Apply a transformation to one variable

The explanatory variable is named for a square transformation. What should happen first?

Reuse the reciprocal-input experiment. Predict how the horizontal position changes before moving x from 2 to 6. The response stays on its original scale

What changes when x grows from 2 to 6?

Transforming one axis

A one-axis transformation changes the selected explanatory or response coordinate and keeps the other coordinate unchanged. Check the domain and relabel the changed axis before judging the transformed association

1. 1Name the variable Read whether the explanatory variable or response variable is named before changing any values \(\displaystyle x=\text{explanatory},\quad y=\text{response}\)
2. 2Transform one axis only Apply the selected operation to the named variable and leave the other variable on its original scale \(\displaystyle x'=x^{2},\quad y'=y\)
3. 3Check the domain A base-10 logarithm needs a positive input, and a reciprocal needs a non-zero input \(\displaystyle \log_{10}(100)=2,\quad\frac{1}{5}=0.2\)
4. 4Relabel the changed axis The new label must show the transformation used for the displayed coordinate \(\displaystyle x'=x^{2}\Rightarrow\text{label the axis }x^{2}\)

Data or rule

\(\displaystyle x=4,\quad t=x^2\)

Square the named explanatory value

Data or rule

\(\displaystyle (t,y)=(4,12),(16,36)\)

Calculate the slope in the transformed coordinate system

For x=4 and y=30, square the explanatory variable x only. Which line is correct?

The response can be the changed axis

General Units 3–4 permits a square, base-10 logarithm or reciprocal on either axis, one at a time. Keep x as the explanatory variable even when the response becomes y², log₁₀(y) or 1/y. An equation for transformed response does not yet predict y on its original scale

1. 1Domain before calculation A square accepts any real value but loses its sign; a logarithm needs a strictly positive value; a reciprocal excludes zero. A positive-scale shape guide has narrower assumptions than the algebraic domain
2. 2Units and precision If y is measured in cm, y² is in cm² and 1/y in cm⁻¹. Log a dimensionless ratio to a stated common reference: log₁₀(y/1 cm). Keep full precision for later fitting unless the task explicitly defines rounded displayed coordinates as its data
3. 3Reproducible tool record Name the original column, state the formula (for example Y^2, LOG10(Y) or 1/Y), fill every paired row and preserve the other column. Use an XY scatterplot with numerical spacing; relabel the changed axis. Verify a first and last entry by hand

Data or rule

\(\displaystyle (x,y)=(9,6),\quad v=y^2\)

Square the response only; report the new vertical coordinate

Which equation describes all four transformed points?

Data or rule

\(\displaystyle (x,v)=(1,4),(4,16),(9,36),(16,64)\)

Check the final consecutive slope after squaring the response

Squaring maps responses −6 and 6 to 36. What must be recorded?

Data or rule

\(\displaystyle y=0.01\,\mathrm{cm},\quad v=\log_{10}(y/(1\,\mathrm{cm}))\)

Transform the positive response using the stated reference

Data or rule

\(\displaystyle y=250,\quad v=\log_{10}(y)\)

Use base-10 logarithms and report the transformed response to three decimal places

A response value is zero. Which of these transformations is defined for that observation?

Data or rule

\(\displaystyle (x,y)=(3,4),\quad v=1/y\)

Replace the response by its reciprocal

How should logged lengths be compared?

Produce and compare the transformed plots

Use the exact observations (x,y)=(1,18),(2,9),(3,6). Make a table for t=x² and t=1/x, keeping y unchanged. Draw both scatterplots with labelled axes and useful windows. Compare consecutive slopes, select the linearising candidate, explain why the decreasing raw trend alone is insufficient, and distinguish the origin on the reciprocal line from an observed input

1. 1Write before continuing Produce your complete table, diagram or explanation independently. The next screen is a model for self-review; the complete written response is not automatically marked

Model response: Produce and compare the transformed plots

Compare your working, labelled axes, numerical checks and conclusion with this model

1. 1Square candidate The transformed pairs are (1,18),(4,9),(9,6). Slopes are −3 and −3/5, so they are not collinear. A window 0≤t≤10, 0≤y≤20 contains all points; label t=x² horizontally
2. 2Reciprocal candidate The transformed pairs are (1,18),(1/2,9),(1/3,6). Both consecutive slopes are 18. A window 0≤t≤1.1, 0≤y≤20 shows the line y=18t; label t=1/x horizontally \(\displaystyle (-9)/(-1/2)=18,\quad(-3)/(-1/6)=18\)
3. 3Conclude with limits The reciprocal candidate fits every supplied observation; a decreasing trend on its own does not establish that. All original inputs are positive. The transformed origin t=0 is approached as x grows, but no finite positive x maps to it. Three observations do not prove a practical law for all x.

Construct two response-axis transformations

Use original synthetic observations (x,y)=(1,10),(2,100),(3,1000),(4,10000), with both variables dimensionless. Formulate the question “Can a one-axis transformation make these observations linear?” Independently construct tables and scatterplots for v=log₁₀(y) and v=1/y, leaving x unchanged. Choose a tool, document its formulas and axis windows, compare every consecutive slope and write a qualified conclusion

1. 1Check the full dataset State the domains, verify the first and last transformed values, and explain why the increasing raw trend alone cannot choose between these candidates. Retain exact fractions for reciprocals
2. 2Save your work before continuing Produce the complete table, labelled plots, tool entries, numerical checks and explanation before opening the model response. Independent drawing, writing and actual calculator or spreadsheet use are not automatically marked; review them yourself or with a teacher

Model response: Construct two response-axis transformations

Compare the complete construction and reasoning with your saved work. Other justified windows or tools may also be suitable

1. 1Log-response candidate All three slopes are one. Every given positive response is a power of ten, so no rounding is needed \(\displaystyle (x,v)=(1,1),(2,2),(3,3),(4,4),\quad 0\le x\le5,\quad 0\le v\le5,\quad v=\log_{10}(y)=x\)
2. 2Reciprocal-response candidate The consecutive slopes disagree, so this candidate does not give an exact line. Unequal vertical changes must not be hidden by categorical spacing \(\displaystyle (x,v)=(1,\frac{1}{10}),(2,\frac{1}{100}),(3,\frac{1}{1000}),(4,\frac{1}{10000}),\quad s_1=-\frac{9}{100},\ s_2=-\frac{9}{1000},\ s_3=-\frac{9}{10000},\quad 0\le x\le5,\quad 0\le v\le0.11,\quad v=\frac{1}{y}\)
3. 3Conclude and verify Both transformations are defined because every response is positive. The log-response candidate linearises these four supplied exact points. It is one transformed response, not simultaneous log–log work, and it does not prove a law for unseen data. Use LOG10 on the response column, preserve x, check LOG10(10)=1 and LOG10(10000)=4 independently, and save the full table and report