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Linear Graphs and their Applications

on Dec 09, 2011

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Linear Graphs and their ApplicationsDocument Transcript

• Page 1 of 5Linear Graphs and their ApplicationsLinear GraphsThe simple but yet challenging question that I am going to answer here will be “What is a LinearGraph?”. Well, let me start first by explaining what a Cartesian Plane (or a Cartesian coordinatesystem) is.The Cartesian Plane 1The Cartesian plate is formed when two lines, one horizontal and the other vertical, intersect eachother and form four quadrants which we call quadrant I, II, III, and IV, as shown in the diagram above.The numbers that you see beside the axes are called coordinates – they are simply numbers that canbe represented on the number line.The vertical line is called the y-axis – when drawn on a graph paper, the numerals on the right handside of the y-axis increases in value from bottom to top.The horizontal line is called the x-axis – when drawn on graph paper, the numerals on the bottom ofthe axis increases in value from left to right.The point at which both line crosses each other, or intersect, is called the origin. The coordinates ofthe origin has only one possible answer: (0,0).Copyright©2011 – Hubert Lawrence Yeo
• Page 2 of 5The QuadrantsThe four differing quadrants have, of course, different characteristics to them. Let me list themdown for you:Quadrant 1:Every single coordinate that can be represented using numerals on the axes in this quadrant hasboth a positive x- and y- axis value.Quadrant 2:Every single coordinate that can be represented using numerals on the axes in this quadrant has apositive y-axis value but a negative x-axis value.Quadrant 3:Every single coordinate that can be represented using numerals on the axes in this quadrant hasboth a negative x- and y- axis value.Quadrant 4:Every single coordinate that can be represented using numerals on the axes in this quadrant has apositive x-axis value and a negative y-axis value.Writing CoordinatesYou would have most probably seen coordinates on a Cartesian plane (or linear graph) being writtenin such a form:(value1, value2)Value1 represents the value of the chosen numeral on the x axis, whereas value2 represents thevalue of the chosen numeral on the y axis.Locating Coordinates2Therefore, if you would want to locate a point on the linear graph given a specific coordinate, youhave to do it in the following manner: 1. Taking the value of the x-coordinate, find that numeral on the x-axis of the linear graph. 2. Use a ruler and draw a straight line from the point on the x-axis which represents that coordinate to the perimeter of the linear graph. 3. Take the value of the y-coordinate and find that numeral on the y-axis of the linear graph. 4. Use a ruler and draw a straight line from the point on the y-axis which represents that coordinate to the perimeter of the linear graph. 5. The exact point at which both lines intersect is the point on the linear graph represented by the pair of numbers (or coordinates) given to you.Copyright©2011 – Hubert Lawrence Yeo
• Page 3 of 5The General Equation of Linear GraphsHere is the general equation ( or general form of the equation) for linear graphs presented in all itsglory:Y = mx + cWhere M is the value of the gradient. The gradient or slope of a line is a measure of the steepness of the line; the bigger the gradient the steeper the line. The gradient of a line can either by positive or negative, depending on which way the line slopes. 3 C is the value of the intercept. The intercept of a line is simply the y value at the point where the line crosses the y-axis (or the y-intercept).Finding the Equation of a Straight LineIn order to find the equation of a straight line graph (or linear graph), all you have to do is find thegradient, m, and the y-intercept, c . Once you’ve got the values, substitute them into the generalequation y=mx+c to give the equation of the line.The GradientThe gradient of a straight line, m, can be given from this formula:Gradient =Basically, you have to do the following. 1. Pick out two points on the straight line and plot their coordinates. 2. Obtain the y values for BOTH coordinates. 3. Obtain the x values for BOTH coordinates. 4. Take the value of the first y value minus the second y value divided by the value of the first x value minus the second x value.4 5. Voila! You’ve got your gradient.Copyright©2011 – Hubert Lawrence Yeo
• Page 4 of 5Three Special GraphsThe Graph of x = ‘A NUMBER’The graphs of these equations are all vertical lines, i.e. they all go straight up and down.For the graph x = ‘a number’, the x coordinates of all points on the line are always the same andequal to the ‘number’. However, all the points will have different y coordinates.The Graph of y = ‘A NUMBER’The graphs of these equations are all horizontal lines, i.e. they all go straight across.For the graph y=’a number’ the y coordinates of all points on the line are always the same and equalto the ‘number’. This time, however, all the points will have different x coordinates.The Graph of y=x, and y=-xThe graphs of these equations are both diagonal lines but in opposite directions. Both lines alwayspass through the origin (0,0) and have a gradient of 1 and -1 respectively.For the graph y=x, the x and y coordinates of a particular point on the line will be the samenumerically and of the same sign (both positive or both negative).For the graph y= -x, the x and y coordinates of a particular point on the line will be the samenumerically, but of opposite signs.Footnotes and Annotations1 Arrows are usually drown at the top of the y-axis and on the furthest right of the x-axis to show thatthe line has the potential to continue and represent more numbers (the amount of numbers you canshow on a linear graph is usually restricted by the size of the graph paper that you are using).2The concept of locating coordinates using the method given above can be used to plot points aswell.3 Positive(+) gradients goes UP from left to right (as we read) whereas negative (-) gradient goesDOWN from left to right.Copyright©2011 – Hubert Lawrence Yeo
• Page 5 of 54 Note that it doesn’t matter which coordinate you choose – whether the one above first, or the onebelow first – the gradient, at the end, will still be the same.Copyright©2011 – Hubert Lawrence Yeo