Math project
- 1. Starring: Je Olive Kathleen Ballener Jasmine Montes Yani Mae P ita Kiesheen May Martonia Marisol Aguilar Honey Grace Tinaco Dianne Joy Cosares
- 2. 5.1 “ The Cartesian Coordinate Plane “ These two number lines make up what we call the coordinate plane as shown here. The number lines intersect at the point called the origin denoted by the letter O. The horizontal number line called the x-axis and the vertical number line is called the y-axis. Arrowheads At each end of both axes indicate the infinity of the set of real numbers. Notice that the axes divide the plane into four regions or quadrants labeled with Roman numerals. I through IV in counterclockwise direction. The first number x is called x coordinate or abscissa and the second number y is called the y Coordinate or ordinate . The plane described is often called the rectangular coordinate system or CartesianCoordinate System . The word Cartesian is used in Rene Descartes, the 17 th -century French philosopher and mathematician who first devised the coordinate system . 4 3 2 1 -1 -2 -3 -4 -1 -2 -3 -4 1 2 3 4 y x I 11 111 1V
- 3. 5.2 “ Points in the Cartesian Coordinate Plane” There exist a one-to-one correspondence between points in the plane and the ordered pairs of real numbers. 5.2.1 “The coordinates of a point” The distance of a point from the x- and y-axes is measured in units from the point along the line perpendicular to the respective axis. 5.2.2 “Plotting of Points” If points in a coordinate plane can be named. Points can also be plotted in the plane given their coordinates. To locate the P represented by the ordered pair (3,4). Point the tip of your pencil at the origin (0,0) move it 3 units to the right, then 4 units upward your point is at P is 3 and the y-coordinate x-coordinate is 4. The process of locating a point in the coordinate plane is called plotting the point. 5.2.3 “ Points in a Quadrants” Given coordinates of a point, the quadrant where it is located can be determined. In a quadrant I, both abscissa and ordinate are positive (x, y). In quadrant II, the abscissa is negative while the ordinate is positive, (x, y). In quadrant III, both abscissa and ordinate are negative, (x, y). Points found on any of the axes are not considered to be in any quadrants.
- 4. LINEAR EQUATIONS IN TWO VARIABLES : Ax + By = C The graph of a line can be drawn using ordered pairs of numbers in the form (x, y). The abscissa is the x-coordinate and the ordinate is the y-coordinate. The ordered pairs (-1,4),(0,2),(1,0),(2,-2) are some of its solutions, that is substituting each pair in the sentence will give a true statement. If the domain is the set of real numbers, the paragraph is a line. This line is the graph of the equation and the sentence is the equation of the line. The mathematical sentence 2x + y = 2 is an example of a linear equation. 5.3 “The graph of Ax + by = c based on the table of values” Values and representing the domain (x) and the range (y), the graph an equation can drawn. 5.3.2 “ Intercepts, Slope, Domain, and Range” Study the graph of the equation 3x – 2y = 6 Notice that the line crosses the x- and y- axes at (2, 0) and (0, -3). The x-intercept is the abscissa of the point (2, 0) where the graph crosses the x- axis. They y- intercept is the ordinate of the point (0, -3) where the graph crosses the y- axis. 3x + 2y = 6 x (2, 0) (0, -3) 2x + y = 2 y y
- 5. 5.3.3 “ Properties of the Graph of a Linear Equation Ax + By = C” The graph of every equation of the form Ax + By = C, where a and b are not both zero, is a line. In studying of this equation, there are properties of the graph that have to be considered. 5.3.3a “ The Intercepts” The set of ordered pairs which satisfies the equation are (-3, 4), (3, 0), (6, -2).The point whose coordinates consist the number pair (0, 2) intersects the y-axis, thus the ordinate 2 point whose coordinates consist of the number pair (3, 0) intersects the x-axis, thus the abscissa 3 of this point is called the x-intercepts. Remember: The x- intercept is the abscissa of the point (a, 0) Where a graph intersects the axis. The y-intercept is the ordinate of the point (0, b) Where a graph intersects the y-axis. 5.3.3b “ The Slope of a Line” The graphs of the equations y = 3x + 1 and y = x + 1 are drawn. Using the marked points on the graph, the ratio of the vertical distance to the horizontal distance between two points can be found. y x y = x + 1 y = 3x + 1 2 (rise) 2 (run) If you take the points (0, 1) and (-1, -2) on the line y = 3x + 1, the vertical distance is 3 units and the horizontal distance is 1 Unit or vertical distance (rise ) = 3 or 3 horizontal distance (run) 1 The equation y = 3x + 1 where 3/1 stands for the slope.
