Graphs of Quadratic Expressions - Examples

Solved Example 1: Plot the graph of \(Q\left( x \right)\,\,:\,\,2{x^2} + 5x + 2\).

Solution:   Step-1: \(a = 2 > 0\), so the parabola will open upward.

Step-2: \(2{x^2} + 5x + 2 = \left( {2x + 1} \right)\left( {x + 2} \right)\), so the zeroes of the expression are \(x =  - \frac{1}{2},\,\,x =  - 2\). These are the points where the parabola will cross the horizontal axis.

Step-3: The discriminant is \(D = 9\) (verify). The coordinates of the vertex are \(V \equiv \left( { - \frac{b}{{2a}}, - \frac{D}{{4a}}} \right)\) or \(V \equiv \left( { - \frac{5}{4}, - \frac{9}{8}} \right)\). This will be the lowermost point on the parabola.

Step-4: We calculate \(Q\left( x \right)\) for some specific values of x. For example, \(Q\left( 0 \right) = 2\), \(Q\left( { - 3} \right) = 5\), etc.

Using all this information, the graph can now be plotted easily: