fry Texas A&M University! Math 150! Spring 2015!!! Unit 4!!! 7 Symmetry about the y-axis On the figure below, label the vertices in the second and third quadrants. Then reflect the image through the y-axis to create a polygon that is symmetric about the y-axis. Notice that if (a, b) is on the polygon, then so is A graph is symmetric about the y- axis iff for each (x, y) on the graph, The test to see if the graph of an equation will be symmetric about the y-axis: Substitute -x for x in the equation. Simplify the equation. If the resulting equation is to the original equation, then the graph will be 2. Show that x 4 + y = 9 is symmetric about the y-axis.

8 Fry Texas A&M University! Math 150! Spring 2015!!! Unit 4!!! 8 Symmetry about the origin Informally, a graph is symmetric about the origin if when it is rotated 180 about the origin, the graph looks the same. By definition, a graph is symmetric about the origin iff for each (x, y) on the graph, is also on the graph. The test to see if the graph of an equation will be symmetric about the origin: Substitute both -x for x and -y for y in the equation. Simplify the equation. If the resulting equation is to the original equation, then the graph will be 3. Show that x 2 = y is symmetric about the origin. 4. Determine the symmetry of y 3 = 6x 4 x a) Test for symmetry about the x-axis by substituting for Equivalent or Not Equivalent? Symmetric or Not Symmetric about x-axis?

9 Fry Texas A&M University! Math 150! Spring 2015!!! Unit 4!!! 9 b) Test for symmetry about the y-axis by substituting for Equivalent or Not Equivalent? Symmetric or Not Symmetric about y-axis? c) Test for symmetry about the origin by substituting both! for and for Equivalent or Not Equivalent? Symmetric or Not Symmetric about origin? Suggested Problems:! Text: 8-14 My Previous Exams:!! Fall A: 5, 12!!!!! Fall A: 5 iii, iv, viii, 6 iii, iv, vii, 8 a & b!!!!!! Spring 13 2A: 5a iii, iv, 5b iii, iv, 5c iii, iv, and 13

10 Fry Texas A&M University! Math 150! Spring 2015!!! Unit 4!!! 10 Chapter 7B Systems of Non-Linear Equations 1. Plot xy = 1 and x + y = 2 on the same graph. Then use algebra to determine the exact point(s) of intersection, if there are any. 2. Plot y = x 2 and y = 8 x 2 on the same graph. Then use algebra to determine the exact point(s) of intersection, if there are any.

11 Fry Texas A&M University! Math 150! Spring 2015!!! Unit 4!!! Plot x 2 + y 2 = 9 and x 2 y = 9 on the same graph. Then use algebra to determine the exact point(s) of intersection, if there are any.

12 Fry Texas A&M University! Math 150! Spring 2015!!! Unit 4!!! Plot x 2 + y 2 = 17 and x + y = 3 on the same graph. Then use algebra to determine the exact point(s) of intersection, if there are any.

13 Fry Texas A&M University! Math 150! Spring 2015!!! Unit 4!!! Plot y = 3x 2 + 6x + 6 and y = x 2 + 4x + 5 on the same graph. Then use algebra to determine the exact point(s) of intersection, if there are any.

14 Fry Texas A&M University! Math 150! Spring 2015!!! Unit 4!!! Plot x 2 + y 2 = 25 and (x 6) 2 + y 2 = 25 on the same graph. Then use algebra to determine the exact point(s) of intersection, if there are any. Suggested Problems for 7B are on page 4.

15 Fry Texas A&M University! Math 150! Spring 2015!!! Unit 4!!! 15 Chapter 4A - Introduction to Functions Difference Quotient. Assume that the function represented by this graph is a position function. The horizontal axis is time given in seconds. The vertical axis is distance an object has moved from the starting point given in feet. It will be helpful to remember that d = rt so r = What is the position of the object at t = 0? What is the position of the object at t = 4? How far did the object move during those 4 seconds? What is the average speed of the object during those 4 seconds? What is the position of the object at t = 8? How far did the object move during those 8 seconds? What is the average speed of the object during those 8 seconds? How far did the object move between t = 4 and t = 8 What is the average speed of the object during those 4 seconds? Connect the points on the graph for t = 4 and t = 8. What is the slope of that line segment? The slope of the secant line for a position function is

16 Fry Texas A&M University! Math 150! Spring 2015!!! Unit 4!!! 16 How might you go about calculating an estimate for the instantaneous velocity at a point, say at t=4? You might calculate the average velocity between t=4 and t=5. Or better, you might calculate the average velocity between t=4 and t=4.5. Or better yet, you might calculate the average velocity between t=4 and t=4.1 The smaller the interval of time, the closer the average velocity would be to instantaneous velocity. Here we have the graph of some function f (x) Choose a general point (x, f (x)) on the graph. Let h > 0 be some teeny-tiny number. x + h would be just to the right of x. (Imagine that we ve zoomed in on the graph so it looks big, but it s really small.) Plot the point at (x + h, f (x + h)). Connect those two points with a line segment. Calculate the slope of that line segment: This is what is called the. In calculus, you will study what happens as h gets smaller and smaller. In the meantime, we will practice the sometimes-messy algebra that often surrounds the Difference Quotient.

17 Fry Texas A&M University! Math 150! Spring 2015!!! Unit 4!!! If f (x) = x 2 +1, determine the difference quotient: 2. If f (x) = x 1, determine the difference quotient: 3. If f (x) = x, determine the difference quotient:

18 Fry Texas A&M University! Math 150! Spring 2015!!! Unit 4!!! 18 Do a job problems I think of them this way: Amount of the job completed = (rate the job is completed) * (time working) Also remember that if it takes someone 3 hours to do a job, then the rate is 1 3 job per hour. 4. Amy can clean the house in 6 hours. Brian can do the same job in 9 hours. In how many hours can they do the job if they work together? 5. Amy and Brian can rake their entire yard in 2 hours when working together. If Brian requires 6 hours to do the job alone, how many hours does Amy need to do the job alone? 6. One pipe can fill a tank 1.25 times faster than a second pipe. When valves to both pipes are opened, they fill the tank in five hours. How long would it take to fill the tank if only the second pipe is used?

19 Fry Texas A&M University! Math 150! Spring 2015!!! Unit 4!!! Suppose there is a piece of sheet metal that is rectangular: 1 foot long and 2 feet wide. Four congruent squares are cut from the corners, so that the resulting piece of metal can be folded and welded into a box. If the length of the sides of the squares is s, write a function that describes the volume of the box in terms of s. 8. Suppose there is a wire 40 inches long. Suppose the wire is cut into 2 pieces, not necessarily equal in length. If each of the pieces of wire is bent to form a square, write a function that describes the sum of the areas of the two squares. Suggested Problems: Text: 1, 2, 6, 7, 10 c-g, 11c-g, 12 c-g, 13, 14, My Previous Exams:!Fall A : 7, 8, 13, 14!!!! Fall A: 3, 4, 9 d & e,! Spring A: 3 & 4,!!!!! Fall A: 1