Graphs

Linear formulas and equations: Formulas

Graphs

Graph

We can create a table corresponding to the formula #\blue{y=5x+10}# :

\[\begin{array}{l|c|c|c|c|c}
x & 0 & 1 & 2 & 3 & 4\\
\hline
y & 10 & 15 & 20 & 25 & 30
\end{array}\] We made this table by calculating the value of #y# corresponding to the chosen value of #x# in the upper row. The values of @y@ are in the bottom row.

We can make a graph corresponding to this table. The upper row with values for #x# corresponds to the horizontal axis, and the bottom row with values for #y# corresponds to the vertical axis.

If #x=1#, then #y=15#. This corresponds to the point #\rv{1,15}#. On the right you see how you can find this point by drawing perpendicular lines from the axes. From the table it follows that the graph goes through the following points: #\rv{0,10}#, #\rv{1,15}#, #\rv{2,20}#, #\rv{3,25}# and #\rv{4,30}#. These points are drawn on the right and connected by a smooth line, which in this case is a straight line.

Draw a graph corresponding to the formula:
\[y=-3\cdot x^2+3\cdot x+1\]

First of all, we compose a table corresponding to the graph. For this we substitute the #x#-values in the formula. For #x=-4# we find #y=\left(-3\right)\cdot \left(-4\right)^2+3\cdot \left(-4\right)+1=-59#. The other #x#-values follow the same pattern. We eventually find the table below.

#\boldsymbol{x}#	#-4#	#-3#	#-2#	#-1#	#0#	#1#	#2#	#3#	#4#
#\boldsymbol{y}#	#-59#	#-35#	#-17#	#-5#	#1#	#1#	#-5#	#-17#	#-35#

To compose the graph, we draw a coordinate system. Then we draw the points from the table in the coordinate system, because these are points in the graph. The #x#-value is on the horizontal axis and the #y#-value is on the vertical axis. We then connect these points with a smooth curve. Doing so gives us the graph below. The points from the table are colored red.

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