Course: 10.2 截取表中的赤道函数图

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截取表单中的二次函数图

Graphs of Quadratic Functions in Intercept Form
::截取表单中的二次函数图

Now it’s time to learn how to graph a parabola without having to use a table with a large number of points.
::现在该是学习如何用图解来绘制抛物线的时候了,

Let’s look at the graph of $\begin{aligned} y = x^{2} - 6 x + 8 \end{aligned}$ .
::让我们看看y=x2-6x+8的图表。

There are several things we can notice:
::有几件事我们可以注意到:

The parabola crosses the $\begin{aligned} x - \end{aligned}$ axis at two points: $\begin{aligned} x = 2 \end{aligned}$ and $\begin{aligned} x = 4 \end{aligned}$ . These points are called the $\begin{aligned} x - \end{aligned}$ intercepts of the parabola.
::抛物线在两个点( x=2 和 x=4) 跨越 x - 轴。这些点被称为 parbola 的 x - 截取点。

The lowest point of the parabola occurs at (3, -1).
- This point is called the vertex of the parabola.
  ::此点被称为抛物线的顶点。
- The vertex is the lowest point in any parabola that turns upward, or the highest point in any parabola that turns downward.
  ::顶点是任何向上翻转的抛物线中的最低点,或任何向下翻转的抛物线中的最高点。
- The vertex is exactly halfway between the two $\begin{aligned} x - \end{aligned}$ intercepts. This will always be the case, and you can find the vertex using that property .
  ::顶点恰好介于两x- 界面之间的一半。情况总是如此, 您可以使用该属性找到顶点。
::parbola 的最低点出现在 (3, - 1) 。这个点被称为 parbola 的顶点。顶点是任何 parbola 向上转的顶点, 或任何 parbola 向下转的顶点。顶点恰好在 2x - intercuts 之间的一半。这将总是这样, 您可以使用该属性找到顶点。

The parabola is symmetric. If you draw a vertical line through the vertex, you see that the two halves of the parabola are mirror images of each other. This vertical line is called the line of .
::parbola 是对称的。如果您在顶端画一条垂直线, 您可以看到 parbola 的两半是彼此的镜像。此垂直线被称为。

We said that the general form of a quadratic function is $\begin{aligned} y = a x^{2} + b x + c \end{aligned}$ . When we can factor a quadratic expression , we can rewrite the function in intercept form :
::我们说四方函数的一般形式是 y= ax2+bx+c。当我们可以将二次表达式乘以时, 我们可以以截取形式重写此函数 :

\begin{aligned} y = a (x - m) (x - n) \end{aligned}

::y=a(x-m(x-n))

This form is very useful because it makes it easy for us to find the $\begin{aligned} x - \end{aligned}$ intercepts and the vertex of the parabola. The $\begin{aligned} x - \end{aligned}$ intercepts are the values of $\begin{aligned} x \end{aligned}$ where the graph crosses the $\begin{aligned} x - \end{aligned}$ axis; in other words, they are the values of $\begin{aligned} x \end{aligned}$ when $\begin{aligned} y = 0 \end{aligned}$ . To find the $\begin{aligned} x - \end{aligned}$ intercepts from the quadratic function, we set $\begin{aligned} y = 0 \end{aligned}$ and solve:
::此窗体非常有用, 因为它使我们很容易找到 parbola 的 x- inter 和 parbola 的顶点。 x- inter 是指图形横跨 x - 轴的 x 值; 换句话说, 它们是 y= 0 的 x 值。要从等阶函数中找到 x- inter , 我们设置 y=0 并解析 :

\begin{aligned} 0 = a (x - m) (x - n) \end{aligned}

::0=a(x-m)(x-n)

Since the equation is already factored, we use the zero-product property to set each factor equal to zero and solve the individual linear equations:
::由于方程已经计算在内,我们使用零产品属性将每个系数设定为零,并解决单个线性方程:

\begin{aligned} x - m = 0 & x - n & = 0 \\ or \\ x = m & x & = n \end{aligned}

::x- m=0x- n=0orx=mx=n

So the $\begin{aligned} x - \end{aligned}$ intercepts are at points $\begin{aligned} (m, 0) \end{aligned}$ and $\begin{aligned} (n, 0) \end{aligned}$ .
::因此,x-截取点是点(m,0)和点(n,0)。

