课程： 12.14 承认赤道功能

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选择节承认二次函数

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承认二次函数
Travis found the following equation in his math book.
::Travis在他的数学书中发现了以下方程。

$\begin{aligned} d = r t - 16 t^{2} \end{aligned}$

::d=rt - 16t2

It is an equation to calculate velocity. In fact, it is a function . Being an avid sports player, Travis was very interested in figuring out how to use the equation, but he isn’t even sure what kind of a function it is. Can you identify this function?
::这是一个计算速度的方程式。事实上,这是一个函数。 Travis作为一名热情的运动运动员,非常想了解如何使用方程式,但他甚至不确定它是什么函数。你能识别这个函数吗?

In this concept, you will learn to recognize a quadratic function as an equation in two variables with a specific form.
::在此概念中,您将学会将二次函数识别为两个具有特定形式的变数的等式。

Quadratic Functions
::二次曲线函数

A parabola is a U shaped figure whose equation is a quadratic equation . Let’s start with quadratic equations and standard form .
::抛物线是一个U形图,其方程式是二次方程。让我们从二次方程和标准形式开始。

To graph a quadratic equation, you need input values, oftentimes $\begin{aligned} x \end{aligned}$ values, to calculate corresponding $\begin{aligned} y \end{aligned}$ values. Your input values are known as the domain , while the output values are known as the range . These are also called the independent variable ( $\begin{aligned} x \end{aligned}$ ) and dependent variable ( $\begin{aligned} y \end{aligned}$ ).
::要绘制二次方程, 您需要输入值, 通常是乘以 x 值, 来计算相应的 y 值。您的输入值被称为域, 而输出值被称为区域。这些也被称为独立的变量 (x) 和依附变量。

A function is a relation that assigns exactly one value of the domain to each value of the range.
::函数是一个关系,为范围的每一值指定一个精确的域值。

So, when you say quadratic function , you are referring to any function that can be written in the form $\begin{aligned} y = a x^{2} + b x + c \end{aligned}$ , where $\begin{aligned} a, b \end{aligned}$ , and $\begin{aligned} c \end{aligned}$ are constants and $\begin{aligned} a \neq 0 \end{aligned}$ . This is standard form .
::所以,当您说二次函数时, 您指的是以 y=ax2+bx+c 的形式写出的任何函数, 其中 a, b, c 是常数和 a+++0 。这是标准格式。

Why can’t ‘ $\begin{aligned} a \end{aligned}$ ’ equal zero?
::为什么“一个”不能等于零?

If the ‘ $\begin{aligned} a \end{aligned}$ ’ value is zero, you might notice that it would make the first term $\begin{aligned} a x^{2} \end{aligned}$ disappear because anything times zero is zero. You would be left with simply $\begin{aligned} y = b x + c \end{aligned}$ . Although this is still a function, it is no longer quadratic. This is a linear function . All quadratic functions are to the 2 ^nd degree .
::如果“a”值为零,你可能会注意到,它会使第一学期的Ax2消失,因为任何值为零的值为零。你会被留下简单的y=bx+c。虽然它仍然是一个函数,但它不再是二次函数。这是一个线性函数。所有二次函数都达到二级。

Let’s take a look at some examples.
::让我们来看看一些例子。

Identify if the following equations are quadratic functions. If they are, place them in standard form and identify ‘ $\begin{aligned} a, b \end{aligned}$ and $\begin{aligned} c \end{aligned}$ ’ values.
::确定以下方程式是否为二次函数。如果为二次函数, 请按标准格式列出, 并标明 'a、 b和 c ' 值。

1. $\begin{aligned} y = x^{2} - 3 x + 5 \end{aligned}$
::1. y=x2-3x+5

Yes, it is a quadratic function.
::是的,它是一种二次函数。

The standard form is $\begin{aligned} y = x^{2} - 3 x + 5 \end{aligned}$ .
::标准表格是y=x2-3x+5。

