Identify when quadratic or exponential models are appropriate for data. | Two-variable data: Models and scatterplots | SAT Math

核心知识

Linear models fit straight patterns. If the data curves like a U (or inverted U), think quadratic. If it grows by a constant percentage or levels off, think exponential.

深入理解

Not every scatterplot follows a line. The SAT expects you to recognize which model shape matches the data.

Linear — Constant rate of change. The points follow a straight path. Equation form: $𝑦 = 𝑚 𝑥 + 𝑏$ .

Quadratic — The data curves, forming a parabola. It might go down then up (U-shape) or up then down (inverted U). Think projectile paths or area problems. Equation form: $𝑦 = 𝑎 𝑥 2 + 𝑏 𝑥 + 𝑐$ .

Exponential — The data grows (or decays) by a constant factor. Early on the change is small, then it accelerates dramatically — or it starts steep and levels off toward a horizontal asymptote. Equation form: $𝑦 = 𝑎 \cdot 𝑏 𝑥$ .

The quickest test: look at how the gaps between consecutive $𝑦$ -values behave. Constant gaps → linear. Gaps that grow (or shrink) at an increasing rate → exponential. Gaps that first increase then decrease (or vice versa) → quadratic.

On the SAT, you'll typically see a scatterplot or a table and be asked which equation type best fits. Focus on the overall shape, not individual points.

分步讲解

Look at the scatterplot's shape. Straight line → linear. U-shape or arch → quadratic. Rapid growth or decay that levels off → exponential.
If you have a table, check the differences between consecutive y-values. Constant differences → linear. Constant ratios → exponential.
For quadratic data, check whether the y-values decrease then increase (or increase then decrease), suggesting a vertex.
Match the shape to the equation form.

常见误解

Forcing a linear model onto curved data because the general direction is upward. Direction alone doesn't determine the model — shape matters.
Confusing exponential growth with quadratic growth. Exponential eventually outpaces quadratic.
Thinking quadratic data must be symmetric. Real-world quadratic patterns often show only one side of the parabola within the data range.

题目

示例解析

A scientist recorded the population of a bacterial colony every hour. At hour 0, there were 100 bacteria. At hour 1, there were 200. At hour 2, there were 400. At hour 3, there were 800. Which type of model best fits these data?

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