Once we find the $\begin{aligned} x - \end{aligned}$ intercepts, it’s simple to find the vertex. The $\begin{aligned} x - \end{aligned}$ value of the vertex is halfway between the two $\begin{aligned} x - \end{aligned}$ intercepts, so we can find it by taking the average of the two values: $\begin{aligned} \frac{m + n}{2} \end{aligned}$ . Then we can find the $\begin{aligned} y - \end{aligned}$ value by plugging the value of $\begin{aligned} x \end{aligned}$ back into the equation of the function.
::一旦我们找到 x - 界面, 找到顶点很简单。顶点的X - 值介于两个 X - 界面之间, 这样我们就可以通过使用 m+n2 这两个值的平均值来找到它。这样我们就可以通过将 x 的值插入函数的方程来找到 y - 值。

Finding the $\begin{aligned} x - \end{aligned}$ Intercepts and the Vertex
::查找 x - 截取器和顶点

Find the $\begin{aligned} x - \end{aligned}$ intercepts and the vertex of the following quadratic functions:
::查找以下二次函数的 x - intertics 和顶点 :

a) $\begin{aligned} y = x^{2} - 8 x + 15 \end{aligned}$
::a) y=x2-8x+15

$\begin{aligned} y = x^{2} - 8 x + 15 \end{aligned}$
::y=x2 - 8x+15 y=x2 - 8x+15

Write the quadratic function in intercept form by factoring the right hand side of the equation. Remember, to factor we need two numbers whose product is 15 and whose sum is –8. These numbers are –5 and –3.
::以截取形式写入二次函数,通过对等方程右侧进行计分。记住,我们要考虑的是,我们需要两个数字,其产品为15,其总和为8。这些数字是-5和-3。

The function in intercept form is $\begin{aligned} y = (x - 5) (x - 3) \end{aligned}$
::截取形式的函数为 Y= (x- 5) (x- 3) 。

We find the $\begin{aligned} x - \end{aligned}$ intercepts by setting $\begin{aligned} y = 0 \end{aligned}$ .
::我们通过设置 y=0 来找到 X - 拦截。

We have:
::我们已:

\begin{aligned} 0 = (x - 5) (x - 3) \\ x - 5 = 0 & x - 3 = 0 \\ or \\ x = 5 & x = 3 \end{aligned}

::0=(x-5)(x-3)(x-3)x-5=0x-3=0orx=5x=3)

So the $\begin{aligned} x - \end{aligned}$ intercepts are (5, 0) and (3, 0).
::因此,X-拦截是(5,0)和(3,0)。

The vertex is halfway between the two $\begin{aligned} x - \end{aligned}$ intercepts. We find the $\begin{aligned} x - \end{aligned}$ value by taking the average of the two $\begin{aligned} x - \end{aligned}$ intercepts: $\begin{aligned} x = \frac{5 + 3}{2} = 4 \end{aligned}$
::顶点介于两个 X - 截面之间的中间点。我们通过使用两个 X - 截面的平均值来发现 x - 值: x=5+32=4

We find the $\begin{aligned} y - \end{aligned}$ value by plugging the $\begin{aligned} x - \end{aligned}$ value we just found into the original equation:
::我们通过连接我们刚发现的 X - 值来找到y - 值。。

\begin{aligned} y = x^{2} - 8 x + 15 \Rightarrow y = 4^{2} - 8 (4) + 15 = 16 - 32 + 15 = - 1 \end{aligned}

::y=x2-8x+15y=42-8(4)+15=16-32+151

So the vertex is (4, -1).
::顶部是( 4, - 1 ) 。

b) $\begin{aligned} y = 3 x^{2} + 6 x - 24 \end{aligned}$
::b) y=3x2+6x-24

$\begin{aligned} y = 3 x^{2} + 6 x - 24 \end{aligned}$
::y= 3x2+6x- 24

Re-write the function in intercept form.
::以截取形式重写此函数。

Factor the common term of 3 first: $\begin{aligned} y = 3 (x^{2} + 2 x - 8) \end{aligned}$
::3 先是 y= 3 (x2+2x-8)

