Linear formula

Linear formulas and equations: Linear functions

Linear formula

A line‌ar formula is a formula of the form #y=\blue{a} x+\green{b}#, in which #\blue{a}# and #\green{b}# are numbers.

The graph of a linear formula is a straight line.

To determine the graph of the linear formula #{y}=\blue 2 {x}+\green 3#, we draw a straight line moving through two points that lay on the graph.

For #{x}={0}# we have #{y}=\blue{2} \cdot {0} + \green{3} = 3#.
Hence, the graph moves through the point #\rv{0, 3}#.

For #{x}={1}# we have #{y}=\blue{2} \cdot {1} + \green{3} = 5#.
Hence, the graph moves through the point #\rv{1, 5}#.

Table

We create a table for the formula #{y}=\blue{2}x +\green{3}#. This is done by substituting the values of #x=-3# until #{x}={3}# in the formula. For example, for #{x}={3}# we have #{y}=2 \cdot {3}+3=9#.

We then obtain the table on the right hand side for the formula #{y}=\blue {2} \cdot {x}+\green{3}#. We see that if the #x#-value increases by #1#, the #y#-value increases by #\blue{2}#.

Table

\[\begin{array}{l|c|c|c|c|c|c|c} x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline y & -3 & -1 & 1 & 3 & 5 & 7 & 9 \end{array}\]

Draw a graph for the formula:
\[y=3\cdot x+1\]

First we create a table for this graph. For this, we substitute the #x#-values in the function. For #x=-4# we find #y=3\cdot \left(-4\right)+1=-11#. Other #x#-values work in the same manner, and can be found in the table below.

#x#	#-4#	#-3#	#-2#	#-1#	#0#	#1#	#2#	#3#	#4#
#y#	#-11#	#-8#	#-5#	#-2#	#1#	#4#	#7#	#10#	#13#

To compose the graph, we draw a coordinate system. Next, we draw the points from this table in the coordinate system, since these are points of the graph. The #x#-valus is shown on the horizontal axis and the #y#-value on the vertical axis. Next, these points are connected through a straight line. We then get the graph below. The points from the table are drawn in red. Please note that since we know that a linear formula has a straight line as graph, it is sufficient to draw just two points.

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