Rates of Change in Linear and Quadratic Functions · 线性与二次函数的变化率
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| quadratic function/kwɒˈdrætɪk ˈfʌŋkʃn/ | 二次函数 | èr cì hán shù |
| linear function/ˈlɪnɪə ˈfʌŋkʃn/ | 线性函数 | xiàn xìng hán shù |
| constant/ˈkɒnstənt/ | 常数 | cháng shù |
| slope/sləʊp/ | 斜率 | xié lǜ |
| average rate of change/ˈævrɪdʒ reɪt ɒv tʃeɪndʒ/ | 平均变化率 | píng jūn biàn huà lǜ |
Two kinds of climb
- An escalator lifts you at a steady pace: every second gains the same height.
- A jet on the runway is different: each second it gains more speed than the last.
- Both are "increasing", but the way they change is completely different.
- Linear and quadratic functions are the first two members of this story.
两种"上升"
- 自动扶梯以稳定的节奏把你抬升:每一秒升高相同的高度。
- 跑道上的喷气机则不同:每一秒获得的速度都比上一秒更多。
- 两者都在"增大",但它们变化的方式完全不同。
- 线性函数和二次函数,就是这个故事的头两位成员。
A linear function: constant rate
- A linear function 线性函数 graphs as a straight line.
- Its slope 斜率 is the same everywhere, so its average rate of change 平均变化率 is the same over any interval.
- Over $[0, 2]$ or $[10, 12]$ or any other stretch — you always get the identical number.
- That single unchanging rate is what "linear" really means.
线性函数:恒定的变化率
- 线性函数(linear function)的图像是一条直线。
- 它的斜率(slope)处处相同,所以它在任何区间上的平均变化率(average rate of change)都相同。
- 在 $[0, 2]$、$[10, 12]$ 或任何其他区间上——你总能得到完全一样的数。
- 那个唯一不变的变化率,正是"线性"真正的含义。
The average rate of change of a linear function over any interval is… · 线性函数在任何区间上的平均变化率是……
A straight line has one fixed slope, so the average rate of change is the same over every interval — that is what makes it linear. · 直线只有一个固定的斜率,所以在每一个区间上的平均变化率都相同——这正是它之所以线性的原因。
A quadratic function: a changing rate
- A quadratic function 二次函数 graphs as a curved parabola.
- Because it bends, its slope is different at different places — the rate of change keeps changing.
- Near the vertex it is nearly flat; far from the vertex it is steep.
- So its average rate of change depends on which interval you pick.
二次函数:变化的变化率
- 二次函数(quadratic function)的图像是一条弯曲的抛物线。
- 因为它是弯的,不同位置的斜率不同——变化率一直在变。
- 在顶点附近它几乎是平的;远离顶点时它很陡。
- 所以它的平均变化率取决于你选的是哪个区间。
One slider turns a parabola into a straight line · 一个滑块就能把抛物线变成直线
y = ax² + bx + c
Set a = 0 to get a straight line — its slope is constant. Make a ≠ 0 and the curve bends, so now the rate of change keeps changing. · 把 a 设为 0 得到一条直线——它的斜率是常数。让 a ≠ 0,曲线就弯了,此时变化率一直在变。
For a quadratic function, the average rate of change is the same over every interval. · 对于二次函数,每个区间上的平均变化率都相同。
A parabola bends, so its slope keeps changing — different intervals give different average rates of change. · 抛物线是弯的,所以斜率一直在变——不同的区间给出不同的平均变化率。
Equal steps reveal the pattern
- Take equal steps in the input and list the outputs of $y = x^2$: $0, 1, 4, 9, 16$.
- The jumps between them — the first differences — are $1, 3, 5, 7$.
- Those jumps are not constant, but they grow by the same amount ($2$) every time.
- So a quadratic's rate of change is not fixed, yet it changes in a perfectly steady, linear way.
