8.17 摘要: 赤道和高危险多面性功能

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• Select section 摘要: 二次曲线和高要求多面函数

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  摘要: 二次曲线和高要求多面函数

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  In this chapter, we learned: 
  ::在本章中,我们学到:

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  How To Solve Quadratic Equations, $\begin{array}{r}a{x}^2+bx+c=0\end{array}$
  ::如何解析赤道等量, 轴2+bx+c=0

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  • By factoring, arrange all the terms on one side of the equation so the other side equals 0, factor the expression, set each factor equal to 0, and solve each equation.
    ::通过乘法,在方程的一边排列所有条件,使另一方等于0,乘以表达式,将每个系数设定为0,并解决每个方程。
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  • By the square root method, isolate the squared term and the constant term on opposite sides of the equation. Then take the square root of both sides, making the side with the constant term plus or minus the square root.
    ::以 平方根 法 将 平方 和 常值 分离 。 然后将 平方 根 法 分离出来 。 然后将 平方 的 平方根 分离出来 , 用 常数 加上 或 减去 平方根 使 侧 成为 常数 。
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  • By completing the square, divide all the terms by the leading coefficient when it is not equal to 1, isolate the variable terms on one side of the equation and the constant terms on the other, complete the square on the variable side, and then take the square root of both sides.   
    ::通过完成正方形,将所有条件除以不等于1时的主要系数,将方程式一边的可变条件和另一边的不变条件分离出来,完成可变方形的正方形,然后取出双方的平方根。
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  • By using the quadratic formula, identify the coefficients in the equation and then substitute those values into the formula  $\begin{array}{r}x=\frac{\text{-}b\pm \sqrt{b^2-4ac}}{2a}.\end{array}$     
    ::通过使用二次公式,确定方程中的系数,然后将这些数值替换为x=-bb2-4ac2a的公式。
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  • The discriminant, $\begin{array}{r}{b}^2-4ac\end{array}$ , indicates the types of solutions of a quadratic equation.
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    • If the discriminant equals 0, then the equation has one real solution—a double root.
      ::如果对立方程式等于 0, 那么方程式就有一个真正的解决方案—— 一个双根。
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    • If the discriminant is less than 0, then the equation has two complex solutions.
      ::如果对立方程式小于0,那么方程式有两个复杂的解决方案。
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    • If the discriminant is greater than 0, then the equation has two real solutions.
      ::如果对立方程式大于0,那么方程式有两个真正的解决方案。
    
    ::dispriminant b2 - 4ac 表示二次方程的解决方案类型。 如果 disriminant 等于 0, 那么方程式就有一个真正的解决方案 — — 一个双根。 如果 disriminant 低于 0, 那么方程式就有两个复杂的解决方案。 如果 disriminant 大于 0, 那么这个方程式就有两个真正的解决方案 。

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  How To Operate With Complex Numbers
  ::如何用复杂数字操作

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  • To simplify a radical with -1 as a factor, rewrite the radical as a product of two radicals—a positive factor and a factor of -1.
    ::为了简化激进,以-1作为因素,将激进改写为两个激进的产物——积极因素和-1因素。
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  • To add and subtract complex numbers, combine the real parts and combine the imaginary parts.
    ::要添加和减去复数,要合并实际部分,要合并假想部分。
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  • To multiply complex numbers, apply the distributive property or the FOIL technique.
    ::乘以复杂数字,应用分配财产或FOIL技术。
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  • To divide complex numbers, multiply the numerator and the denominator by the complex conjugate of the denominator.  
    ::要分隔复杂数字, 将分子和分母乘以分母的复杂组合。

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  How To Graph Quadratic Functions
  ::如何绘制二次曲线函数

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  • To graph quadratic functions either in standard form, $\begin{array}{r}y=a(x-h{)}^2+k\end{array}$ , or general form, $\begin{array}{r}y=a{x}^2+bx+c\end{array}$ , find the vertex, create a table of values with five points, plot those points, and draw a parabola.
    ::要用标准格式 y=a(x-h)2+k 或一般格式 y= ax2+bx+c 来图形二次函数,请找到顶点,创建带有五个点的数值表,绘制这些点,并绘制抛物线。

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  How To Solve a Quadratic Inequality
  ::如何解决赤道不平等问题

