I keep finding goofy things in our new pre-calc text.
It's bad enough that it just presents "facts" without any proof. If I remember correctly (we haven't gotten to this topic yet), it just presents the Law of Sines and the Law of Cosines without proof! It just presents them as being delivered directly from Heaven on tablets. Several important points are presented without proof.
Then there's stuff I'd consider wrong.
For you math types out there, what would you consider to be the standard form of the equation of a parabola? For me, without any hesitation, I'd say standard form is y=ax^2+bx+c. Our new textbook, though, says standard form is y=a(x-h)^2+k, which is similar to what I'd call vertex form, namely (y-k)=a(x-h)^2, a form that would graph a parabola with vertex (h,k).
If we can't agree on standard form, it's not very standard, is it?
10 comments:
As soon as I read your question "what is the standard form of a parabola" I pictured the same eqn as you. That's the standard form, at least up until now. They are losing the forest for the leaves.
I'm with you Darren. You listed the standard form. They've listed the Vertex form. Don't think I learned the vertex form until I hit college math.
As far as terminology goes, yes I agree that y=ax^2+bx+c is the "standard form" of a parabola just as Ax+By=C is the standard form of a line. However going by what does "standard" actually mean as in what is most used/useful, the "standard forms" fall very short. The issue I see is that neither standard form is particularly useful for understanding what the function is actually doing. Standard form of a line doesn't tell you anything about the slope or x or y intercepts. Standard form of a parabola doesn't tell you about the focus, x intercepts, min/max value, etc. The h/k form also ties nicely into the similar equations for the other conics.
They are a bit useful if you are trying to add/subtract equations to find points of intersection... I'll admit I haven't compared ease / time to complete adding equations in standard form versus slope y-intercept form
So yes, that is the standard form, but maybe it's time to change the nomenclature to "old form" or "addition form" or something like that.
That same issue has been bothering me as well. The only answer that I have come up with is that y=ax^2+bx+c is the standard form of a quadratic equation which of course is a parabola whereas (y-k)=4p(x-h)^2 is the standard(conic) form of a parabola. Your new book gives as you said the vertex form of a quadratic equation. Maybe a copyright violation to say otherwise. Either way it is a pain in the latus rectum.
I've seen a version of "vertex form" for a parabola written this way:
4p(y-k)=(x-h)^2, where p is the focal length and (h,k) is the vertex. Alternately, a=1/(4p).
This is perfect timing I just did quadratic yesterday and noticed our new Pre-Calculus says the same thing. I told the students how the book calls it, but in every other book and math class standard for is ax^2 + bx + c and the other is vertex form. Thought it was just me.
y = ax^2 + bx + c is also the standard form in the antipodes.
I have always wondered why the US uses the AX + BY = c as the standard form of the linear equation; down the bottom of the world, we use y = ax + b (or mx + c), which is a closer match for the other standard forms.
Standard form: y = ax^2 + bx + c
Vertex form: y = a(x - h)^2 + k
At least that's the way I see it. :)
CyberChalky,
We don't (or didn't).
y = mx + b
Standard form for a line is Ax+By=C (or, as it has some good uses, Ax+By+C=0).
y=mx+b is the "slope-intercept form" of a line.
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