Quadratic Inequalities solved w/ Graphs & Symbols
- Judah L Schwartz
A quadratic inequality is always of the form . For example, in the inequality we can regard as and as . Solving a quadratic inequality means transforming the original inequality into a new inequality that has the form (where is a constant). We can then take the square root of both sides of the equation and get the solution set of the inequality. The graph of the function is a parabola that is open (concave) upward and just touches (tangent to) the x-axis at . The graph of the constant function is a horizontal line above and parallel to the x-axis. The environment allows you to enter a quadratic function by varying a, b and c sliders and a function by varying A, B and C sliders. Fundamentally this applet allows you to explore the following question - For what values of x is the GREEN function larger than the BLUE function ? You may solve your inequality graphically by dragging the GREEN, BLUE and WHITE dots on the graph. Challenge - Dragging the WHITE dots changes both functions, but dragging the GREEN dot changes only the GREEN function and dragging the BLUE dot changes only the BLUE function. This means that when you drag either the GREEN dot or the BLUE dot you are changing only ONE side of the inequality!! - Why is this legitimate? - Why are we taught that you must do the same thing to both sides of the inequality? - What is true about all the legitimate things you can do to a quadratic inequality? You can also solve your quadratic inequality symbolically by using sliders. The sliders allow you to change the constant term, the linear term and the quadratic term on both sides of the inequality as well as to scale both sides of the equation. Particular attention needs to be paid to the action of the scale slider! - What symbolic operations correspond to the actions you take when solving graphically? - What graphical operations correspond to the actions you take when solving graphically? What other questions could/would you pose to your students based on this applet ?