第一章 图像（Graphs）

本章的主要内容包括坐标系、距离公式、中点公式、二元方程的图形表示、直线、圆等内容。此部分用于复习坐标系和方程图像，是最基本的数学知识，但与人工智能相关的数学关系不大，可以跳过或快速略读。但是为了保证知识的完整性和连贯性，在本系列课程中，仍然保留了本章。

1.1 距离和中点公式(The Distance and Midpoint formulas)

直角坐标系（Rectangular Coordinates）

上图为一个直角坐标系图，其中水平的线为 $x$ 轴($x$-axis)，垂直的线为 $y$ 轴($y$-axis)，两条线的交点为原点 $O$(origin $O$)。

• 对于$x$轴，原点左侧为负数，原点右侧为正数
• 对于$y$轴，原点下部为负数，原点上部为正数

直角坐标系也叫笛卡尔坐标系(Cartesian coordinate system)。

坐标系上的一个点 $P$ 由一个有序对偶（ordered pair）$(x,y)$ 表示，$x,y$均为实数（real numbers）。

一个坐标系被 $x$轴和$y$轴分为四个部分，这四个部分称为象限（Quadrants）。

• 第一象限：$x>0,y>0$
• 第二象限：$x<0,y>0$
• 第三象限：$x<0,y<0$
• 第四象限：$x>0,y<0$

距离公式（Distance Formula）

THEOREM 距离公式（Distance Formula）

两点$P_1 = (x_1,y_1), P_2 = (x_2,y_2)$之间的距离表示为 $d(P_1,P_2)$。则

$$d(P_1,P_2) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

例： 计算$(-4,5),(3,2)$两点之间的距离

Solution

$$\begin{array}{ll} d=\sqrt{[3-(-4)]^2 + (2-5)^2} &= \sqrt{7^2+(-3)^2} \\ &=\sqrt{49+9} \\ &=\sqrt{58} \\ &\approx 7.62 \end{array}$$

中点公式（Midpoint Formula）

中点公式用于求一个 线段(line segment) 的中点。

THEOREM 中点公式（Midpoint Formula）

一条从$P_1 = (x_1,y_1)$到$P_2 = (x_2,y_2)$的线段的中点为$M=(x,y)$。则

$$M = (x,y) = \Big(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\Big)$$

例： 找出线段$P_1={-5,5}$ 到 $P_2=(3,1)$的中点。

Solution

$$\begin{array}{l} \displaystyle x = \frac{x_1+x_2}{2} = \frac{-5+3}{2} = -1,\\ \\ \displaystyle y = \frac{y_1+y_2}{2} = \frac{5+1}{2} = 3 \end{array}$$

因此中点 $M=(-1,3)$。

1.2 二元方程的图像；截距；对称性

通过绘制点来绘制方程图像

Graphs of Equations in Two Variables; Intercepts; Symmetry

两个变量$x$和$y$的方程的图像由$xy$平面上满足方程的坐标点$(x,y)$的集合组成。

The graph of an equation in two variables $x$ and $y$ consists of the set of points in the $xy$-plane whose coordinates $(x, y)$ satisfy the equation.

consists of: to be formed from the people or things mentioned 由…组成

上面的定义读起来有些拗口，我们用更直白一点的表述来解释。

DEFINITION

一个由变量$x$和$y$构成的二元方程（注：这里没有强调是几次，即定义适用于二元$n$次方程）在$xy$平面上的图像，由能满足此方程（即带入后，使方程的等号两边相等）的坐标点$(x, y)$的集合组成。

例： 对于方程 $x^2 + y^2 = 5$的所有坐标点为 $\{(x,y) | x^2 + y^2 = 5\}$。满足此方程的点有 $(1,2),(-1,2),(1,2)...$

判断一个点是否能够满足方程，只需将$x$和$y$带入方程，如果等号两边相等，即为满足。

例： 画出方程 $y = 2x +5$的图像。

Solution

图像的截距

图像与坐标轴相交或相切的点称为图的截距。

The points, if any, at which a graph crosses or touches the coordinate axes are called the intercepts of the graph.