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  • To solve a quadratic inequality, find and arrange the terms on one side of the inequality so the other side equals 0. Then solve the quadratic equation associated with the function. This will give you the x −intercepts. Then choose from the:
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    • Graphical approach: Sketch a graph of the quadratic function, using the x −intercepts and based on whether the parabola opens upward or downward. Read the graph.
      ::图形化方法:使用 x- 界面,根据抛物线向上还是向下打开,绘制二次函数图。读取图表。
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    • Algebraic approach: Pick test points in each interval created by the x −intercepts. Determine if the test points satisfy the inequality.
      ::代数法:在 x- intercuts 创造的每个间隔中选择测试点。确定测试点是否满足了不平等。
    
    ::为解决二次不平等,在二次不平等的一方找到和安排条件,使另一方等于0。然后解决与函数相关的二次方程。这将为您提供 x- intercuts 。然后从以下选项中选择: 图形化方法: 使用 x- intercuts 绘制一个二次函数的图形, 使用 x- intercuts 并依据抛光波拉向上或向下打开。 读下图。 代数法: 在x- intercuts 创造的每个间隔中选择测试点。 确定测试点是否满足了不平等 。

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  How To Analyze a Quadratic Model
  ::如何分析二次曲线模型

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  • To maximize or minimize a quadratic model, find the vertex. If the parabola opens upward, the vertex represents a minimum. If the parabola opens downward, the vertex represents a maximum.
    ::要最大化或最小化二次模型,请找到顶点。如果抛物线向上打开,顶点代表最小值。如果抛物线向下打开,顶点代表最大值。

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  How To Find the Real and Complex Zeros of a Polynomial Function
  ::如何找到一个多面函数的真实和复杂的零点

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  • To find the real zeros of a polynomial function, we can start with the Rational Root Theorem. The possible rational roots are the combinations of the factors of the constant term in the numerator and the factor of the leading coefficient in the denominator.
    ::要找到多元函数的实际零, 我们可以从理性根理论开始。 可能的理性根源是分子中常数因子的组合和分母中主要系数的组合。
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  • We can reduce our search using the Bounds Theorem or Descartes's Rule of Signs.
    ::我们可以使用Bounds Theorem 或Descartes的标志规则 减少我们的搜索。
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  • We can factor polynomials over the complex numbers as linear factors.
    ::我们可以将复杂数字的多元系数作为线性系数。
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  • The Fundamental Theorem of Algebra guarantees a complex root.
    ::代数的基本理论保证了一个复杂的根。
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  • If the coefficients of a polynomial are real numbers and there is a complex root, then its complex conjugate is also a root.
    ::如果多元数值的系数是真实数字,而且有复杂的根,那么其复杂的共产物也是根。

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  How To Sketch the Graph of a Polynomial Function
  ::如何绘制多面函数的图形

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  • Polynomial function graphs are smooth and continuous.
    ::多面函数图形是光滑和连续的。
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  • Power functions are one-term polynomial functions and help us recognize the end behavior of our function, or what happens for large $\begin{array}{r}|x|\end{array}$ .
    ::权力功能是一线多功能, 帮助我们识别我们功能的最终行为, 或大型的 x会发生什么 。
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  • The multiplicity of the zero is the power of the factor that yielded that zero. This determines whether the graph will cross the x -axis or touch the x -axis at that zero.
    ::零的倍数是产生零的系数的功率。这决定图形是跨过x轴,还是触碰零的X轴。
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  • Using the end behavior and the zeros and their multiplicities, we can sketch a graph of a polynomial function.
    ::利用结束行为和零及其多重特性, 我们可以绘制一个多面函数的图形。

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  Looking Back, Looking Forward
  ::回顾,展望未来

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  In this chapter, we covered polynomial functions, specifically quadratic functions. These functions had many applications, including population growth, bridge design, and the stars we see in the sky.
  ::在本章中,我们覆盖了多元函数,特别是二次函数。这些功能有许多应用,包括人口增长、桥梁设计和我们在天空中看到的恒星。

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  A polynomial involves multiplication, addition, or subtraction. The missing operation is division. In the next chapter, we discuss functions that are one polynomial divided by another polynomial, what we call rational functions. 
  ::多式函数涉及乘法、加法或减法。 缺少的操作是分割。 在下一章中, 我们讨论的函数是一个多式函数除以另一个多式函数, 我们称之为理性函数 。

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  Chapter Review
  ::回顾章次审查