Parabola

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

Draw the points in the coordinate system and connect them with a smooth curve.

Take a look at the formula #y=9\cdot x^2+45\cdot x+54#. Does the point #\rv{-9, 381}# lie on the graph of this formula?

We substitute #x=-9# in the formula. This is done in the following way:

\[y=9\cdot (-9)^2+45\cdot (-9)+54=378\]

Hence #\rv{-9, 381}# does not lie on the graph.

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