$\begin{aligned} a = 1, b = - 3, c = 5 \end{aligned}$ .
::a=1,b3,c=5。

2. $\begin{aligned} y = - 7 x^{2} + 4 x \end{aligned}$
::2. y=7x2+4x

Yes, it is a quadratic function.
::是的,它是一种二次函数。

The standard form is $\begin{aligned} y = - 7 x^{2} + 4 x \end{aligned}$ .
::标准表格是y7x2+4x。

$\begin{aligned} a = - 7, b = 4, c = 0 \end{aligned}$ .
::a7,b=4,c=0。

3. $\begin{aligned} y - 6 = x^{2} \end{aligned}$
::3. y-6=x2

Yes, it is a quadratic function.
::是的,它是一种二次函数。

The standard form is $\begin{aligned} y = x^{2} + 6 \end{aligned}$ .
::标准表格是y=x2+6。

$\begin{aligned} a = 1, b = 0, c = 6 \end{aligned}$ .
::a=1,b=0,c=6。

4. $\begin{aligned} 3 x^{2} + y = 3 x^{2} + 4 x - 2 \end{aligned}$
::4. 3x2+y=3x2+4x-2

With this function, you have to rewrite it into standard form. Standard form would have the $\begin{aligned} y \end{aligned}$ -value on the left side of the equals and the $\begin{aligned} a, b \end{aligned}$ and $\begin{aligned} c \end{aligned}$ values on the right side. To accomplish this task, you will have to subtract $\begin{aligned} 3 x^{2} \end{aligned}$ from both sides.
::3⁄4 ̄ ̧漯B

$\begin{aligned} \begin{array}{rcl} 3 x^{2} - 3 x^{2} + y & = & 3 x^{2} - 3 x^{2} + 4 x - 2 \\ y & = & 4 x - 2 \end{array} \end{aligned}$

::3x2 - 3x2+y=3x2 - 3x2+4x-2y=4x-2

The function is not a quadratic function because if you subtract $\begin{aligned} 3 x^{2} \end{aligned}$ from both sides, your $\begin{aligned} a \end{aligned}$ value will be zero. This function is a linear function.
::函数不是一个二次函数,因为如果从两边减去 3x2,则你的数值将是零。此函数是一个线性函数。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about the function.
::早些时候,有人给了你一个有关函数的问题。

Is the function $\begin{aligned} d = r t - 16 t^{2} \end{aligned}$ a quadratic function?
::函数 d=rt - 16t2 是二次函数吗 ?

This is a quadratic function because “ $\begin{aligned} d \end{aligned}$ ” is dependent on the right side of the function. One value will impact the others. The quadratic equation will have one value in the range for each value in the domain. This will make it a quadratic function.
::这是一个二次函数, 因为“ d” 取决于函数的右侧。一个值会影响其他值。二次方程式在域内每个值的幅度中将有一个值。这将使它成为二次函数。

Example 2
::例2

Is this function a quadratic function? If so, write it in standard form.
::此函数是否为二次函数? 如果是, 请以标准格式写入。

$\begin{aligned} 3 y - 9 = 3 x^{2} \end{aligned}$

::3y- 9=3x2

First, you have to get the $\begin{aligned} y \end{aligned}$ value alone. Let’s add 9 to both sides to start.
::首先,你必须单独获得 Y 的价值。让我们从双方增加9 个开始。

$\begin{aligned} \begin{array}{rcl} 3 y - 9 & = & 3 x^{2} \\ 3 y - 9 + 9 & = & 3 x^{2} + 9 \\ 3 y & = & 3 x^{2} + 9 \end{array} \end{aligned}$

::3- 9=3x23y- 9 + 9=3x2+93y=3x2+9

Next, divide both sides by 3.
::其次,将双方除以3。

$\begin{aligned} \begin{array}{rcl} 3 y & = & 3 x^{2} + 9 \\ \frac{3 y}{3} & = & \frac{3 x^{2}}{3} + \frac{9}{3} \\ y & = & x^{2} + 3 \end{array} \end{aligned}$

::3y=3x2+93y3=3x23+93y=x2+3

The answer is that this is a quadratic function and the standard form is $\begin{aligned} y = x^{2} + 3 \end{aligned}$ .
::答案是,这是一个二次函数,标准表格是y=x2+3。

