Fit and use square, reciprocal and logarithmic models

Which coordinates should be used to find the slope of a transformed-linear model?

Modelling with transformed data

A non-linear relationship can become a straight line after the explanatory variable is transformed. Fit the line in the transformed coordinates, then transform a new x value before using the model

1. 1Transform the horizontal variable Replace x with the stated transformed value before fitting or using the line \(\displaystyle x=2,4\Rightarrow t=x^2=4,16\)
2. 2Find k from transformed coordinates Calculate the slope using the transformed horizontal coordinates and their response values \(\displaystyle k=\frac{y_2-y_1}{t_2-t_1}\)
3. 3Find c without rounding k Substitute one transformed point into the model and solve for the vertical intercept \(\displaystyle c=y_1-kt_1\)
4. 4Transform before predicting Find the new transformed input, then substitute it into y = kt + c. \(\displaystyle x=5\Rightarrow t=25\Rightarrow y=k(25)+c\)

Square the input before using the transformed model

Use y = 2t + 3 with t = 16 to find the predicted response

Original observations and transformation

\(\displaystyle (x,y)=(2,11),(4,35),\quad t=x^2\)

Transform the original observations using t=x². Which calculation gives the slope in the transformed coordinates?

Start with inverse variation

The basic reciprocal relationship has no added constant. Plot y against t=1/x: its line passes through the origin. Use positive x and positive k.

1. 1Check the product The observations (2,18) and (6,6) have the same product, so they fit one inverse-variation rule \(\displaystyle y=k/x,\quad xy=k,\quad y=kt\)
2. 2Distinguish an added constant The later examples also practise fitting y=k/x+c. When c is nonzero, y is no longer exactly inversely proportional to x; xy=k+cx varies with x.

Model or data

\(\displaystyle y=k/x,\quad (x,y)=(2,18)\)

Find k for the basic reciprocal model from a known observation

Model or data

\(\displaystyle y=36/x,\quad x=9\)

Use the basic reciprocal model to predict y at x=9.

A reciprocal model with an added constant

For positive x, let t=1/x. Fit y=kt+c using the transformed coordinates. The line intercept t=0 is not an original data point at x=0.

1. 1Use exact transformed values Keep fractions while finding the slope and intercept \(\displaystyle (x,y)=(2,15),(6,7)\quad\Rightarrow\quad(t,y)=(1/2,15),(1/6,7)\)
2. 2Reverse the operations Subtract c before solving for the original input \(\displaystyle y=k/x+c\quad\Rightarrow\quad x=k/(y-c)\)

Model or data

\(\displaystyle (t,y)=(1/2,15),(1/6,7)\)

Find the reciprocal-model slope using the two transformed points

Model or data

\(\displaystyle k=24,\quad (t,y)=(1/2,15),\quad y=kt+c\)

Use the slope and one data point to find the intercept

Model or data

\(\displaystyle y=24/x+3,\quad x=3\)

Predict the response at original input x=3.

Model or data

\(\displaystyle 9=24/x+3\)

Find the positive original input giving response 9.

Move the original input through positive values. The point uses the reciprocal of x as its transformed coordinate. Compare x=2 and x=6 with the observations (2,15) and (6,7). Displayed decimals are rounded; retain exact fractions in calculations

What does the line intercept at t=0 mean?

A base-10 logarithmic model

Here the explanatory variable is logged and the response stays unchanged. Use positive inputs. Recover x with a power of ten when solving in reverse

1. 1Transform the input The data below have exact base-10 logarithms \(\displaystyle (x,y)=(10,6),(1000,14)\quad\Rightarrow\quad(t,y)=(1,6),(3,14)\)
2. 2Return to x A value of t is not the original input \(\displaystyle t=\log_{10}x\quad\Rightarrow\quad x=10^t\)

Model or data

\(\displaystyle (t,y)=(1,6),(3,14)\)

Find the slope in logarithmic input coordinates

Model or data

\(\displaystyle y=4t+c,\quad(t,y)=(1,6)\)

Find the intercept of the transformed line

Model or data

\(\displaystyle y=4\log_{10}(x)+2,\quad x=100\)

Predict y at the original input 100.

Model or data

\(\displaystyle 18=4\log_{10}(x)+2\)

Recover the original input when the response is 18.

Model or data

\(\displaystyle t=\log_{10}x=0\)

Which original input corresponds to the transformed-line intercept?

Model or data

\(\displaystyle 21=2x^2+3,\quad x>0\)

The rule y=2x²+3 is used for a positive physical input. Find x when y=21.

