画出斜截式

阅读本章之前，我们建议您先阅读斜截式入门一章.

让我们来画$y=2x+3$y, equals, 2, x, plus, 3的图像吧.

回想一下,在斜截式方程的一般形式$y=\maroonC{m}x+\greenE{b}$y, equals, start color #ed5fa6, m, end color #ed5fa6, x, plus, start color #0d923f, b, end color #0d923f中,斜率是由$\maroonC{m}$start color #ed5fa6, m, end color #ed5fa6得出的,而$y$y截距则是由$\greenE{b}$start color #0d923f, b, end color #0d923f得出的. 因此,$y=\maroonC{2}x+\greenE{3}$y, equals, start color #ed5fa6, 2, end color #ed5fa6, x, plus, start color #0d923f, 3, end color #0d923f的斜率是$\maroonC{2}$start color #ed5fa6, 2, end color #ed5fa6而$y$y截距则为 $(0,\greenE{3})$left parenthesis, 0, comma, start color #0d923f, 3, end color #0d923f, right parenthesis.

为了画一条线,我们首先需要在这条线上的两个点.我们已经知道点$(0,\greenE{3})$left parenthesis, 0, comma, start color #0d923f, 3, end color #0d923f, right parenthesis在这条线上了.

此外,因为这条线的斜率为$\maroonC{2}$start color #ed5fa6, 2, end color #ed5fa6,我们可以知道点$(0\maroonC{+1},\greenE{3}\maroonC{+2})=(1,5)$left parenthesis, 0, start color #ed5fa6, plus, 1, end color #ed5fa6, comma, start color #0d923f, 3, end color #0d923f, start color #ed5fa6, plus, 2, end color #ed5fa6, right parenthesis, equals, left parenthesis, 1, comma, 5, right parenthesis也在这条线上.

让我们来绘制$y=\maroonC{\dfrac{2}{3}}x\greenE{+1}$y, equals, start color #ed5fa6, start fraction, 2, divided by, 3, end fraction, end color #ed5fa6, x, start color #0d923f, plus, 1, end color #0d923f的图像吧.

就和之前一样,我们可以看出这条线会经过$y$y截距$(0,\greenE{1})$left parenthesis, 0, comma, start color #0d923f, 1, end color #0d923f, right parenthesis和另一个点$\left(0\maroonC{+1},\greenE{1}\maroonC{+\dfrac{2}{3}}\right)=\left(1,1\dfrac{2}{3}\right)$left parenthesis, 0, start color #ed5fa6, plus, 1, end color #ed5fa6, comma, start color #0d923f, 1, end color #0d923f, start color #ed5fa6, plus, start fraction, 2, divided by, 3, end fraction, end color #ed5fa6, right parenthesis, equals, left parenthesis, 1, comma, 1, start fraction, 2, divided by, 3, end fraction, right parenthesis.

尽管点$\left(1,1\dfrac{2}{3}\right)$left parenthesis, 1, comma, 1, start fraction, 2, divided by, 3, end fraction, right parenthesis确实在线上,我们却不能像绘制整数坐标点那样精确地绘制分数坐标点.

我们需要一个方法去找出另一个在这条线上的整数坐标点.为此,我们可以利用一个事实,即在$\maroonC{\dfrac{2}{3}}$start color #ed5fa6, start fraction, 2, divided by, 3, end fraction, end color #ed5fa6的斜率中,使$x$x增长$\maroonC{3}$start color #ed5fa6, 3, end color #ed5fa6个单位也会使$y$y增长$\maroonC{2}$start color #ed5fa6, 2, end color #ed5fa6个单位.

这使我们得到了一个额外的点$(0\maroonC{+3},\greenE{1}\maroonC{+2})=(3,3)$left parenthesis, 0, start color #ed5fa6, plus, 3, end color #ed5fa6, comma, start color #0d923f, 1, end color #0d923f, start color #ed5fa6, plus, 2, end color #ed5fa6, right parenthesis, equals, left parenthesis, 3, comma, 3, right parenthesis.