相等的步长揭示规律
- 在输入上取相等的步长,列出 $y = x^2$ 的输出:$0, 1, 4, 9, 16$。
- 它们之间的跳跃——也就是一阶差——是 $1, 3, 5, 7$。
- 这些跳跃不是常数,但它们每次都增大相同的量($2$)。
- 所以二次函数的变化率不是固定的,但它以一种完全稳定、线性的方式改变。
For · 支持 $y = x^2$, the first differences at $x = 1, 2, 3$ are $3, 5, 7$. By what constant amount do they grow each step? · 对于 $y = x^2$,在 $x = 1, 2, 3$ 处的一阶差是 $3, 5, 7$。它们每一步增大的那个常数量是多少?
$5 - 3 = 2$ and $7 - 5 = 2$. This constant second difference is the fingerprint of a quadratic. · $5 - 3 = 2$,$7 - 5 = 2$。这个恒定的二阶差正是二次函数的指纹。
For a quadratic, equal steps in $x$ give first differences that… · 对于二次函数,$x$ 上相等的步长给出的一阶差会……
The first differences themselves form a linear · 线性 pattern — they grow by the same amount each step. (Doubling each time would be exponential.) · 一阶差本身形成一个线性模式——每步增大相同的量。(每次翻倍则是指数增长。)
Every average rate is a secant slope
- Whichever function you have, the average rate of change over $[a, b]$ is the slope of the secant line through its endpoints.
- For a line, every secant lies right on top of the graph — same slope every time.
- For a parabola, each secant tilts differently — that is the visible sign of a changing rate.
- Reading secant slopes is how you compare how two functions change.
每个平均变化率都是割线的斜率
- 无论是哪种函数,$[a, b]$ 上的平均变化率都是穿过其端点的割线的斜率。
- 对直线来说,每条割线都正好压在图像上——每次斜率都相同。
- 对抛物线来说,每条割线倾斜的程度都不同——这就是变化率在改变的可见标志。
- 读割线的斜率,正是你比较两个函数如何变化的方法。
Put the steps for finding an average rate of change over $[a, b]$ in order. · 把求 $[a, b]$ 上平均变化率的步骤按顺序排列。
Outputs first, then their difference, then divide by the input difference — that is the secant slope. · 先求输出,再求它们的差,最后除以输入的差——这就是割线的斜率。
Select all · 所有 true statements. · 选出所有正确的说法。
A quadratic's rate is not · 不 constant — it changes steadily (its second difference is constant). The other three statements are true. · 二次函数的变化率不是常数——它稳定地改变(二阶差为常数)。其余三条都对。
Do not call a quadratic's rate "constant" just because the pattern is predictable. The rate itself changes from interval to interval; it is the change in the rate that is constant. For a straight line, that change in the rate is zero.
不要因为规律可预测,就把二次函数的变化率称为"常数"。变化率本身在一个个区间之间改变;真正恒定的,是变化率的改变量。对一条直线来说,这个改变量为零。
A dropped ball falls a distance $d = 5t^2$ metres after $t$ seconds.
- Distances at $t = 0, 1, 2, 3$: $0, 5, 20, 45$ metres.
- First differences (distance each second): $5, 15, 25$ — growing by a constant $10$.
- The ball's speed rises linearly even though the distance is quadratic.
一个落下的球在 $t$ 秒后下落的距离是 $d = 5t^2$ 米。
- $t = 0, 1, 2, 3$ 时的距离:$0, 5, 20, 45$ 米。
- 一阶差(每秒下落的距离):$5, 15, 25$——按常数 $10$ 增大。
- 尽管距离是二次的,球的速度却是线性上升的。
A linear function changes at one constant rate — every secant has the same slope. A quadratic function's rate of change is not constant: over equal steps its first differences grow by a fixed amount, so the rate changes steadily. The average rate of change over $[a, b]$ is always the secant slope.
线性函数以一个常数(constant)变化率改变——每条割线的斜率都相同。二次函数的变化率不是常数:在相等的步长上,它的一阶差按固定的量增大,所以变化率稳定地改变。$[a, b]$ 上的平均变化率永远是割线的斜率。