Independent modelling task — before the model answer

On paper or using a spreadsheet/graphing tool, investigate the exact observations (1,27), (2,15), (4,9). Compare t=x², t=1/x and t=log₁₀(x). Produce the three transformed tables or plots; label both axes, choose a useful window, test whether the slopes agree, and select a model. Write the complete original-variable rule, predict y at x=3 and x=8, solve y=11, and explain its positive-input domain and the limits of extrapolation. This written investigation is self-reviewed; the following checks do not automatically mark the complete report

1. 1Use the observations All three pairs must fit the chosen candidate, not just the two points used to find its line
2. 2Explain the choice of tool State which table or plotting function you used and why it helps compare the transformed slopes. Verify one result independently by substitution

Model or data

\(\displaystyle x=4,\quad t=1/x\)

Check the third reciprocal transformed input in the independent task

Model or data

\(\displaystyle (x,y)=(1,27),(2,15),(4,9)\)

Which transformation fits every observation in the independent task?

Model or data

\(\displaystyle y=24/x+3,\quad x=8\)

Use your fitted rule to predict the response at x=8.

Model response to the independent task

The reciprocal transformation fits the three exact observations. A spreadsheet table of each transformed input makes consecutive slopes easy to compare; a scatterplot helps inspect the pattern. Either tool is suitable if the numerical checks are retained

1. 1Compare all three candidates Reciprocal slopes are both 24. Square slopes are −4 and −1/2. Logarithmic input steps are both log₁₀2, but response changes are −12 and −6, so their slopes differ \(\displaystyle \begin{aligned}t=x^2 &: (1,27),(4,15),(16,9)\\t=\frac{1}{x} &: (1,27),(\frac{1}{2},15),(\frac{1}{4},9)\\t=\log_{10}(x) &: (0,27),(\log_{10}2,15),(\log_{10}4,9)\end{aligned}\)
2. 2Fit and state the rule With k=24, c=27−24=3. A useful transformed plotting window is 0≤t≤1.1 and 0≤y≤30; label the horizontal axis t=1/x. The intercept is a line extrapolation \(\displaystyle y=24t+3\quad\Rightarrow\quad y=24/x+3,\quad x>0\)
3. 3Predict, reverse and verify At x=3, y=11. At x=8, y=6. Solving the model for the original input gives 3. Substitution independently verifies each result \(\displaystyle 24/3+3=11,\qquad24/8+3=6\)
4. 4Limit the conclusion The model fits all supplied exact observations. Three points do not prove a relationship for every input. The x=3 prediction is inside the observed range; x=8 is outside. A practical application needs a context, assumptions and further observations

Produce an actual transformation and model record

Re-run the earlier independent investigation for the exact observations (1,27),(2,15),(4,9), now using actual numerical, graphical and symbolic technology. Keep the original modelling work: the extra demand is your own tool operation and record, not another copied model. Record the tool, exact inputs, settings, complete outputs and independent checks. Use a symbolic-capable tool for the symbolic steps; paper algebra checks those outputs but does not replace running the tool

1. 1Produce and compare all three candidates Generate the three transformed tables t=x², t=1/x and t=log₁₀(x), retaining y unchanged. Produce each scatterplot with labelled axes and your own useful window. Calculate consecutive slopes using unrounded values. Select the candidate supported by all three exact observations and derive its original-variable rule
2. 2Generate a family in your own tool Generate tables at x=1,2,4,8 and graphs for y=k/x with k=12,24,36. Keep x positive. On the same display add the earlier fitted extension y=24/x+3 and explain why it is not pure inverse variation. Deliberately restrict the plotted domain to 1≤x≤8, state each restricted range and record a window showing the complete restricted curves. Compare this deliberate domain restriction with the unrestricted positive-input models
3. 3Test windows rather than trusting a crop Record one full transformed-data view, then crop its horizontal bounds to hide a supplied observation. Restore it and explain why the cropped pattern is insufficient evidence about the complete exact dataset. For the original-variable family, choose bounds that show every supported endpoint with room for labels; record the mathematical domain separately from the display bounds
4. 4Run symbolic checks and compare notation Use symbolic technology to solve 11=24/x+3 for positive x and to simplify the difference between 24/x+3 and (24+3x)/x. Preserve the domain condition. Compare both with 24/(x+3) at x=1. Write formal expressions beside your actual input syntax, explaining brackets, the division bar, explicit multiplication and the variable being solved for. Keep exact outputs as well as any decimal approximations

Review the technology family and exact checks

The earlier complete model response still applies: reciprocal slopes are 24 and 24, while square slopes are −4 and −1/2. Log-input steps are equal but response differences are −12 and −6, so those slopes differ. A plot must show every observation, not only the two used to fit a line. Compare the following expected values with your saved outputs; they are not evidence that your tool ran

