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Figure 1
Figure 1
Figure 2
Figure 2
To understand this page you must know how to plot coordinates and substitute values in equations, check out the coordinates page and substitution page to understand.
Plotting straight lines is just a further extension of plotting points.
Question 1:
If y = 2x - 1, find the values of y, when x equals -3, -2, -1, 0, 1, 2 and 3. Plot y = 2x -1, for the values of x from -3 to 3.
when x = -3, y = 2(-3) - 1 = -6 -1 = -7when x = -2, y = 2(-2) - 1 = -4 - 1 = -5when x = -1, y = 2(-1) - 1 = -2 - 1 = -3when x = 0, y = 2(0) - 1 = 0 - 1 = -1when x = 1, y = 2(1) - 1 = 2 - 1 = 1when x= 2, y = 2(2) - 1 = 4 - 1 = 3when x = 3, y = 2(3) - 1 = 6 - 1 = 5
Using coordinates we can represent our inputs and outputs on a 2 dimensional graph; we can represent the equation y = 2x -1, for when x takes on values from -3 to 3.
We can write the inputs and outputs as coordinates. Once we attain coordinates, we are able to plot them. The inputs are the x values and the outputs are the y values.
Why is x the input? Well x is substituted with numbers to find the value of y, in other words we input numbers into 2x -1 by substituting x with numbers to find the value of y; the values of y are the outputs.
You can see above when we input the values of x from -3 to 3, we get outputs, so lets now write the inputs and outputs in form of (x,y) which is in coordinate form.
when x = -3, y = -7, so in coordinate form we can write (-3,-7)when x = -2, y = -5, so in coordinate form we can write (-2,-5)when x = -1, y = -3, so in coordinate form we can write (-1,-3)when x = 0, y = -1, so in coordinate form we can write (0,-1)when x = 1, y = 1, so in coordinate form we can write (1,1)when x = 2, y = 3, so in coordinate form we can write (2,3)when x = 3, y = 5, so in coordinate form we can write (3,5)
We get
(-3,7), (-2,-5), (-1,-3), (0,-1), (1,1), (2,3), (3,7). Figure 1 shows the above coordinates plotted on a 2-dimensional graph.
As shown in figure 2, when join the the coordinates up they form a straight line. Thus we have plotted the equationy = 2x -1, when x takes on values from -3 to 3.
This shows us an application of using graphs to represent equations. For example, lets say you were given a 2 dimensional graph with a straight line on it, and you needed to find the value of y, when x = 5. You can just substitute 5 into the equation and find y, OR you can look at the graph and find the point where x = 5, and see what the y value is. Remember we are plotting the input and outputs as coordinates.
ONE THING TO NOTE IS when we write coordinates (x,y), x (the first letter) is always the input value and y (the second letter) is always the output value. It is not necessary that the letters x and y are used. Another way to think that in a 2 dimensional graph the horizontal axis ALWAYS has our input values on it, these values are represented by the first number in a coordinate and the vertical axis ALWAYS has our output values on it, these values are represented by the second number in a coordinate. You can understand why we call the horizonal axis, the x axis because when people write equations they normally represent the input (the letter we substitute values with to find an output) with the letter x. The letter y is normally used to represent the output, so that is why the vertical axis is normally called the y axis.
However, lets say we have the function f(x) = 2x - 1, here x is the input and f(x) is the output.If we wanted to plot f(x), for when x takes on values from -3 to 3, we would still attain the inputs and outputs, put them in coordinates form and plot the coordinates.
In this case though, the output has been called f(x) instead of y, but this does not change anything because the inputs and outputs would be the same. When we write coordinates for a 2 dimensional graph, in the form of (x,y), the first letter represents the input and the second letter represents the output. However, if we are working with functions, f(x) is the output, so we can write
(x,f(x))
We would get the exact same coordinates, no matter if we write y = 2x -1 or f(x) = 2x -1. However, when we plot f(x) = 2x -1, we would just call the y axis the f(x) axis, both in this case represent the output. You can understand this in video 2's explanation.
THE MAIN THING IS THAT IN A 2 DIMENSIONAL GRAPH, THE HORIZONTAL AXIS CONTAINS THE INPUTS AND THE VERTICAL AXIS CONTAINS THE OUTPUTS.
