Nonlinear functions, transformations and growth
| English | Español |
|---|---|
| vertex form/ˈvɜːteks fɔːm/ | vertex form |
| growth factor/ɡrəʊθ ˈfæktə/ | growth factor |
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:
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:
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 ____.
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.
Which answer fits this case?
Interpret zeros, vertices and parameters from useful function forms
Equal additive differences prove exponential growth.
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.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.