- 6. 5.4 “ Rewriting the Linear Equation Ax + By = C in the form Y = Mx + B And Vice Versa” One convenient way so an equation can Be also be used to graph the Same line is to solve for Y In terms of X. When the equation Is transformed into y = mx + b The independent variable, and b are the consonants. The letter representing elements from the domain Is called the independent variable. For example, In Y= 3x-2, X is the independent variable. The letter representing elements from the range Is called the dependent variable. It’s value depends on X. Any equation of the form Ax + By = C Can be transformed to an equivalent linear Equation Y= Mx+ B, which is also the Y-form. Illustrative examples Simply the equation, by solving for y In terms of x 3x + 4y = 12 Solution: 3x + 4y = 12 3x + (-3x) + 4y = (-3x) + 12 Addition property of equality 4y = -3x + 12 Additive inverse property ( 1 / 4 ) 4y = (-3x + 12) 1 / 4 Multiplication property of equality Y= -3 / 4 x + 3 Multiplicative inverse property On the other hand, any linear equation Of the form y = mx + b can be transformed To Ax + By = C “ Graph of a Linear Equation in two Variables” The graph of a linear equation can Be drawn in the coordinate plane using The x- and y- intercepts of the Line, any two points on the line Or the slope and a given point. The equation y = mx + b is known as the Slope-intercept form of the equation of A line, where m is the slope And b is the y-intercept. They intercept Is the point where the line intersects The y-axis and the x intercept is the Point where the line intersects the x-axis. To find the y-intercept in a given Equation solve for y when x=0. Similarly, To find the x-intercept in a given Equation, solve for x, when y=0. Illustrative Example Draw the graph of the equation 4x + 3y=12 Solution: If y=0, 4x + 3(0) =12 4x = 12 X= 4, y-intercept With x-intercept 3, and y-intercept 4, the Graph in the line that connects the Points (3,0) and (0,4) is the coordinate Plane shown above. y x 4x + 3y = 12 0 -2 -2 2 2 -4 -4 4 4
- 7. Increasing/Decreasing Graph of y = mx + b The graph of the linear equation may either be increasing or decreasing, depending upon the trend of the line. ILLUSTRATIVE EXAMPLES A. Consider the graphs of y=3x-2 and y= -2x+3 y=3x-2 y=-2x+3 X Y What is the slope of y = -2x+3? What is the slope of y=3x-2? What is the relation of the slope to the trend of the line? When the slope is positive , as in Y=3x-2, the graph of the line is Increasing, or the line rises uniformly from Left to right . When the slope is Negative as in y=-2x+3, the graph of The line is decreasing, or the line Falls uniformly from left to right. 4 2 -4 -2 0 2 4 -2 -4
- 8. “ Obtaining the Equation of a Line” The graph of Ax+By=C (A & B not both 0) is a line in the coordinate plane. Its basic characteristics have also been identified. the geometric conditions used to describe any given line in the coordinate plane will be useful in finding the equation of a line. An equation for a line can be obtained given: 1. the slope and one point on the line 2. two points on the line 3. the slope and its y-intercept ILLUSTRATIVE EXAMPLES A. The slope of a line is -2 and one point on the line is (2,3). Find the equation of the line. (2,3) (x,y) X Solution : let (x,y) be any point on the line other than (2,3). Using the slope formula, m=y-y1/x-x1 and replacing with the given values, -2=y-3/x-2. Simplifying, -2 (x-2) = y-3 -2x + 4 = y-3 -2x –y =-7 or 2x +y= 7 Check: does (2,3) satisfy the equation? 2 (2) + 3 = 7 4 + 3 = 7 Therefore, 2x + y = 7 is the desired equation 4 2 -4 -2 0 2 4 -2 -4
- 9. Problem involving linear equations Solution: The table shows the relationship between the perimeter of a square and its sides. Use the relation y= 4x Many of the problems that are encountered in daily life involve linear relations. The perimeter of a square depends upon the length of its side. Show how the perimeter changes as the length of a side of the square changes. s (x) 2 4 6 8 10 12 14 P (y) 8 16 24 32 - - -
- 10. Consider the number line below. 0 4 4 1 1 2 2 3 3 Measuring a distance of 4 units from the origin, two correct answers, -4 and +4 are obtained. Since no specific direction is given, count in either direction. Remember: The distance between 0 and any real number n on the number line is called the absolute value of the number. It is denoted by /n/, read as “ absolute value of n”. 5.5.1 “ Graph of Absolute Value “ Using the corresponding table of values, the accompanying graphs are obtained. y = x x y 2 -2 2 -2 4 4 -4 -4 y = x Using the table of values for y = /x/, the graph of the basic absolute value function is drawn below. Unless otherwise specified, the domain of the function is the set of real numbers. x y y = /x/ The shape of the graph of the absolute value y = /x/ above reminds us that the value of every real number n is always nonnegative . x -3 -2 -1 0 1 2 3 y -3 -2 -1 0 1 2 3 X -3 -2 -1 0 1 2 3 y 3 2 1 0 1 2 3