Desmos regression from a supplied table

Some SAT questions hand you a table of paired values and ask for an equation, a coefficient, or a prediction from a model. A regression finds the equation that best matches a set of paired values, and Desmos runs one in seconds. You bring three things: the right columns, the right model, and a careful read of what the question wants.

Here's how that plays out on a typical question.

Fast Desmos solution: Put $t$ in x_1 and $A(t)$ in y_1. Then select Add Regression and choose Exponential. Desmos finds $A(t)=500(0.8)^t$. Plug in $t=5$ and you get $163.84$, which rounds to $\boxed{164}$ milligrams.

You never had to work out the $0.8$ yourself. Desmos found it from the table.

Once the table is in, select Add Regression next to it and pick the model:

The squiggle, $\sim$, is called a tilde. It tells Desmos to fit a model to the table, and it's not an equals sign. These are Desmos's automatic regressions: the menu types each line for you, so there's nothing to memorize yet.

If the question names a model, use that one. If it doesn't, the context, the answer choices, and the table's pattern tell you. Check the pattern when the inputs are evenly spaced:

• If the outputs change by about the same amount each step, like $3$, $7$, $11$, $15$, try linear.
• If the differences themselves change by about the same amount each step, try quadratic. For $1$, $4$, $9$, $16$, the differences are $3$, $5$, $7$, and those go up by $2$ each time.
• If each output is about the same multiple of the one before, like $2$, $6$, $18$, $54$, try exponential.

Measured data usually won't follow any pattern perfectly, and that's fine. Go with the pattern the numbers stay closest to. What you shouldn't do is grab a fancier curve just because it can hit every point. A quadratic can pass through any three points with different $x$-values, but that doesn't make it the right model. The question and the pattern decide, not how many letters the equation has.

Desmos remembers the fitted values, so you can use them on a new line. In the medicine example, once the exponential regression reports $a$ and $b$, you type a*b^5 to get the model's value at $t=5$. Type the letters, not the decimals you see on screen. Desmos keeps more digits than it shows, so the letters give you the full fitted values instead of rounded copies. That means you never have to copy a long decimal. Round once, at the very end.

Then reread the question. Desmos hands you the whole model, but the question often wants just one piece of it. The medicine table gives $A(t)=500(0.8)^t$, and different questions about it want different things. A question about the amount at $t=5$ wants a prediction, so you plug in $5$ and round $163.84$ to $164$. A question about the multiplier wants $b=0.8$, not the $20\%$ drop per hour it stands for. Lines work the same way: a question about the slope wants $m$ alone, not the whole equation. And read the units: if the outputs are in thousands of dollars, so is the model's prediction.

How far should you trust the fit? That depends on where the table came from.

Real measurements rarely sit exactly on the model, and that's normal. When the question asks for the "best estimate," the model's value gives you the answer. It's an estimate, so don't expect it to match every row.

A table from an exact function calls for a quick check. Desmos shows the fitted values as decimals, which may be rounded, so plug a row or two into the equation you picked before you trust it as exact. The medicine table passes: at $t=2$, $500(0.8)^2=320$, which matches the table.