Then factor completely: $\begin{aligned} y = 3 (x + 4) (x - 2) \end{aligned}$
::然后完全因数: y=3(x+4)(x-2)

Set $\begin{aligned} y = 0 \end{aligned}$ and solve:
::设置 Y=0 并解析 :

\begin{aligned} x + 4 = 0 & x - 2 = 0 \\ 0 = 3 (x + 4) (x - 2) \Rightarrow & or \\ x = - 4 & x = 2 \end{aligned}

::x+4=0x-2=00=3(x+4)(x-2)orx=4x=2

The $\begin{aligned} x - \end{aligned}$ intercepts are (-4, 0) and (2, 0).
::x- 拦截为( 4, 0) 和 (2, 0) 。

For the vertex,
::对于顶部,

$\begin{aligned} x = \frac{- 4 + 2}{2} = - 1 \end{aligned}$ and $\begin{aligned} y = 3 (- 1)^{2} + 6 (- 1) - 24 = 3 - 6 - 24 = - 27 \end{aligned}$
::x4+221和y=3(-1)2+6(-1)-24=3-6-2427

The vertex is: (-1, -27)
::顶点是-1, -27)

Knowing the vertex and $\begin{aligned} x - \end{aligned}$ intercepts is a useful first step toward being able to graph quadratic functions more easily. Knowing the vertex tells us where the middle of the parabola is. When making a table of values, we can make sure to pick the vertex as a point in the table. Then we choose just a few smaller and larger values of $\begin{aligned} x \end{aligned}$ . In this way, we get an accurate graph of the quadratic function without having to have too many points in our table.
::了解顶点和 x - intercents 是向更方便地绘制二次函数图的第一步。了解顶点可以告诉我们抛物线中间的位置。在绘制一个数值表时, 我们可以确定选择顶点作为表格中的一个点。然后我们只选择几个较小和更大的 x 值。这样, 我们就可以获得一个精确的二次函数图, 而不必在表格中设置过多的点。

Graphing Functions
::图图函数

Find the $\begin{aligned} x - \end{aligned}$ intercepts and vertex. Use these points to create a table of values and graph each function.
::查找 x- 界面和顶点。使用这些点来创建数值表格, 并绘制每个函数的图形。

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

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

Let’s find the $\begin{aligned} x - \end{aligned}$ intercepts and the vertex:
::让我们找到 X - 界面和顶端 :

Factor the right-hand side of the function to put the equation in intercept form:
::函数的右侧乘以函数的右侧,将方程式以截取形式放入截取表单 :

\begin{aligned} y = (x - 2) (x + 2) \end{aligned}

::y=(x- 2)(x+2)

Set $\begin{aligned} y = 0 \end{aligned}$ and solve:
::设置 Y=0 并解析 :

\begin{aligned} 0 = (x - 2) (x + 2) \\ x - 2 = 0 & x + 2 = 0 \\ or \\ x = 2 & x = - 2 \end{aligned}

::0=(x- 2)(x+2)x-2=0x+2=0orx=2x=2x2

The $\begin{aligned} x - \end{aligned}$ intercepts are (2, 0) and (-2, 0).
::x- 拦截为(2,0)和(-2,0)。

Find the vertex:
::查找顶点 :

\begin{aligned} x = \frac{2 - 2}{2} = 0 y = (0)^{2} - 4 = - 4 \end{aligned}

::x=2-22=0y=(0)2-44

The vertex is (0, -4).
::顶点是( 0, - 4 ) 。

Make a table of values using the vertex as the middle point. Pick a few values of $\begin{aligned} x \end{aligned}$ smaller and larger than $\begin{aligned} x = 0 \end{aligned}$ . Include the $\begin{aligned} x - \end{aligned}$ intercepts in the table.
::以顶点作为中点, 绘制数值表。选择小于或大于 x=0 的数个数值。表格中包含 x- interviews 。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y = x^{2} - 4 \end{aligned}$
$\begin{aligned} - 3 \end{aligned}$	$\begin{aligned} y = (- 3)^{2} - 4 = 5 \end{aligned}$
–2	$\begin{aligned} y = (- 2)^{2} - 4 = 0 \end{aligned}$	$\begin{aligned} x - \end{aligned}$ intercept
–1	$\begin{aligned} y = (- 1)^{2} - 4 = - 3 \end{aligned}$
0	$\begin{aligned} y = (0)^{2} - 4 = - 4 \end{aligned}$	vertex
1	$\begin{aligned} y = (1)^{2} - 4 = - 3 \end{aligned}$
2	$\begin{aligned} y = (2)^{2} - 4 = 0 \end{aligned}$	$\begin{aligned} x - \end{aligned}$ intercept
3	$\begin{aligned} y = (3)^{2} - 4 = 5 \end{aligned}$

