비선형 함수, 변환 및 성장
도입됨| English | 의미 |
|---|---|
| vertex form/ˈvɜːteks fɔːm/ | 꼭짓점 형식 |
| growth factor/ɡrəʊθ ˈfæktə/ | 성장 계수 |
A decision before an answer
- A population increasing by twenty percent each hour adds different numbers each hour, because the base keeps changing.
- Your goal: Interpret zeros, vertices and parameters from useful function forms.
Read the relationship
- Vertex form a(x-h)²+k identifies the vertex (h,k) and opening direction; factored form identifies zeros. The same function can have both forms. In context, determine which features are meaningful: a maximum height occurs at the vertex of a downward parabola, while a zero can represent returning to ground level.
- Distinguish additive linear change from multiplicative exponential change.
In 200(0.8)^t, the decay rate per unit is:
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1-0.8=0.2.
Use the defining rule
- For y=A b^t, A is the value at t=0 and b is the factor per unit time. A growth rate r gives b=1+r; decay gives b=1-r. Equal differences indicate a linear model, while equal ratios indicate an exponential model for equal input steps. Neither model should be extrapolated beyond a justified domain without a contextual reason.
- Transform a graph while tracking its input and output.
The vertex of (x+3)²-7 is:
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x+3=x-(-3), so h=-3 and k=-7.
Check the conditions
- An output shift f(x)+k moves every point vertically by k. An input shift f(x-h) moves the graph right by h: the same old input is reached when the new x is larger. A multiplier outside scales output; a negative outside reflects across the x-axis. Determine the transformation algebraically instead of memorising a direction without checking a point.
- Transform a graph while tracking its input and output.
h(t)=-4(t-3)²+36 has a maximum 36 at t=3, with zeros t=0 and 6. N(t)=80(1.25)^t starts at 80 and grows 25% per unit; N(2)=125. A quantity doubling every three hours is 80·2^(t/3), so after six hours it is 320. If f(x)=x², f(x-2)+5 has vertex (2,5).
100 grows by 10% twice. The result is ____.
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100·1.1²=121.
Apply the task format
- For an exponential factor given over several time units, distinguish that interval from one unit. Doubling every three hours means 2^(t/3), not 2^t. For a quadratic, use the requested representation to read an extremum or solve for an input. Keep initial values, growth factors and growth percentages separate.
- Transform a graph while tracking its input and output.
A 1.25 factor means 25% growth, not 125% growth; f(x-2) shifts right, and a repeated percentage uses the updated amount.
다음 중 이 사례에 부합하는 답안은 무엇입니까?
유용한 함수 형태로부터 영점, 꼭짓점 및 매개변수 해석
Equal additive differences prove exponential growth.
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Constant differences indicate a linear relationship; equal ratios support exponential growth.
Keep the distinctions
- growth factor 增长因子 — The multiplier applied in one stated interval.
- vertex form 顶点式 — A quadratic representation a(x-h)²+k exposing its vertex.
- Interpret zeros, vertices and parameters from useful function forms.
- Distinguish additive linear change from multiplicative exponential change.
- Transform a graph while tracking its input and output.
Match each term with its precise meaning in this lesson.
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Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
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The order follows the stated process; check each stage before the next.