Slope and intercept

Linear formulas and equations: Linear functions

Slope and intercept

SlopeIn the linear formula of the form \[y=\blue a \cdot x + \green b\] with #\blue a# and #\green b# being numbers #\blue a# is called the slope (or gradient). The slope indicates the direction of the line.

The slope can be calculated in a graph by dividing the vertical change #{\Delta y}# by the horizontal change #{\Delta x}#:

\[ \blue {a} =\frac{{\Delta y}}{{\Delta x}}\]

If #\Delta x = 1#, then #\Delta y = \blue a#. In other words, if we move #1# to the right in the graph, we increase by #\blue a#. This can be seen in the graph on the right-hand side.

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Determining the slope from two points

Assume that the linear formula moves through the points #\blue{\rv{1, 3}}# and #\green{\rv{3,7}}#. Then, the slope will be equal to the vertical change divided by the horizontal change. On the horizontal axis, the #x#-value changes from #1# to #3#, while on the vertical axis,s the #y#-value changes from #3# to #7#. This gives: \[\frac{{\Delta y}}{{\Delta x}}=\frac{7-3}{3-1}=\frac{4}{2}=2\]

In general, if a linear formula moves through the points #\blue A# with coordinates #\blue{\rv{x_A, y_A}}# and #\green{B}# with coordinates #\green{\rv{x_B,y_B}}#, then the slope is equal to:

\[\frac{{\Delta y}}{{\Delta x}}=\frac{y_B-y_A}{x_B-x_A}\]

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Intercept

In the linear formula of the form #y=\blue a \cdot x +\green b# with #\blue a# and #\green b# numbers, the number #\green b# is the #\green{\text{intercept}}#.

The intercept indicates at which #y#-value the graph of the formula intersects with the #y#-axis. Hence, the point #\rv{0,\green b}# lies on the graph of the formula.

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Recognizing the intercept in a table

Since the intercept indicates for which #y#-value the formula intersects the #y#-axis, the intercept is equal to the value of the formula at #x=0#. Hence, in a table, we can read the intercept at #x=0#.

Example

The following table belongs to a linear formula.

#x#	#-2#	#-1#	#0#	#1#	#2#
#y#	#-6#	#-8#	#-10#	#-12#	#-14#

The intercept can be found by looking at the value of #x=0#. In this case, we have #y=-10#. This means that the intercept of this formula is equal to #-10#.

Determine the slope of the line given by the formula #y={{5\cdot x}\over{7}}+{{9}\over{7}}#.

The slope is: #{{5}\over{7}}#

The slope of the line from the formula #y=a\cdot x+b#, in which #a# and #b# are numbers, is equal to #a#.

The given formula #y={{5\cdot x}\over{7}}+{{9}\over{7}}# already has the form #y=a\cdot x+b#, which means you can directly determine the slope: it is the coefficient of #x#, which is equal to #{{5}\over{7}}#.

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