1. 1Family table and restrictions At x=1,2,4,8, the k=12 row is 12,6,3,1.5; k=24 gives 24,12,6,3; k=36 gives 36,18,9,4.5. The extension gives 27,15,9,6. On [1,8] their ranges are [1.5,12], [3,24], [4.5,36] and [6,27]. A possible original-variable window is 0.5≤x≤8.5 and 0≤y≤40, with each curve explicitly restricted to [1,8]. Other clear windows are valid
2. 2Do not confuse the family and extension Each pure model has xy=k and unrestricted positive-input range (0,∞). The extension has xy=24+3x and range (3,∞) for x>0. Its added constant is retained modelling extension work, not the named pure inverse-variation relationship. Restricting to [1,8] changes the model domain; changing viewing bounds alone does not
3. 3Symbolic result and grouping The equation 11=24/x+3 gives x=3, verified by 24/3+3=11. The difference between 24/x+3 and (24+3x)/x simplifies to zero when x≠0; the current model uses x>0. At x=1, these expressions equal 27, whereas 24/(x+3)=6. Brackets around the denominator change the expression; a tool must not silently be credited with a different formal rule \(\displaystyle \frac{24}{x}+3=\frac{24+3x}{x}\quad(x\ne0),\qquad\frac{24}{x+3}\ne\frac{24}{x}+3\)
4. 4Verify graphical evidence The full reciprocal transformed view can use 0≤t≤1.1 and 0≤y≤30. Cropping to 0.3≤t≤1.1 hides (1/4,9). The remaining two points do not test the third observation. Restore the complete view and retain its settings alongside the numerical slope check and the original positive-input domain

Generate the logarithmic example and counterexample

Use the accepted 9D exact data (0.1,2),(1,5),(10,8),(100,11). In your own technology calculate t=log₁₀(x), produce the complete transformed table and scatterplot, and verify every consecutive slope. Save the actual operation and output, not only the model numbers. Then change only the final response from 11 to 12, repeat the calculation and plot, and explain why increasing original data need not become exactly linear after logging x.

1. 1Record the domain and a useful view Use positive original inputs and label the transformed horizontal axis t=log₁₀(x). Choose and record horizontal and vertical bounds showing all four original and changed points with margin. Explain why a negative t is allowed although a negative original x is not
2. 2Use symbolic output and compare log notation Use a symbolic solver for 8=3log₁₀(x)+5 over positive real x, then verify by an exact power-of-ten calculation. Record how your actual tool specifies base 10, the entire log argument, multiplication and the equation. Compare its notation with the formal rule. Do not assume a function named log always uses base 10; check its documented meaning or its value at 10, and use an explicit base if needed
3. 3Retain the complete comparison Include both transformed tables/plots, all slopes, the symbolic solution and inverse check, the changed datum and the mathematical conclusion. Agreement with a line for the original exact sample is an example, not proof of a universal practical law. A single failed transformed slope is enough to reject exact collinearity of the modified sample

Model response: Test the logarithmic transformation independently

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

1. 1Table and plot The transformed points are (−1,2),(0,5),(1,8),(2,11). Use horizontal axis t=log₁₀(x), vertical y and, for example, −1.5≤t≤2.5, 0≤y≤13. All three consecutive slopes are three
2. 2Counterexample Changing the last response to 12 makes the last slope four, while the other two remain three. The modified exact observations are not collinear after logging x.
3. 3Interpretation Original x must be positive; negative t is permitted. The response values were retained unchanged. Agreement at four exact observations supports this candidate for the supplied dataset but does not establish a universal practical law

Review the actual-operation record and log syntax

The original transformed inputs are −1,0,1,2. With responses 2,5,8,11, all slopes are 3 and y=3t+5. Changing only the final response to 12 gives slopes 3,3,4. A window −1.5≤t≤2.5 and 0≤y≤13 shows both datasets; retain your own justified settings. Solving 8=3log₁₀(x)+5 gives log₁₀(x)=1 and x=10. Verify 3×1+5=8 and 10¹=10.

1. 1Compare representations The formal expression 3log₁₀(x)+5 may need an explicit multiplication symbol and a base-specific logarithm function in your tool. The subscript 10 states the base; it is not multiplication by 10. Log the explanatory x-values only, not y. At x=0.1, t=−1 is a valid output; x itself is still positive
2. 2Compare your actual record Check that the saved command used the complete argument, base and positive-real assumption, and that the symbolic result, numerical table and plot describe the same inputs. Investigate any disagreement before accepting an output. The copied expected response is not a replacement for the actual tool record

A technology plot uses transformed inputs −1,0,1,2 and responses 2,5,8,12. Does this exact dataset lie on one straight line?