论坛 › 数学欣赏 › 欣赏一些函数图象

manthanein 发表于 2021-10-5 21:26:39

欣赏一些函数图象

函数\(y=x\)

https://www.desmos.com/calculator/dj7vjfsj6s?lang=zh-CN

manthanein 发表于 2021-10-5 21:27:05

函数\(y=x^2\)

https://www.desmos.com/calculator/13on7bph8f

manthanein 发表于 2021-10-5 21:27:34

本帖最后由 manthanein 于 2021-10-5 22:38 编辑

函数\(y=x^3\)和\(y=Ax^3\)

https://www.desmos.com/calculator/bvjzo3eh34

manthanein 发表于 2021-10-5 21:28:27

本帖最后由 manthanein 于 2021-10-5 22:48 编辑

函数\(y=x^5\)和\(y=(Ax+B)^5\)

https://www.desmos.com/calculator/rxlnnsslq0

函数\(y=x^e\)

https://www.desmos.com/calculator/geabxblici

函数\(\D y=\frac{1}{x^{2}}\)和\(\D y=\frac{1}{x^{3}}\)和\(\D y=\frac{1}{x^{5}}\)
https://www.desmos.com/calculator/horbfwrqht

manthanein 发表于 2021-10-5 21:29:41

本帖最后由 manthanein 于 2021-10-6 00:33 编辑

双曲线

\(\D xy=t\\implies y=\frac{t}{x}\)
\(\D (x+a)y=t \implies y=\frac{t}{x+a}\)
\(\D (x+a)(y+b)=xy+bx+ay+ab=t \implies y=\frac{t-ab-bx}{x+a}\ \)

https://www.desmos.com/calculator/xoopyshebs

\(\D y=\frac{x^{2}}{ax+b}\)
https://www.desmos.com/calculator/z8daqz7x4q

\(\D y=\frac{t}{ax+b}\)
\(\D \implies a(x+\frac{b}{a})y=t\)
\(\D \implies y=\frac{\D \frac{t}{a}}{\D x+\frac{b}{a}}\)
https://www.desmos.com/calculator/gyk4kzswue



\(\D y=\frac{x}{ax+b}\)
\(\D \implies a(ax+b)y=ax\)
\(\D \implies a(ax+b)y+b=ax+b\)
\(\D \implies (ay-1)(ax+b)+b=0\)
\(\D \implies (y-\frac{1}{a})(x+\frac{b}{a})=\frac{-b}{a^2}\)
https://www.desmos.com/calculator/466zvkho5g

\(\D y=\sqrt{x^{2}+a^{2}}\)
\(\implies y^{2}-x^{2}=a^{2}\)

https://www.desmos.com/calculator/5bmz3iewxk

\(\D y=\sqrt{x^{2}-a^{2}}\)
\(\implies x^{2}-y^{2}=a^{2}\)
https://www.desmos.com/calculator/cu8rtah1ac

manthanein 发表于 2021-10-5 21:35:04

本帖最后由 manthanein 于 2021-10-5 22:56 编辑

\(\D y=\frac{1}{x^{2}\left(ax+b\right)}\)

https://www.desmos.com/calculator/mi97vkzrwp

manthanein 发表于 2021-10-5 22:10:42

本帖最后由 manthanein 于 2021-10-6 01:19 编辑

\(\D (ax^2+bx+c)y=t \implies \D y=\frac{t}{ax^2+bx+c}\)
\(\D \implies (4a^2x^2+4abx+4ac)y=4at\)
\(\D \implies (2ax+b)^2+(4ac-b^2)=\frac{4at}{y}\)
\(\D \implies 4a^2(x+\frac{b}{2a})^2+(4ac-b^2)=\frac{4at}{y}\)
\(\D \implies (x+\frac{b}{2a})^2+\frac{4ac-b^2}{4a^2}=\frac{t}{ay}\)

\(\D \implies y=\frac{\D \frac{t}{a}}{\D (x+\frac{b}{2a})^2+\frac{4ac-b^2}{4a^2}}\)

https://www.desmos.com/calculator/hd78xfykwl
https://www.desmos.com/calculator/pnk76ijkxd
https://www.desmos.com/calculator/v5yrhutnbj

\(\D y=\sqrt{\frac{x-a}{x-b}} \)
\(\left(x-b\right)y^{2}=x-a\)
\(xy^2-by^2=x-a\)
\(x(y^2-1)=by^2-a\)
\(x(y^2-1)=by^2-b+b-a\)
\(x(y^2-1)=b(y^2-1)+b-a\)
\((x-b)(y^2-1)=b-a\)

https://www.desmos.com/calculator/7urca6snua

\(\D y=\sqrt{\frac{x-a}{b-x}} \)
\(y^2(b-x)=x-a\)
\(by^2-xy^2=x-a\)
\(x(y^2+1)=by^2+a\)
\(x(y^2+1)=by^2+b-b+a\)
\(x(y^2+1)=b(y^2+1)-b+a\)
\((x-b)(y^2+1)=a-b\)

https://www.desmos.com/calculator/em1zz5owdo

manthanein 发表于 2021-10-5 22:26:31

本帖最后由 manthanein 于 2021-10-5 23:03 编辑

\(\D y=\frac{x}{\left(ax+b\right)^{2}}\)

https://www.desmos.com/calculator/axgp3xzdge

manthanein 发表于 2021-10-5 22:28:29

本帖最后由 manthanein 于 2021-10-5 23:05 编辑

\(\D y=\frac{x^{2}}{\left(ax+b\right)^{2}}\)
https://www.desmos.com/calculator/ojudqajifu

manthanein 发表于 2021-10-5 22:29:32

本帖最后由 manthanein 于 2021-10-5 23:06 编辑

\(\D y=\frac{1}{x\left(ax+b\right)^{2}}\)

https://www.desmos.com/calculator/ftrjwemsj1

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