Linear Graphs - Revision Notes

In these revision notes for Linear Graphs, we cover the following key points:

• What are linear graphs?
• What are the elements of a linear graph?
• What special feature do horizontal graphs have?
• What special feature do vertical graphs have?
• How do we calculate the distance between two horizontal or vertical graphs?
• What features do steeped graphs have?
• What is the domain of a linear graph?
• What is the domain of the range of linear graphs.
• What is the procedure used to plot a linear graph with limited domain?

Linear Graphs Revision Notes

By definition, a linear graph is the graphical representation of a straight line.

Linear graphs only have meaning in coordinate systems. Such systems must be at least two-dimensional in order to contain at least two variables, where one of them is called independent (named so because the values of this variable are provided by the user, who is independent in his/her decision on which values to consider). The other variable depends on the values of the independent variable and the formula that gives the relationship between the two variables (hence the name 'dependent'). All combinations between these two variables produce a line in the X-Y system, which may be straight or not, depending on the type of relation between the variables involved.

In other words, a linear graph is an illustration of a first-order equation with two variables, where the coordinates of each point of the graph make the equation true when substituted into the equation.

If a linear graph is horizontal, this means that the y-coordinate does not change. This means that all points of the graph have the same y-coordinate for whatever value of x. In this way, we don't need an x-coordinate in horizontal graphs; the only important information in such graphs is their y-coordinate which shows how far above or below the horizontal axis the graph of the line lies.

The general equation of horizontal lines (in a 2-D figure) is

where a and b are coefficients and c is a constant. In this specific case, the independent variable is not necessary in the equation so we can hide it from the view. This is achieved by making the coefficient a preceding the variable x zero. Thus, first, we obtain

then

Solving it for the variable y yields

The above value indicates the vertical position (quote) of the graph with respect to the horizontal axis - an axis that acts as a ground-level (or as a reference position or origin). Thus, if y is positive the graph is above the origin, if y = 0 the graph lies according to the origin represented by the X-axis and if y < 0, the graph lies below the origin.

If a linear graph is vertical, this means that the x-coordinate does not change. This means that all points of the graph have the same x-coordinate for whatever value of y. In this way, we don't need a y-coordinate in vertical graphs; the only important information in these graphs is their x-coordinate, which shows how much on the left or on the right of the vertical axis the graph of the line lies.

Hopefully, it is obvious now that, this time, the only variable present in the equation is the independent variable x. the equation of a line in two dimensions is

but since the y-variable is not important, we omit it by multiplying the variable y by zero. This means b = 0. Therefore, the line equation becomes

or

Solving it for the variable x yields

If a graph line is steep, this means that none of the variables are zero because both variables change when shifting along the graph. We can identify a steep graph by looking at the coefficient a and b. If both of them are different from zero, the graph is steep. The general formula of these graphs in two dimensions is

The constant c plays a role in the vertical shift of a linear graph (but not only). On the other hand, the coefficient a are equal in two parallel lines; so are the coefficients b as well. If the coefficient b is not provided we can review the y-intercept of the line as it corresponds to the value of the constant c. We can therefore calculate the value of the constant c by taking x = 0 and solving the rest of the equation

Not all linear graphs are unlimited in space; some of them have certain restrictions that are determined by the values of domain and range.

• The domain is the set of all possible values the independent variable x can take.
• The range is the set of the corresponding y-values of the dependent variable.

If the domain is limited within a certain region of the X-axis, this usually brings a limitation of the graph's line, which in return causes some limitation in the corresponding y-values as well.

The procedure followed to plot the graph if only two limit values of the independent variable x are known, is as follows:

Step 1: Substitute the two limit values x_A and x_B in the line's equation. In this way, you will obtain the two corresponding y-values of the limit points A and B.

Step 2: Connect these points in the shortest way (in a straight line) to plot the graph.

When plotting a linear graph with a limited domain and range (but not only), we make use of one of the main line's properties, which states that two points determine a single line. That's why it is sufficient to have only two points of a linear function available in order to plot its graph. To avoid unnecessary operations we choose these two values to be the limit values which determine the domain and range of a linear function.