Question 15·Hard·Two-Variable Data: Models and Scatterplots
A scatterplot shows the relationship between the temperature (in degrees Celsius) and the number of smoothies sold (in one day) at a shop. A line of best fit for the data is
The shop owner rewrites the model using temperature measured in degrees Fahrenheit, where
Which choice gives an equivalent line of best fit relating to ?
When a line of best fit is rewritten in a new x-unit, treat it like an algebra substitution problem: (1) solve the unit-conversion equation for the original variable, (2) substitute into the model, and (3) simplify. Be careful: if the new variable is a scaled-and-shifted version of the old one (like Fahrenheit), both the slope and the intercept change, not just the slope.
Hints
Isolate first
Use the Fahrenheit-to-Celsius relationship to rewrite in terms of (so the model will only have in it).
Substitute carefully
After you solve for , substitute that entire expression into using parentheses.
Watch what happens to the intercept
Because includes a "+32", converting units changes more than just the slope; make sure the constant term is updated too.
Desmos Guide
Enter the Celsius-to-Fahrenheit conversion
Enter as an expression (you can use a different variable name in Desmos, such as replacing with ).
Build the rewritten model
Enter and let Desmos simplify it visually.
Compare to the choices by graphing
Graph each answer choice as a line in terms of (for example, , etc.). The correct choice will lie exactly on top of the simplified line you graphed in the previous step for all .
Step-by-step Explanation
Solve for Celsius in terms of Fahrenheit
From
subtract 32 and multiply by :
Substitute into the line of best fit
Substitute into :
Simplify the coefficient correctly
Write as a fraction: . Then
so
Distribute and combine constants
Distribute and combine constants:
So the equivalent line of best fit is .