Example 3
::例3

Identify whether or not the function is a quadratic function.
::确定该函数是否为二次函数。

$\begin{aligned} y - 8 = x^{2} \end{aligned}$

::y-8=x2 y-8=x2

In standard form $\begin{aligned} y - 8 = x^{2} \end{aligned}$ becomes $\begin{aligned} y = x^{2} + 8 \end{aligned}$ and this is a quadratic function.
::在标准表y-8=x2变为y=x2+8,这是一个二次函数。

Example 4
::例4

Identify whether or not the function is a quadratic function.
::确定该函数是否为二次函数。

$\begin{aligned} y + 2 x^{2} = 2 x^{2} + 1 \end{aligned}$

::y+2x2=2x2+1 y+2x2=2x2+1

In standard form $\begin{aligned} y + 2 x^{2} = 2 x^{2} + 1 \end{aligned}$ becomes $\begin{aligned} y = 1 \end{aligned}$ and this is not a quadratic function.
::在标准格式 y+2x2=2x2+1 变为y=1, 这不是二次函数。

Example 5
::例5

Identify whether or not the function is a quadratic function.
::确定该函数是否为二次函数。

$\begin{aligned} y + 4 = 2 x^{2} \end{aligned}$

::y+4=2x2 y+4=2x2

In standard form $\begin{aligned} y + 4 = 2 x^{2} \end{aligned}$ becomes $\begin{aligned} y = 2 x^{2} - 4 \end{aligned}$ and this is a quadratic function.
::在标准表y+4=2x2变为y=2x2-4,这是一个二次函数。

Review
::回顾

Identify whether the following equations are quadratic functions. If they are, identify the ‘ $\begin{aligned} a, b \end{aligned}$ and $\begin{aligned} c \end{aligned}$ ’ values.
::确定以下方程式是否为二次函数。如果为二次函数,请标明`a、b和c ' 值。

$\begin{aligned} y = 3 x^{2} - x + 4 \end{aligned}$
::y= 3x2 - x+4 y= 3x2 - x+4

$\begin{aligned} y = 2 x^{2} + 4 \end{aligned}$
::y= 2x2+4 y= 2x2+4

$\begin{aligned} 2 y = 4 x^{2} + 4 \end{aligned}$
::2y=4x2+4

$\begin{aligned} 3 y = 6 x^{2} + 12 \end{aligned}$
::3y=6x2+12

$\begin{aligned} 4 y = 2 x^{2} - 12 \end{aligned}$
::4y=2x2-12

$\begin{aligned} 3 y - 1 = 6 x^{2} + 11 \end{aligned}$
::3- 1=6x2+11

$\begin{aligned} 2 y + 2 = 2 x^{2} + 4 \end{aligned}$
::2 y+2=2x2+4

$\begin{aligned} y + 2 x^{2} = 2 x^{2} + 4 \end{aligned}$
::y+2x2=2x2+4 y+2x2=2x2+4

$\begin{aligned} y - 2 x^{2} = 2 x^{2} + 4 \end{aligned}$
::y-2x2=2x2+4 y-2x2=2x2+4

$\begin{aligned} y = 2 x^{2} - 3 x + 4 \end{aligned}$
::y=2x2 - 3x+4

$\begin{aligned} y = 4 x - 18 + x^{2} \end{aligned}$
::y=4x-18+x2 y=4x-18+x2

$\begin{aligned} y + x^{2} - 6 = x^{2} + 4 x - 5 \end{aligned}$
::y+x2- 6=x2+4x- 5

$\begin{aligned} 3 y + 3 = 9 x^{2} - 12 x \end{aligned}$
::3y+3=9x2 - 12x

$\begin{aligned} 6 y = 3 x^{5} + 3 x^{4} + 6 x + - 18 \end{aligned}$
::6y=3x5+3x4+6x18

$\begin{aligned} 4 y + 3 x = 8 x^{2} + 3 x - 12 \end{aligned}$
::4+3x=8x2+3x-12

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

章节大纲

常规

承认二次函数

Quadratic Functions ::二次曲线函数

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Quadratic Functions
::二次曲线函数

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)