Desmos Tables, Parameters and Model Checks

Definition In the current format of the digital SAT, the Desmos graphing calculator is built into the testing app. The version in the testing app may not include every feature of the classroom or website version. If the app also offers a scientific calculator mode, do not count on entries typed in one mode being available in the other. The features to rely on are the ones you have actually used in the official practice tests in the testing app.

Definition A table in Desmos is a list of ordered pairs entered in columns, and each row is plotted as a point. In a new table the columns are named $x_1$ and $y_1$ by default, and a regression refers to the data by these names. A table of values of a function lists inputs $x$ and the matching outputs $f(x)$. A function $f$ fits a table exactly when $f(x) = y$ holds in every row, which you can confirm by substitution.

Definition A model is a function, usually with numerical constants, used to describe how one quantity depends on another, such as a linear model $y = mx + b$ or an exponential model $y = ab^x$ with $a \ne 0$, $b > 0$ and $b \ne 1$. A model can fit some data exactly, or only approximately when the data are noisy measurements. Its predictions are most trustworthy within the range of the data used to build it.

Definition A coefficient is a constant number multiplying a variable or a power of a variable in an expression. In a model each coefficient has a meaning in context. In $y = mx + b$, the coefficient $m$ is the change in $y$ for each increase of $1$ in $x$, and the constant term $b$ is the value of $y$ when $x = 0$. In $y = ab^x$, the coefficient $a$ is the value at $x = 0$ and $b$ is the factor by which $y$ is multiplied each time $x$ increases by $1$.

Definition A regression finds the model of a chosen form that fits a set of data points best by a fixed rule. For a linear regression the rule is least squares, which chooses $m$ and $b$ to make the sum of the squared vertical distances from the points to the line as small as possible. In Desmos you enter the data in a table and type a line such as

with the tilde symbol in place of the equals sign. Desmos then shows the fitted values of the constants. For noisy data, regression coefficients are estimates, and a fitted model does not prove that its form is the true relationship.

Definition A parameter is a constant in a formula that can take different values, where each value gives a different function or equation from the same family. In $y = x^2 + c$ the variable is $x$, and $c$ is a parameter. A question with a parameter usually asks which value of the parameter produces some exact property, such as one solution, no solution or a given intercept.

Definition In Desmos, when you type an expression that uses a letter with no value yet, such as $a$ in $y = a(x - 2)^2 + 1$, the calculator offers to add a slider for it. A slider lets you change the value of that letter by dragging and watch the graph respond. A slider shows numbers to limited precision, so it can suggest an exact parameter value but does not prove it. The exact value comes from algebra.