Plotting straight lines is just a further extension of plotting points.
Question 1:
If y = 2x - 1, find the values of y, when x equals -3, -2, -1, 0, 1, 2 and 3. Plot y = 2x -1, for the values of x from -3 to 3.
when x = -3, y = 2(-3) - 1 = -6 -1 = -7when x = -2, y = 2(-2) - 1 = -4 - 1 = -5when x = -1, y = 2(-1) - 1 = -2 - 1 = -3when x = 0, y = 2(0) - 1 = 0 - 1 = -1when x = 1, y = 2(1) - 1 = 2 - 1 = 1when x= 2, y = 2(2) - 1 = 4 - 1 = 3when x = 3, y = 2(3) - 1 = 6 - 1 = 5
Using coordinates we can represent our inputs and outputs on a 2 dimensional graph; we can represent the equation y = 2x -1, for when x takes on values from -3 to 3.
We can write the inputs and outputs as coordinates. Once we attain coordinates, we are able to plot them. The inputs are the x values and the outputs are the y values.
Why is x the input? Well x is substituted with numbers to find the value of y, in other words we input numbers into 2x -1 by substituting x with numbers to find the value of y; the values of y are the outputs.
You can see above when we input the values of x from -3 to 3, we get outputs, so lets now write the inputs and outputs in form of (x,y) which is in coordinate form.
when x = -3, y = -7, so in coordinate form we can write (-3,-7)when x = -2, y = -5, so in coordinate form we can write (-2,-5)when x = -1, y = -3, so in coordinate form we can write (-1,-3)when x = 0, y = -1, so in coordinate form we can write (0,-1)when x = 1, y = 1, so in coordinate form we can write (1,1)when x = 2, y = 3, so in coordinate form we can write (2,3)when x = 3, y = 5, so in coordinate form we can write (3,5)
We get
(-3,7), (-2,-5), (-1,-3), (0,-1), (1,1), (2,3), (3,7). Figure 1 shows the above coordinates plotted on a 2-dimensional graph.
As shown in figure 2, when join the the coordinates up they form a straight line. Thus we have plotted the equationy = 2x -1, when x takes on values from -3 to 3.
This shows us an application of using graphs to represent equations. For example, lets say you were given a 2 dimensional graph with a straight line on it, and you needed to find the value of y, when x = 5. You can just substitute 5 into the equation and find y, OR you can look at the graph and find the point where x = 5, and see what the y value is. Remember we are plotting the input and outputs as coordinates.
ONE THING TO NOTE IS when we write coordinates (x,y), x (the first letter) is always the input value and y (the second letter) is always the output value. It is not necessary that the letters x and y are used. Another way to think that in a 2 dimensional graph the horizontal axis ALWAYS has our input values on it, these values are represented by the first number in a coordinate and the vertical axis ALWAYS has our output values on it, these values are represented by the second number in a coordinate. You can understand why we call the horizonal axis, the x axis because when people write equations they normally represent the input (the letter we substitute values with to find an output) with the letter x. The letter y is normally used to represent the output, so that is why the vertical axis is normally called the y axis.
However, lets say we have the function f(x) = 2x - 1, here x is the input and f(x) is the output.If we wanted to plot f(x), for when x takes on values from -3 to 3, we would still attain the inputs and outputs, put them in coordinates form and plot the coordinates.
In this case though, the output has been called f(x) instead of y, but this does not change anything because the inputs and outputs would be the same. When we write coordinates for a 2 dimensional graph, in the form of (x,y), the first letter represents the input and the second letter represents the output. However, if we are working with functions, f(x) is the output, so we can write
(x,f(x))
We would get the exact same coordinates, no matter if we write y = 2x -1 or f(x) = 2x -1. However, when we plot f(x) = 2x -1, we would just call the y axis the f(x) axis, both in this case represent the output. You can understand this in video 2's explanation.
THE MAIN THING IS THAT IN A 2 DIMENSIONAL GRAPH, THE HORIZONTAL AXIS CONTAINS THE INPUTS AND THE VERTICAL AXIS CONTAINS THE OUTPUTS.
Try this question.
Draw the line y = x, for values of x from -5 to 5.
Draw the line y = -x, for the values of x from -5 to 5.
Question 1 Answer
Question 1 Answer
Question 2 Answer
Question 2 Answer
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