Characteristics of a Quadratic Function in Standard Form (Example)

This video discusses the characteristics of a quadratic function in standard form by looking at a specific example and determining the domain and range of the function, the vertex and the axis of symmetry of the graph of the function, as well as other behavior of the function.

[jan-willemreens]

Good evening James, After thoroughly reading your unambiguous clear explanation a few times, where the equation of the axis of symmetry comes from, it has finally become completely clear to me! With the help of your explanation I have graphically represented the three cases (D<0, D=0, and D>0) with pencil and paper, actually the six cases (a<0 and also a>0 of course), with all relevant parts that follow from the quadratic formula. What also drawing a simple graph can do, can't it? Especially looking at case D=0, where (-b/2a, 0) is the vertex and x=-b/2a is the vertical axis of symmetry of the Parabola, just like you mentioned. Last comment I want to make is, I think we could probably also say regarding case D=0, that you have two equal x-intercepts (x1, 2=-b/2a), given that we are dealing here with 2nd degree functions, and so must solve 2nd degree equations to find real or complex x-intercepts, maybe I'm wrong. Thank you very much James for your precious time and extensive clear explanation, and I will continue watching your next videos with great enthusiasm. Jan-W

[jan-willemreens]

Good day James, Maybe I'm a bit too far ahead of the theory, but suddenly, during a walk, I thought of a possible connection between the equation of the axis of symmetry of the Parabola: x=-b/2a, and the quadratic formula for 2nd degree Polynomials: x1=(-b-sqrt(D))/2a or x2=(-b+sqrt(D))/2a, where D=b^2-4ac. If both fractions x1 and x2 are split separately into two new fractions each, you get a common fraction: x=-b/2a. Maybe that explains the equation for the axis of symmetry? After all x=-b/2a is the middle x-value, and thus the axis of symmetry equation?! ----> ( ). All the time I'm wondering where x=-b/2a comes from! An annoyance for me is learning formulas without understanding them. I immediately forget formulas learned in this way! Finally, the content of your video was easy to follow. Thank you, Jan-W