グローバルビジネスで役立つ数学（83）２次関数の係数の符号とグラフ（英語版）

Hi there!

Now people are so concerned about whether they are tested positive or negative on Coronaviruses; it sounds to me just as if you want to know whether you’re living in the air or not. I don’t understand why people are so sensitive to viruses and the results of the PCR test. We, humans, are constantly surrounded by viruses.

The bottom line is whether you are resistant to viruses or not. If not, we have no choice but to maintain and improve strength and energy without resorting to any medicine and vaccines. I don’t want to be rude, but with this coronaviruses pandemic, we are challenged in our usual cultural level, i.e., a person with irregular habits has to face the music.

To be candid, I haven’t received any vaccine shots because I don’t want any foreign object to be administered into my body. That’s it. These days, I’ve seen more and more people who contracted COVID-19 even though they’ve got two to three vaccine shots. Naturally, they are exposed to contracting the disease because the viruses constantly mutate, which everyone knows.

That being said, I’m concerned about whether the coefficients of a quadratic function are positive or negative. So let me challenge today to identify positive or negative on the word problem below.

Identify positive or negative the following coefficients, given the graph of the quadratic function \(y = ax^{2} + bx + c\). (1) \(a\) (2) \(b\) (3) \(c\) (4) \(b^{2} – 4ac\) (5) \(a – b + c\). (Ref: Yellow Chart Math Ⅰ＋Ａ P88, slightly changed the quadratic function)

Here is my solution to the problem.

First of all, from the condition given, we get

\(ax^{2} + bx + c = a(x + \frac{b}{2a})^{2} – \frac{b^{2} -4ac}{4a}\)

So, regarding the parabola \(y = ax^{2} + bx + c\),
the axis is the straight line \(x = -\frac{b}{2a}\),
the y-coordinate of its vertex becomes \(-\frac{b^{2} – 4ac}{4a}\), and
the y-coordinate of the intersection with the y-axis is \(c\).

Needless to say, at \(x = -1\), \(y = a(-1)^{2} + b(-1) + c = a – b + c\).

(1) Since the graph shows convex upward, \(a < 0\).

(2) Since the axis is situated at \(x < 0\), \(-\frac{b}{2a} < 0\). From (1), we know \(a < 0\), so \(b < 0\).

(3) Since the graph shows the intersection with the positive point of the y-axis, \(c > 0\).

(4) Since the y-axis of the vertex is positive,
\(- \frac{b^{2} – 4ac}{4a} > 0\).
From (1), we know \(a < 0\), so \(-(b^{2} - 4ac) < 0\), i.e., \(b^{2} - 4ac > 0\)

(5) \(a – b + c\) shows the value of y at \(x = -1\).
From the above graph, \(y > 0\) at \(x = -1\).
So, \(a – b + c > 0\).

So, the quadratic function of the graph is \(y = -x^{2} -x + 2\).

Stay tuned, and expect to see my next post.

Keep well.

Frank Yoshida

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【グローバルビジネスで役立つ数学】でもっと学習する b＾＾)
【参考図書】『もう一度高校数学』（著者：高橋一雄氏）株式会社日本実業出版社
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