When to use regression on SAT Math in Desmos

Topic: Regression
By: SolvePlot

Use regression when a question gives data or constraints for a model with unknown parameters. Enter the data, use a tilde (~) to fit the stated model, and evaluate the quantity requested. A fit is only useful if the model and its assumptions match the question.

Original worked examples

An equals sign defines an equation. A tilde asks Desmos to fit free parameters to data. For a quadratic model, three distinct input values can determine its three coefficients when the data are consistent.

Fitting three points does not establish that an unknown real-world relationship is quadratic. In the example below, the question explicitly tells us that the function is quadratic.

How do you fit a quadratic from a table?

Keep the input and output values in matching order. You can use a table, or enter two lists called $x_{1}$ and $y_{1}\text{.}$ The quadratic model uses a, $b\text{,}$ and c as the parameters to fit.

After the fit, define a function using those parameters and evaluate it. This lets Desmos do the arithmetic while keeping the connection to the original model visible.

Original example

Work through the question

A quadratic function has values $f(0)=3\text{,}$ $f(1)=6\text{,}$ and $f(2)=11\text{.}$ What is $f(4)\text{?}$

Answer $f(4)=27$

The fitted curve x² + 2x + 3 passes through (0, 3), (1, 6), and (2, 11), and reaches (4, 27). Illustration; follow the rows below in Desmos.

What to enter in Desmos

Use one expression row for each entry. Underscores create subscripts; the tilde ~ fits parameters.

```
x_1=[0,1,2]
```
Enter the three inputs in the order given. The underscore makes the subscript.
```
y_1=[3,6,11]
```
Enter the corresponding outputs in the same order.
```
y_1~ax_1^2+bx_1+c
```
Use a tilde, not an equals sign. Desmos fits a = 1, b = 2, and c = 3.
```
f(x)=ax^2+bx+c
```
Define the fitted quadratic as a function.
```
f(4)
```
Evaluate the requested input. The result is 27.

Read the result

Read 27 beside f(4). The fitted curve passes through all three data points. The three fitted coefficients describe this specified quadratic; do not round them before evaluating the function.

Check it

The fitted function is $f(x)=x^2+2x+3\text{.}$ It returns $3\text{,}$ $6\text{,}$ and $11$ at inputs $0\text{,}$ $1\text{,}$ and $2\text{.}$ At $4\text{,}$ it gives $16+8+3=27\text{.}$

Try the entries in Desmos

The algebraic solution

Write $f(x)=ax^2+bx+c\text{.}$ Since $f(0)=3\text{,}$ $c=3\text{.}$
The other values give $a+b=3$ and $4a+2b=8\text{.}$ Subtract twice the first equation from the second to get $2a=2\text{,}$ so $a=1$ and $b=2\text{.}$
Then $f(4)=4^2+2(4)+3=27\text{.}$

When is each method useful?

Use Desmos when

Regression avoids solving for the coefficients by hand when a model is given and enough independent data determine it. It is especially helpful with less convenient values.

Use algebra when

Algebra may be preferable when the question asks for an exact symbolic relationship, a proof, or why a parameter is not determined. A model with too many free parameters can fit the givens without fixing the requested answer.

Common mistakes to avoid

Start with a blank graph so a, b, and c are free parameters, rather than values left over from another problem.
Use ~ for a fit. Using = may graph a relationship or create sliders instead.
Keep x_1 paired with y_1; reversing just one list changes the data.
Use the model named by the problem. A good-looking fit is not evidence that you chose the right model.
A fit to approximate data is approximate. Do not treat a rounded regression coefficient as an exact proof.

Try one yourself

A quadratic has $g(0)=2\text{,}$ $g(1)=5\text{,}$ and $g(2)=10\text{.}$ Find $g(3)$

See the answer and steps

$g(3)=17$

The fitted model is $g(x)=x^2+2x+2\text{,}$ so $g(3)=9+6+2=17\text{.}$ The three given outputs agree with this model.

Desmos entries:

x_1=[0,1,2]
y_1=[2,5,10]
y_1~ax_1^2+bx_1+c
g(x)=ax^2+bx+c
g(3)

Calculator references

The examples above are original. These official resources explain the calculator features and test-day rules.