Then plot the graph:
::然后绘制图表:

b) $\begin{aligned} y = - x^{2} + 14 x - 48 \end{aligned}$
:b) yx2+14x-48

$\begin{aligned} y = - x^{2} + 14 x - 48 \end{aligned}$
::yx2+14x- 48

Let’s find the $\begin{aligned} x - \end{aligned}$ intercepts and the vertex:
::让我们找到 X - 界面和顶端 :

Factor the right-hand-side of the function to put the equation in intercept form:
::将方程式以截取形式置于函数右侧的系数 :

\begin{aligned} y = - (x^{2} - 14 x + 48) = - (x - 6) (x - 8) \end{aligned}

::y(x2- 14x+48) (x-6)(x-8)

Set $\begin{aligned} y = 0 \end{aligned}$ and solve:
::设置 Y=0 并解析 :

\begin{aligned} 0 = - (x - 6) (x - 8) \\ x - 6 = 0 & x - 8 = 0 \\ or \\ x = 6 & x = 8 \end{aligned}

::0(x- 6)(x-8)x- 6=0x-8=0orx=6x=8

The $\begin{aligned} x - \end{aligned}$ intercepts are (6, 0) and (8, 0).
::x- 拦截是(6,0)和(8,0)。

Find the vertex:
::查找顶点 :

\begin{aligned} x = \frac{6 + 8}{2} = 7 y = - (7)^{2} + 14 (7) - 48 = 1 \end{aligned}

::x=6+82=7y(7)2+14(7)-48=1

The vertex is (7, 1).
::顶点为( 7, 1 ) 。

Make a table of values using the vertex as the middle point. Pick a few values of $\begin{aligned} x \end{aligned}$ smaller and larger than $\begin{aligned} x = 7 \end{aligned}$ . Include the $\begin{aligned} x - \end{aligned}$ intercepts in the table.
::以顶点作为中点, 绘制数值表。选择小于或大于 x=7 的数个数值。在表格中包含 x- interviews 。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y = - x^{2} + 14 x - 48 \end{aligned}$
4	$\begin{aligned} y = - (4)^{2} + 14 (4) - 48 = - 8 \end{aligned}$
5	$\begin{aligned} y = - (5)^{2} + 14 (5) - 48 = - 3 \end{aligned}$
6	$\begin{aligned} y = - (6)^{2} + 14 (6) - 48 = 0 \end{aligned}$
7	$\begin{aligned} y = - (7)^{2} + 14 (7) - 48 = 1 \end{aligned}$
8	$\begin{aligned} y = - (8)^{2} + 14 (8) - 48 = 0 \end{aligned}$
9	$\begin{aligned} y = - (9)^{2} + 14 (9) - 48 = - 3 \end{aligned}$
10	$\begin{aligned} y = - (10)^{2} + 14 (10) - 48 = - 8 \end{aligned}$

Then plot the graph:
::然后绘制图表:

Applications of Quadratic Functions to Real-World Problems
::将赤道函数应用于现实世界问题

As we mentioned in a previous concept, parabolic curves are common in real-world applications. Here we will look at a few graphs that represent some examples of real-life application of quadratic functions.
::正如我们在前一个概念中所提到的,抛物线曲线在现实世界的应用中很常见。在这里,我们将查看几个图表,这些图表代表了二次函数实际应用的一些实例。

Real-World Application: Fencing
::真实世界应用程序:

Andrew has 100 feet of fence to enclose a rectangular tomato patch. What should the dimensions of the rectangle be in order for the rectangle to have the greatest possible area?
::Andrew有100英尺长的栅栏可以围成一个长方形番茄补丁。矩形的尺寸应该有多大才能让矩形有尽可能多的面积?

