Quadratic equations: Parabola
Parabola
Graph
The graph of a quadratic \[y=\blue ax^2+\green bx+\purple c\] is called a parabola.
If #\blue a \gt 0# the graph is an upward opening parabola.
If #\blue a \lt 0# the graph is a downward opening parabola.
An upward opening parabola has a minimum and downward opening parabola has a maximum. In both cases, this point is referred to as the vertex of the graph.
The parabola is symmetrical about the vertical line through the top of the graph. Such a line is also called a #\orange{\textbf{line of symmetry}}#.
geogebra picture
Plotting a graph
To plot the graph of the quadratic #y=-x^2+2x+1#, start by constructing a table. Next, draw the points in a coordinate system and connect the dots with a smooth curve.
Table
#\begin{array}{l|c|c|c|c|c|c|c|c|c} x & -3 & -2 & -1 & 0 & 1 & 2 & 3&4&5\\ \hline y & -14 & -7 & -2 & 1 & 2 & 1 & -2 &-7 & -14 \end {array}#
We find the values of #y# by plugging the values of #x# into the formula.
Chart
To plot the graph of the quadratic #y=-x^2+2x+1#, start by constructing a table. Next, draw the points in a coordinate system and connect the dots with a smooth curve.
Table
#\begin{array}{l|c|c|c|c|c|c|c|c|c} x & -3 & -2 & -1 & 0 & 1 & 2 & 3&4&5\\ \hline y & -14 & -7 & -2 & 1 & 2 & 1 & -2 &-7 & -14 \end {array}#
We find the values of #y# by plugging the values of #x# into the formula.
Chart
Draw the points in the coordinate system and connect them with a smooth curve.
Take a look at the formula #y=8\cdot x^2+88\cdot x+192#. Does the point #\rv{-7, -32}# lie on the graph of this formula?
Yes
We substitute #x=-7# in the formula. This is done in the following way:
\[y=8\cdot (-7)^2+88\cdot (-7)+192=-32\]
Hence #\rv{-7, -32}# does lie on the graph.
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