Explorations of Quadratic Equations:

Fix the values for a and b and vary the variable c for the quadratics equation and discuss your exploration and results.

Let a= 1 and b=1 and vary c between -3 and 3, which means -3 ≤ c ≤ 3. The quadratic equation, with the aforementioned fixed values for a and b is as follows:

The family of parabolas for produces vertical translations. Each value of c is the y-intercept for the particular equation (i.e the y-intercept for the equation is (0,1)).
Let’s consider when from the quadratic equation.
When c > 0, we get , which means we won’t get any real roots for x because the value under the square root will be negative, producing an imaginary number.
When c < 0, we get , we will get real roots for x because the value under the square root is positive and will not produce an imaginary number.

I want to figure out the pattern about the locus of the vertices of the parabolas. I’m going to use the general quadratic equation again.
First we must find out the value for x, which can be computed by taking the derivative of the quadratic equation.

Set , therefore , and
For the purpose of my equation, we know the variables a and b equal 1, therefore we can determine the value of x. when a = 1 and b =1.

So any value of c we choose, when a =1 and b = 1, we will get the point ().

We know for the equation , by plugging in , the point will be

() = ( ) will always be on the vertical line .

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