A is any set of ordered pairs of real numbers. This is a set of ordered pairs of real numbers, so it is a.

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1 Fry Texas A&M University!! Math 150!! Chapter 3!! Fall 2014! 1 Chapter 3A Rectangular Coordinate System A is any set of ordered pairs of real numbers. A relation can be finite: {(-3, 1), (-3, -1), (0, 5), (1, -3), (2, 3)} This is a set of ordered pairs of real numbers, so it is a. Each ordered pair of real numbers corresponds to a on the Cartesian plane. The set of all points corresponding to a relation is the graph of the relation. Sketch the graph the relation given above. 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. What is the domain of this relation?! What is the range of this relation?!

2 Fry Texas A&M University!! Math 150!! Chapter 3!! Fall 2014! 2 A relation can be infinite: 1. Consider {(x, y) -3 x < 2, 1< y 4} List 2 different elements of that set.!! Sketch the graph the relation What is the domain of this relation? What is the range of this relation? Equations can be used to define relations 2. Consider the set {(x, y) x + y = 3} List several elements of that set. Sketch the graph the relation What is the domain of this relation? What is the range of this relation?

3 Fry Texas A&M University!! Math 150!! Chapter 3!! Fall 2014! 3 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.

4 Fry Texas A&M University!! Math 150!! Chapter 3!! Fall 2014! 4 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.

5 Fry Texas A&M University!! Math 150!! Chapter 3!! Fall 2014! 5 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. 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?

6 Fry Texas A&M University!! Math 150!! Chapter 3!! Fall 2014! 6 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? The Midpoint of a Line Segment. Consider the points A (-3, 2) and B ( 4, -4). Determine the midpoint of the segment connecting these 2 points. Remember: The number half between a and b is the average of a and b Given a line segment with endpoints (x 1, y 1 ) and (x 2, y 2 ), the midpoint of that segment is

7 Fry Texas A&M University!! Math 150!! Chapter 3!! Fall 2014! 7 Suggested Problems: Text: 4, 5, 6a, 11-13, 14a, 15a, 16, 17a, My Previous Exams:!! S14 3A:6,!!! F13 2A: 2, 6,!!!!!! S13 2A: 1, 2,!! F12 2A: 3 Dr. Scarborough s Previous Exams:!F13 2: p2: 4, p8:8 Dr. Scarborough s Fall 2013 WIR 4: 1, 3, 5 Dr. Kim s Fall 2014 WIR:

8 Fry Texas A&M University!! Math 150!! Chapter 3!! Fall 2014! 8 Chapter 3B Graphs of Equations Graphs of Ten Relations that you should know: y = c, c! Example : y = 3 x = c, c! Example : x = 2 y = x 2 y 2 = x

9 Fry Texas A&M University!! Math 150!! Chapter 3!! Fall 2014! 9 y = x y = x x 2 + y 2 = 4 y = 9 x 2

10 Fry Texas A&M University!! Math 150!! Chapter 3!! Fall 2014! 10 y = x y = x 3 xy = 1

11 Fry Texas A&M University!! Math 150!! Chapter 3!! Fall 2014! 11 Variations on the Basic Equations y = x y = 2x 2 y = 2x 2 + 5

12 Fry Texas A&M University!! Math 150!! Chapter 3!! Fall 2014! 12 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.

13 Fry Texas A&M University!! Math 150!! Chapter 3!! Fall 2014! 13 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. We can test an equation to see if its graph 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.

14 Fry Texas A&M University!! Math 150!! Chapter 3!! Fall 2014! 14 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, We can test an equation to see if its graph 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.

15 Fry Texas A&M University!! Math 150!! Chapter 3!! Fall 2014! 15 Symmetry about the origin Consider the following graphs: 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. Interestingly, if a graph is symmetric about the origin, then for each point on the graph, there is a corresponding point on the graph such that the line segment connecting these two points has the origin as its midpoint. We can test an equation to see if its graph 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.

