Desmos for Equations and Intersections

Definition A graphing calculator draws the set of points $(x, y)$ that satisfy an equation or inequality you type, and computes numerical values of expressions. In the current format of the digital SAT, the Desmos graphing calculator is built into the testing app. You may also bring your own calculator if it is approved. Some calculators are not permitted, so check the current official list well before test day. The version inside the testing app can differ from the classroom or website version, so practice with the calculator in the official practice tests rather than assuming every feature you know from class is there.

Definition In Desmos, an expression line of the form $y = f(x)$ draws the graph of $f$, and a line such as $2x + 3y = 12$ draws every point that satisfies that equation. To solve an equation $L(x) = R(x)$ in one variable, graph

on two separate lines. A real number $a$ is a solution exactly when the point $(a, L(a))$ lies on both graphs, so the solutions are the $x$-coordinates of the points where the graphs meet.

Definition An intersection point of two graphs is a point $(a, b)$ that lies on both of them. For a system of two equations in $x$ and $y$, the real solutions are exactly the intersection points of the two graphs. In Desmos, clicking a graph usually marks points of interest such as intersections, and clicking one of those points shows its coordinates, rounded to a few decimal places.

Definition A zero of a function $f$ is a real number $a$ in its domain with $f(a) = 0$. Zeros are the $x$-coordinates of the $x$-intercepts of the graph of $y = f(x)$. Any equation $L(x) = R(x)$ can be rewritten as $L(x) - R(x) = 0$, so its solutions are the zeros of $L - R$. In Desmos, $x$-intercepts are usually among the points of interest marked when you click a graph.

Definition The viewport is the rectangle of the coordinate plane shown on screen, set by a range of $x$-values and a range of $y$-values. Points of a graph outside the viewport exist but are not drawn. You can change the viewport by zooming in or out and by dragging the graph. A missing intersection on screen proves nothing until you have checked a larger and a closer view.

Definition An exact answer is the true value written with integers, fractions, radicals, $\pi$ or other exact symbols, such as $\frac{2}{3}$ or $1 + \sqrt{2}$. An approximation is a decimal close to that value, such as $0.667$ or $2.414$. A coordinate that a graphing calculator displays is a rounded decimal. It can suggest an exact value, but it never proves one, since many different numbers round to the same decimal.

Definition Algebraic steps such as squaring both sides or clearing a denominator can produce candidates that do not satisfy the original equation. A candidate is a solution only if it is in the domain of the original equation and makes it true. Graphing the two sides of the original equation shows only true solutions, because each graph uses the original expression and its real domain.

Definition The graph is most useful for checking a solution, for counting solutions, for nonlinear systems that are slow to solve by hand, and for matching a decimal to an answer choice. Direct algebra is usually quicker and more reliable for short linear equations, for answers that must be exact, for questions with a parameter that has no numerical value, and for questions about an expression such as $2x + 1$ rather than $x$ itself.