A quadratic equation is always of the form . For example, in the equation
we can regard as and as .
Solving a quadratic equation means transforming the original equation into a new equation that has the form . We can then take the square root of both sides of the equation and get and
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.
This applet allows you to enter a quadratic function by varying a, b and c sliders and a function by varying A, B and C sliders.
You may solve your equation graphically by dragging the GREEN, BLUE and WHITE dots on the graph in order to produce a 'solution equation' of the form The solution set of the equation can then be gotten by taking the square root of both sides.
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 equation!!
- Why is this legitimate?
- Why are we taught that you must do the same thing to both sides of the equation?
- What is true about all the legitimate things you can do to a quadratic equation?
You can also solve your quadratic equation 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 equation as well as to scale both sides of the equation.
- 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 ?