Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Chapter 3 Section 3.1 Graphs Graphs A graph is a way to draw a "picture" of an equation. A location on the picture is called a point and is marked with a dot and is a pair of numbers in parenthesis: ๐ฅ, ๐ฆ โ Point ๐ฅ ๐๐ ๐กโ๐ ๐ฅ โ ๐๐๐๐๐๐๐๐ก๐ ๐ฆ ๐๐ ๐กโ๐ ๐ฆ โ ๐๐๐๐๐๐๐๐ก๐ The graph is drawn on a coordinate system that at 2 perpendicular number lines intersecting at the origin (0 position) of each number line. The location is found by moving left or right by the x-coordinate and up or down by the y-coordinate. 7 6 B 5 D 2nd 1st 4 C A 3 2 1 E 7 6 5 4 3 2 1 1 2 3 4 5 6 7 1 2 G 3 3rd 4th 4 F 5 6 7 D: โ3,5 C: 0,3 F: โ3, โ5 E: โ5,0 Axis A: 5,3 B: 3,5 G: 5, โ3 Points on Equations A point is on the graph of an equation if when you plug the x and y coordinates into the equation you get a true equation (i.e. equal values). If the point does not work in the equation then the point is not on the graph. Points on the graph of 9๐ฅ 2 + ๐ฆ 2 = 25 9x2 y2 25 5 Point 1,4 9 1 2 + 4 2 = 25 9 + 16 = 25 25 = 25 4 3 2 1 5 4 3 2 1 1 1 2 3 4 5 2 3 4 5 Point 0, โ5 9 0 2 + โ5 2 = 25 0 + 25 = 25 25 = 25 Points not on the graph of 9๐ฅ 2 + ๐ฆ 2 = 25 Point 1, โ2 9 1 2 + โ2 2 = 25 9 + 4 = 25 13 = 25 Point โ2,3 9 โ2 2 + 3 2 = 25 36 + 9 = 25 45 = 25 Graphing (by Point Plotting) One way to graph an equation is to make a table of x and y values which become points that are plotted. Try to use enough points to find the overall shape. y x 2 ๐ฆ = ๐ฅ โ2 ๐ฅ ๐ฆ 0 โ2 1 โ1 โ1 โ1 0 2 0 โ2 5 5 4 5 4 3 2 4 3 3 2 2 1 1 1 1 1 2 3 2 3 4 5 5 4 3 2 1 1 1 2 3 4 4 5 5 2 3 4 5 If the equation is solved for y continue to pick enough values for x so that as you plot the points you can begin to understand what the shape of the graph is going to be. 2 y x2 2x ๐ฆ = ๐ฅ โ 2๐ฅ ๐ฅ 0 1 โ1 2 3 ๐ฆ 0 โ1 3 0 3 5 5 4 5 4 3 2 4 3 3 2 2 1 1 1 1 2 3 4 5 5 4 3 2 1 1 1 1 2 2 3 4 3 4 5 5 2 3 4 5 An equation that is written as ๐ฆ = ๐๐ฅ + ๐ for numbers a and b is a line. To graph a line you only need 2 points. A third point can serve as a check you did it right. y 3x 1 5 5 4 ๐ฆ = 3๐ฅ + 1 ๐ฅ ๐ฆ 0 1 1 4 โ1 โ2 4 3 3 2 2 1 5 4 3 2 1 1 1 1 2 2 3 4 5 5 4 3 2 1 1 1 2 3 3 4 4 5 5 2 3 4 5 An equation that is written as ๐๐ฅ + ๐๐ฆ = ๐ for numbers a, b and c is also a line. The graph can be found by filling in a table of at least 3 values. To do this you will need to plug the values in and solve an equation. 2x 3 y 12 2๐ฅ โ 3๐ฆ = 12 ๐ฅ 0 6 3 ๐ฆ โ4 0 โ2 For ๐ฅ = 0 2 0 โ 3๐ฆ = 12 0 โ 3๐ฆ = 12 โ3๐ฆ = 12 1 ๐ฆ = โ 12 3 ๐ฆ = โ4 8 For ๐ฆ = 0 2๐ฅ โ 3 0 = 12 2๐ฅ โ 0 = 12 2๐ฅ = 12 1 ๐ฅ = 12 2 ๐ฅ=6 7 6 5 4 3 2 1 8 7 6 5 4 3 2 1 1 1 2 For ๐ฅ = 3 2 3 โ 3๐ฆ = 12 6 โ 3๐ฆ = 12 โ3๐ฆ = 12 + โ6 โ3๐ฆ = 6 1 ๐ฆ=โ 6 3 ๐ฆ = โ2 3 4 5 6 7 8 2 3 4 5 6 7 8