与$x$轴相交或相切的点的$x$坐标，称为 $x$-截距。与$y$轴相交或相切的点的$y$坐标，称为 $y$-截距。

The $x$-coordinate of a point at which the graph crosses or touches the $x$-axis is an $x$-intercept, and the $y$-coordinate of a point at which the graph crosses or touches the $y$-axis is a $y$-intercept.

如：点$(3,0)$是一个截距，数字$3$就是 $x$-截距

例：找出一个方程的截距

找出方程$y=x^2-4$的 $x$-截距 和 $y$-截距。

Solution

解：要找出 $x$-截距，则令 $y=0$，得到以下方程

$$\begin{array}{rl} x^2-4 &=0 \\ (x+2)(x-2) &=0 \\ x+2=0 \quad \text{or} \quad x-2 &=0 \\ x = -2 \qquad \text{or} \quad x&=2 \end{array}$$

因此方程的 $x$-截距为 $-2$ 或 $2$。

要找出 $y$-截距，则令 $x=0$，得到以下方程

$$\begin{array}{rl} y &=x^2 -4 \\ &=0^2 -4 = -4 \\ \end{array}$$

因此方程的 $y$-截距为 $-4$。

方程关于原点、$x$轴、$y$轴的对称性(Symmetry)

DEFINITION 关于x轴对称

如果对于图像上的每一个点$(x,y)$，都有 $(x,-y)$也位于图像上，则称图像关于$x$轴对称。

英文定义原文

DEFINITION Symmetry with Respect to the x-Axis

A graph is symmetric with respect to the $x$-axis if, for every point $(x,y)$ on the graph, the point $(x,-y)$ is also on the graph.

Example: $(3,2)$关于 $x$轴对称的点为 $(3,-2)$

关于$x$轴对称，相当于上下翻转，因此点在$x$轴的位置不变，在$y$轴的位置变到另一边。

DEFINITION Symmetry with Respect to the y-Axis

A graph is symmetric with respect to the $y$-axis if, for every point $(x, y)$ on the graph, the point $(-x,y)$ is also on the graph.

DEFINITION Symmetry with Respect to the Origin

A graph is symmetric with respect to the origin if, for every point $(x, y)$
on the graph, the point $(-x, -y)$ is also on the graph.

Tests for Symmetry

To test the graph of an equation for symmetry with respect to the

• $x$-Axis Replace $y$ by $-y$ in the equation and simplify. If an equivalent equation results, the graph of the equation is symmetric with respect to the $x$-axis.
• $y$-Axis Replace $x$ by $-x$ in the equation and simplify. If an equivalent equation results, the graph of the equation is symmetric with respect to the $y$-axis.
• Origin Replace $x$ by $-x$ and $y$ by $-y$ in the equation and simplify. If an equivalent equation results, the graph of the equation is symmetric with respect to the origin.

Example: Test $4x^2 + 9y^2 = 36$ for symmetry.

Solution

• $x$-Axis: To test for symmetry with respect to the $x$-axis, replace $y$ by $-y$. Since $4x^2 + 9(-y)^2 = 36$ is equivalent to $4x^2 + 9y^2 = 36$, the graph of the equation is symmetric with respect to the x-axis.
• $y$-Axis: To test for symmetry with respect to the $y$-axis, replace $x$ by $-x$. Since $4(-x)^2 + 9y^2 = 36$ is equivalent to $4x^2 + 9y^2 = 36$, the graph of the equation is symmetric with respect to the y-axis.
• Origin: To test for symmetry with respect to the origin, replace $x$ by $-x$ and $y$ by $-y$. Since $4(-x)^2 + 9(-y)^2 = 36$ is equivalent to $4x^2 + 9y^2 = 36$, the graph of the equation is symmetric with respect to the origin.