Drawing a picture will help us find an equation to describe this situation:
::绘制图片将有助于我们找到一个方程式来描述这种情况:

If the length of the rectangle is $\begin{aligned} x \end{aligned}$ , then the width is $\begin{aligned} 50 - x \end{aligned}$ . (The length and the width add up to 50, not 100, because two lengths and two widths together add up to 100.)
::如果矩形的长度为x,则宽度为50-x。 (长度和宽度加50,而不是100,因为两长和两个宽加100。 )

If we let $\begin{aligned} y \end{aligned}$ be the area of the triangle, then we know that the area is length $\begin{aligned} \times \end{aligned}$ width, so $\begin{aligned} y = x (50 - x) = 50 x - x^{2} \end{aligned}$ .
::如果我们让 y 是三角形的区域, 那么我们就会知道这个区域是长x宽, 所以 y=x( 50- x) = 50x- x2 。

Here’s the graph of that function, so we can see how the area of the rectangle depends on the length of the rectangle:
::以下是此函数的图示, 所以我们可以看到矩形区域如何取决于矩形的长度 :

We can see from the graph that the highest value of the area occurs when the length of the rectangle is 25. The area of the rectangle for this side length equals 625. (Notice that the width is also 25, which makes the shape a square with side length 25.)
::从图中可以看出,当矩形的长度为25时,区域的最高值即为区域的最高值。矩形的边长面积等于625。 (注意宽度为25,使形形形为正方形,侧长为25。 )

This is an example of an optimization problem. These problems show up often in the real world, and if you ever study calculus, you’ll learn how to solve them without graphs.
::这是一个优化问题的例子。这些问题经常出现在现实世界中,如果你研究微积分,你就会学会如何在没有图表的情况下解决这些问题。

Example
::示例示例示例示例

Example 1
::例1

Anne is playing golf. On the $\begin{aligned} 4^{t h} \end{aligned}$ tee, she hits a slow shot down the level fairway. The ball follows a parabolic path described by the equation $\begin{aligned} y = x - 0.04 x^{2} \end{aligned}$ , where $\begin{aligned} y \end{aligned}$ is the ball’s height in the air and $\begin{aligned} x \end{aligned}$ is the horizontal distance it has traveled from the tee. The distances are measured in feet. How far from the tee does the ball hit the ground? At what distance from the tee does the ball attain its maximum height? What is the maximum height?
::安妮正在打高尔夫。在第四炮上, 她缓慢地击落了平流道。球沿着以 y=x- 0.04x2 公式描述的抛物线路径前进, 方程式为y=x- 0.04x2, 方程式在空中的高度是y, x 是球与球之间的水平距离。距离是用脚来测量的。球打到地面的距离有多远? 球离梯子的距离是多少? 球达到最大高度是多少? 最大高度是多少?

Let’s graph the equation of the path of the ball:
::让我们绘制球路径的方程式 :

$\begin{aligned} x (1 - 0.04 x) = 0 \end{aligned}$ has solutions $\begin{aligned} x = 0 \end{aligned}$ and $\begin{aligned} x = 25 \end{aligned}$ .
::x(1 - 0.04x) =0 有溶液 x=0 和 x=25 。

From the graph, we see that the ball hits the ground 25 feet from the tee. (The other $\begin{aligned} x - \end{aligned}$ intercept, $\begin{aligned} x = 0, \end{aligned}$ tells us that the ball was also on the ground when it was on the tee!)
::从图中,我们可以看到球击中地表距地表25英尺。 (另一个X-interaction, x=0,告诉我们球在地上时也在地上! )

We can also see that the ball reaches its maximum height of about 6.25 feet when it is 12.5 feet from the tee.
::我们还可以看到,球在距离球体12.5英尺时达到最高高度约6.25英尺。

Review
::回顾

For 1-4, rewrite the following functions in intercept form. Find the $\begin{aligned} x - \end{aligned}$ intercepts and the vertex.
::1-4, 重写以下截取格式的函数。查找 x- 界面和顶点。

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

$\begin{aligned} y = - x^{2} + 10 x - 21 \end{aligned}$
::yx2+10x-21

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

$\begin{aligned} y = 3 (x + 5) (x - 2) \end{aligned}$
::y=3( x+5)( x-2)

For 5-8, the vertex of which parabola is higher?
::对于5 -8, 哪个抛物线的顶端更高?

