I attended the Northwest Math Conference in Tacoma this past weekend and learned so much. I will be sharing various ideas over the next few days. Ideas I think are awesome.One session shared with us a slightly different way of graphing quadratics without completing the square or using a graphing calculator but he started with the idea that a quadratic is composed of two linear equations multiplied together. This is a unique way of looking at quadratics.
In addition he stated any polynomial is made up of a specific number of linear terms multiplied together. So a 3rd degree is made up of three linear equations multiplied together, etc.
The graphing method is so much simpler and so much clearer than traditional ways while combining finding the zeros and the vertex and line of symmetry.
Lets look at y = x^2 + 3x - 7
Step one: (x^2 + 3x) - 7. Place the first two terms inside parenthesis.
Step two: x(x+ 3) - 7. Factor out a common term from the terms inside the parenthesis.
Step three: x = 0 and x+3 = 0. Set the terms equal to zero and solve. So we
Step four. x = 0, -3. These are the two points of the equation that produce the symmetrical points.
Step five: These points give (0, -7) and (-3, -7) when you plug the x values into the equation.
Step six: (0 + -3)/2 = -1.5. Add the two points together and divide by two to find the x value of the vertex.
Step six: (-1.5)^2 + 3(-1.5) - 7 = 2.25 -4.5 -7 = - 9.25. produces the vertex of (-1.5,-9.25)
Now place dots on (0,-7) (-3, -7) and (-1.5, -9.25)
According to the speaker, this method works on all equations, even the ones with irrational roots. One does not have to apply the quadratic formula, complete the square, or any other method. I've not had a chance to investigate these claims but I agree it is much easier to graph the equations this way as long as you are not trying to find the roots.
Give it a try and let me know what you think, I'd love to hear your opinion. This is just one of the new things I've learned. Have a great day.
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