Coordinate Geometry

By Roy Richard Sawyer

In a Cartesian coordinate system each point is specified by the distance from perpendicular lines that are denoted as X- axis and Y-axis.

The simplest function Y = X is represented by a straight line that intersects the origin (the point of intersection X-axis and Y-axis) and bisects the straight angle between X-axis and Y-axis.

To plot a function on the coordinate system, assign any value to X and calculate the corresponding value for Y. For example, for the function Y =X, if X = 0 then Y =0. If X = 10 then Y = 10.

Two points are enough to draw a graph for a linear function that is a straight line on the coordinate system.

For the function Y = -X, the graph will have a different orientation.

In Figure 1, you can see how the points were calculated. Note that for any linear function

y = aX + b where a and b are constants and are represented by real numbers,

changing 'a' to a negative number will change the graph orientation.

linear function

Figure 1 Graph of the functions: Y = X and Y = - X

Figure 2.

On the figure 2, you can see that the graph for the function Y = 3X has a mirror orientation of the graph for the function Y = - 3X.

Also, you can see that the steepness of the line for the function Y = 3X is greater than that of the line for the function Y = X

The slope of the line is the ratio of vertical changes to the horizontal changes.

slope

Figure 2. Slope

Parallel lines for functions with the same slope

Figure 3. Parallel lines for functions with the same slope.

Lines for functions Y = X, Y = X+5 and Y=X-5 are parallel because these functions have the same coefficient in front of x.  It is true for any linear function: Y = aX + b.

y = ax + b is the equation of the line with slope a and y-intercept b.

For Y = X + 5, a =1 and b=5. On the graph you can see that y-intercept for the function equals 5.

For the function Y = X - 5, a =1 and b= -5. So, the y- intercept is -5.

The slope for both functions is 1.

Question: Find the slope and X, Y-intercept for the function Y = 3X + 7

Answer: Since the coefficient in front of X is 3, then the slope =3

And since b is 7, the y-intercept will be 7.

How to find X intercept?  Assign Y to 0, then 0 = 3X + 7

0 - 3X = 3X + 7 - 3X.  Subtract 3X from both sides of the equation

-3X = 7

X = -7/3; X-intercept is - 2.33

You can plot the graph and check if the solution is correct.

Take a look at the following graphs:

Linear functions with a different slope

Figure 4 Graphs for linear functions with a different slope.

Note that for a graph of a function Y = aX + b, the slope will increase with an increasing coefficient in front of X. You can see it in figure 1.

The greater the coefficient in front of X, the greater is the slope of the function.

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