$\begin{aligned} y = x^{2} + 4 \end{aligned}$ or $\begin{aligned} y = x^{2} + 1 \end{aligned}$
::y=x2+4 或y=x2+1

$\begin{aligned} y = - 2 x^{2} \end{aligned}$ or $\begin{aligned} y = - 2 x^{2} - 2 \end{aligned}$
::y2x2 或 y2x2-2

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

$\begin{aligned} y = 5 - 2 x^{2} \end{aligned}$ or $\begin{aligned} y = 8 - 2 x^{2} \end{aligned}$
::y= 5 - 2x2 或y= 8 - 2x2

For 9-14, graph the following functions by making a table of values. Use the vertex and $\begin{aligned} x - \end{aligned}$ intercepts to help you pick values for the table.
::对于 9-14, 请通过绘制值表来绘制以下函数。使用顶点和 x- interview 来帮助您为表格选择值。

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

$\begin{aligned} y = - x^{2} + x + 12 \end{aligned}$
::yx2+x+12 yx2+12

$\begin{aligned} y = 2 x^{2} + 10 x + 8 \end{aligned}$
::y=2x2+10x+8

$\begin{aligned} y = \frac{1}{2} x^{2} - 2 x \end{aligned}$
::y=12x2-2x

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

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

Nadia is throwing a ball to Peter. Peter does not catch the ball and it hits the ground. The graph shows the path of the ball as it flies through the air. The equation that describes the path of the ball is $\begin{aligned} y = 4 + 2 x - 0.16 x^{2} \end{aligned}$ . Here $\begin{aligned} y \end{aligned}$ is the height of the ball and $\begin{aligned} x \end{aligned}$ is the horizontal distance from Nadia. Both distances are measured in feet.
1. How far from Nadia does the ball hit the ground?
  ::球从Nadia到地面有多远?
2. At what distance $\begin{aligned} x \end{aligned}$ from Nadia, does the ball attain its maximum height?
  ::距离Nadia多远,球能达到最大高度吗?
3. What is the maximum height?
  ::最大高度是多少?
::Nadia 向 Peter 扔球。 Peter 不抓球, 球会击中地面。图表显示球在空中飞过时的路径。描述球路径的方程式是 y= 4+2x-0.16x2 。这里是球的高度, x 是与 Nadia 的水平距离。两条距离都是用脚测量的。球在距离Nadia 有多远的地方打到地面 ? 球在距离 Nadia 的哪个距离 ? 球是否达到最高高度 ? 最大高度是多少 ?

Jasreel wants to enclose a vegetable patch with 120 feet of fencing. He wants to put the vegetable against an existing wall, so he only needs fence for three of the sides. The equation for the area is given by $\begin{aligned} A = 120 x - x^{2} \end{aligned}$ . From the graph, find what dimensions of the rectangle would give him the greatest area.
::Jasreel想用120英尺长的栅栏围上一个蔬菜块,他想把蔬菜放在现有的墙上,所以他只需要三个侧面的栅栏。这个区域的方程由A=120x-x2给出。从图中,找出矩形的尺寸会给他带来最大的面积。

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Section outline

General

截取表单中的二次函数图

Graphs of Quadratic Functions in Intercept Form ::截取表单中的二次函数图

Finding the x − Intercepts and the Vertex ::查找 x - 截取器和顶点

Graphing Functions ::图图函数

Applications of Quadratic Functions to Real-World Problems ::将赤道函数应用于现实世界问题

Real-World Application: Fencing ::真实世界应用程序:

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)

Graphs of Quadratic Functions in Intercept Form
::截取表单中的二次函数图

Finding the $\begin{aligned} x - \end{aligned}$ Intercepts and the Vertex
::查找 x - 截取器和顶点

Graphing Functions
::图图函数

Applications of Quadratic Functions to Real-World Problems
::将赤道函数应用于现实世界问题

Real-World Application: Fencing
::真实世界应用程序:

Example
::示例示例示例示例

Example 1
::例1

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::回顾

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::回顾(答复)