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Key terms
Regression
A fitted rule that estimates the relationship in paired data.
Residual
Observed output minus predicted output.
Slider
A draggable parameter used to test how a graph changes.
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Step 2 of 7·Try it firstDone
Try it first
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Take a shot before the lesson. Write one relationship from the stem, even if you do not finish.
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Step 3 of 7·LearnDone
The big idea
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See the structure
A table is a set of exact input-output pairs. In Desmos, press the plus button and choose Table, then enter inputs in the first column and outputs in the second.
Key idea
Fit first, interpret second
Write what each number measures. A correct calculation with the wrong quantity or unit still misses the question.
Build a reliable method
Use regression only when the data are meant to be modeled. Type y_1~mx_1+b; the tilde asks Desmos to estimate, while an equals sign draws an equation.
Explore in Desmos
Fit a line to measured data
Type the two lists, using the subscript key for x_1 and y_1. On line 3, type a tilde, not an equals sign. Read m and b, then say what one more x-unit predicts for y.
The calculator is live: change anything and see what happens.
Fit a line to measured data
Degree mode
Press Escape to leave the calculator (twice if a menu is open).
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Stretch to a 36
Regression parameters have units and meaning. The fitted m is the predicted change in output for one input unit. A model can fit the data and still be unreasonable far outside the observed domain.
Trap
A fitted model is not a law
A small residual inside the data range does not justify a wild extrapolation. A temperature model fitted from 10 to 30 minutes may be useful at 25 minutes and absurd at 10,000 minutes.
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Step 4 of 7·Worked examplesDone
Worked examples
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Skip check · 2 questions
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Worked example
Example 1: Fit a line from a table
Problem
What linear regression fits the practice-hour and score-gain data (1,8), (2,11), (3,15), (4,18), and (5,22), and what does its slope mean?
Press +, choose Table, and enter 1,2,3,4,5 in the x1 column and 8,11,15,18,22 in the y1 column.
On a new line, type y_1~mx_1+b. Which symbol tells Desmos to fit the data?
Which Desmos input or parameter claim is valid here?
Desmos reports m=3.5 and b=4.3, so the fitted rule is y=3.5x+4.3.
The slope predicts about 3.5 additional score points per extra practice hour.
Which statement correctly interprets or verifies the Desmos output?
Worked example
Example 2: Use a slider to meet a condition
Problem
What value of a makes y=ax2+2 pass through (3,20), and how can substitution verify it?
Type y=ax^2+2, choose add slider for a, and type the point (3,20) on the next line.
Drag a until the parabola passes through the point. What value should you test exactly?
Which Desmos input or parameter claim is valid here?
Set a=2. The point lies on y=2x2+2 because 2(32)+2=20.
The slider finds the candidate; substitution certifies it.
Which statement correctly interprets or verifies the Desmos output?
Faded example · you finish it
Example 3: You interpret the residual
Problem
For y=3.5x+4.3, what is observed minus predicted when the observed value at x=4 is 18?
The predicted value is 3.5(4)+4.3=18.3.
Observed minus predicted is
The negative residual means
Desmos way
Regression and residuals
Enter both lists, then type y_1~mx_1+b. Read m=3.5 and b=4.3.
Type y_1-(mx_1+b) to calculate every residual. Read signs carefully: negative means the point is below the fitted line.
Regression and residuals
Degree mode
Press Escape to leave the calculator (twice if a menu is open).
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Step 5 of 7·PracticeDone
Practice
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Step 6 of 7·Spot the mistakeDone
Spot the mistake
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Spot the mistake
Where did it go wrong?
One line has an error. Click the line where it first goes wrong.
What went wrongThe coefficient 4 multiplies the input: 4(6)+7=31.
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Step 7 of 7·Exit ticketDone
Exit ticket
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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36
These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.