1.3 Lines

Calculate and interpret the Slope of a line

Slope：斜率

是描述与度量该线“方向”和“陡度”的数字，斜率也用来计算斜坡的“斜度”（倾斜程度）。一直线的斜率在其上任一点皆相等。

DEFINITION Slope

Let $P = (x_1, y_1)$ and $Q = (x_2, y_2)$ be two distinct points. If $x_1 \ne x_2$, the slope m of the nonvertical line $L$ containing $P$ and $Q$ is defined by the formula

$$\boxed{\qquad\qquad m=\frac{y_2-y_1}{x_2-x_1} \qquad x_1 \ne x_2 \qquad\qquad}$$

If $x_1 = x_2$, then $L$ is a vertical line and the slope $m$ of $L$ is undefined (since this results in division by 0).

斜率的角度公式：$m=\dfrac{\Delta x}{\Delta y} = \tan{\theta}$，其中$\theta$为直线与$x$-轴的夹角。

$$m=\frac{y_2-y_1}{x_2-x_1} = \frac{\text{Rise}}{\text{Run}} \quad \text{or as } \quad m=\frac{y_2-y_1}{x_2-x_1} = \frac{\text{Change in $y$}}{\text{Change in $x$}}=\dfrac{\Delta x}{\Delta y}$$

That is, the slope m of a nonvertical line measures the amount $y$ changes when $x$ changes from $x_1$ to $x_2$. The expression $\dfrac{\Delta x}{\Delta y}$ is called the average rate of change of $y$ with respect to $x$.

• Since any two distinct points can be used to compute the slope of a line, the average rate of change of a line is always the same number.
• The slope of a line may be computed from $P = (x_1, y_1)$ to $Q = (x_2, y_2)$ or from $Q$ to $P$ because

$$\frac{y_2-y_1}{x_2-x_1} = \frac{y_1-y_2}{x_1-x_2}$$

Find the Equation of a Vertical Line

THEOREM Equation of a Vertical Line

A vertical line is given by an equation of the form

$$x=a$$

where $a$ is the $x$-intercept.

Use the Point–Slope Form of a Line; Identify Horizontal Lines

THEOREM Point-Slope From of an Equation of a Line

An equation of a nonvertical line with slope m that contains the point $(x_1, y_1)$ is

$$y-y_1 = m(x - x_1)$$

Point-Slope From：点斜式。

通过方程的一个点和方程的斜率，可以还原出方程。

Example： Find the point-slope form of an equation of the line with slope $4$, containing the point $(1, 2)$.

Solution

An equation of the line with slope $4$ that contains the point $(1, 2)$ can be found by using the point-slope form with $m = 4, x_1 = 1\text{, and }y_1 = 2$.

$$\begin{array}{rcl} y - y_1 & = &m(x-x_1) \\ y-2 & = & 4(x-1) \qquad \color{blue} m=4,x_1 = 1,y_1=2 \\ y & = & 4x-4+2 \\ y & = & 4x-2 \\ \end{array}$$

Finding the Equation of a Horizontal Line

THEOREM Equation of a Horizontal Line

A horizontal line is given by an equation of the form

$$y=b$$

where $b$ is the $y$-intercept.

Use the Slope-Intercept Form of a Line

THEOREM Slope–Intercept Form of an Equation of a Line

An equation of a line with slope $m$ and $y$-intercept $b$ is

$$y = mx+b$$

Slope–Intercept Form: 斜截式

$$\begin{array}{lll} y=&5x+&2 \\ &\color{blue}\uarr&\color{blue}\uarr\\ &\color{blue}\text{slope} & \color{blue}y\text{-intercept} \end{array}$$

Graph Lines Written in General Form Using Intercepts

DEFINITION General Form

The equation of a line is in general form* when it is written as

$$Ax+By=C$$

where $A, B$, and $C$ are real numbers and $A$ and $B$ are not both $0$.