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Algebra 2 Modelling with functions

Making predictions from a model

20 practice questions 0 video lessons Theory + worked examples

Making Predictions from a Model

Texas Algebra II (TEKS) • Standard 2A.8(C) • Modelling with Functions

Making Predictions from a Model is a topic in Modelling with Functions in the Texas Essential Knowledge and Skills (Algebra II, §111.40). It is aligned to Standard 2A.8(C), which requires students to predict and make decisions and critical judgments from a given set of data using linear, quadratic, and exponential models.

A model predicts by substituting an input or solving for one; interpolation inside the data is safer than extrapolation outside it.

Texas Algebra II (TEKS) › Modelling with Functions › Making Predictions from a Model  —  Standard 2A.8(C)

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Theory

A model lets you predict:

  • Substitute an \(x\)-value to predict \(y\).
  • Solve for \(x\) to predict when \(y\) reaches a value.
  • Interpolation (inside the data) is more reliable than extrapolation (outside).
Predictions far outside the data assume the pattern holds — treat them with caution.
Predicting from a model A model predicts inside the data (interpolation) more safely than outside it (extrapolation). x interpolate extrapolate
Interpolating inside vs extrapolating outside the data.
Making predictions Making predictions Making predictions substitute x → predict y solve for x → predict when interpolate: inside the data (safer) extrapolate: outside (less certain)
Ways to predict with a model.

Predict a value or a time:

\[y=f(x)\ \text{(predict } y),\qquad f(x)=y\ \text{(solve for } x)\]
substitute to predict y, or solve to find the input
State units and check the prediction is reasonable.

How to predict

  1. Identify the model.
  2. To find \(y\), substitute the \(x\)-value.
  3. To find when, set the model equal and solve.
  4. Judge whether the answer is reasonable.
Example 1 — Predict a value
A model \(y=2x+3\) fits sales. Predict \(y\) at \(x=6\).
Solution

Substitute \(x=6\).

\(y\)\(=\)\(2(6)+3\)
\(=\)\(15\)
the prediction is 15
Example 2 — Predict when
Using \(y=2x+3\), when does \(y=25\)?
Solution

Solve for \(x\).

\(25\)\(=\)\(2x+3\)
\(x\)\(=\)\(11\)
y reaches 25 at x equals 11
Example 3 — Exponential prediction
A population \(P=100(1.2)^t\). Predict \(P\) at \(t=3\).
Solution

Substitute \(t=3\).

\(P\)\(=\)\(100(1.2)^3\)
\(\approx\)\(172.8\)
about 173
Example 4 — Interpolate vs extrapolate
Why is predicting far outside the data risky?
Solution
Extrapolation assumes the pattern continues, which may fail far from the data; interpolation inside the range is safer.
extrapolation far from the data is less reliable

Common pitfalls

Extrapolation is uncertain — the pattern may not continue.
Solve, don't just substitute, to predict a time.
Check reasonableness and units.

Frequently asked questions

How do you predict a value from a model?

Substitute the input into the model.

How do you predict when something happens?

Set the model equal to the target value and solve.

What is the difference between interpolation and extrapolation?

Interpolation predicts inside the data; extrapolation predicts outside it.

Why is extrapolation risky?

The pattern may not hold far from the observed data.

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