16 Fry Texas A&M University!! Math 150!! Chapter 3!! Fall 2014! 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? 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?

17 Fry Texas A&M University!! Math 150!! Chapter 3!! Fall 2014! 17 Suggested Problems: Text: 8-14 My Previous Exams:!! F13 2A: 5 iii, iv, viii, 6 iii, iv, vii, 8 a & b!!!!!! S13 2A: 5a iii, iv, 5b iii, iv, 5c iii, iv, and 13 Dr. Scarborough s Previous Exams:!F13 2: p7: 7, p8:9 Dr. Scarborough s Fall 2013 WIR 4: 2, 6a, 7, 8 and!! WIR 6: 11 Dr. Kim s Fall 2014 WIR:

18 Fry Texas A&M University!! Math 150!! Chapter 3!! Fall 2014! 18 Chapter 3C -- Linear Equations in Two Variables Linear Equations Which of the following is a linear equation in 2 variables? 3x + 4y = 5!!! πx + 4y = 5!!! x + y + z = 0!! y = x 9 3x + xy + 4y = 5!! x + 4y = 2!!! x 2 + 4y = 2!! y = 2 x v = 32t +16!!! y = 2!!!! x = 5!! x = y Any equation that can be written in the form where A and B are not both is called a Slope: Label the graphs of!! y 1 = 4x! y 2 = 5x When m is very large, the slope of the line is very steep. For extremely large values of m, the graph of the line may look vertical. When the slope is positive, the graph is (from left to right). When the slope is negative, the graph is (from left to right)

19 Fry Texas A&M University!! Math 150!! Chapter 3!! Fall 2014! 19 On the Cartesian plane below graph the vertical lines x = 3 and x = 7 2 Notice that there is no on x, and does not appear in the equation. Vertical lines have slope. Sometimes it said that the slope for vertical lines is Label the graphs of!y 1 = 1 4 x!! y = x When m is very small, the slope of the line is very shallow. For extremely small values of m, the graph of the line may look horizontal.

20 Fry Texas A&M University!! Math 150!! Chapter 3!! Fall 2014! 20 Look at the horizontal lines on the graph below: What are the equations for these lines? Notice that does not appear in the equation. The slope of a horizontal line is Calculate the slope m of the line on the graph below.

21 Fry Texas A&M University!! Math 150!! Chapter 3!! Fall 2014! 21 On the graph below, label one point on the line (x 1, y 1 ) and second point on the line (x 2, y 2 ). Then calculate the slope of the line. The slope of a line through the points (x 1, y 1 ) and (x 2, y 2 ) is defined as or as!! Note Δ is the Greek letter. Δ or D is for Some people like the phrase slope = So for the line y = 2x 3. The slope is and the y-intercept is. This form of a linear equation is known as the form. What is the equation of a line with slope -3 and y-intercept of 4?

22 Fry Texas A&M University!! Math 150!! Chapter 3!! Fall 2014! 22 Graph the lines y = 3x +1 and y = 3x 2 Parallel lines have the same. Parallel lines do not. When 2 lines are perpendicular their slopes are. Determine the equation of the line through (-1, -2) that is perpendicular to y = 3x + 4. Determine the equation of the line through (-1, -2) and (4, 3).

23 Fry Texas A&M University!! Math 150!! Chapter 3!! Fall 2014! 23 Suggested Problems: Text: 17, 21a, 22 g, h, i, 23d, My Previous Exams:!! S14 1A: 1, 2, 13f,! S13 2A: 10!,! F12 2A: 5 Dr. Scarborough s Previous Exams:!F13 II: p3:5! F12 II: p2:2, 3 Dr. Scarborough s Fall 2013! WIR 4: 4, 9, 10, 11, 17!!!!!!!! WIR 5: 1!!!!!!!! WIR 6: 15, 16b, 17, 20 Dr. Kim s Fall 2014 WIR: