Understand the Digital SAT before you study for it

Use this when isolating the variable would require several distributions, fractions, radicals, or nested expressions.

1. Enter the left side as one graph, using y as the output.
2. Enter the right side as a second graph.
3. Select the intersection and read only the coordinate the question asks for.

The intersection is (12, 41), so the equation is true when x = 12. The y-coordinate is only a shared output and is not the requested answer.

Watch out: Do not report the full intersection automatically. SAT questions often ask for x, y, x + y, or a coefficient instead.

Use this for quadratic, cubic, absolute-value, or other equations written as an expression equal to zero.

1. Move every term to one side if the equation is not already equal to zero.
2. Graph the resulting expression as y = that expression.
3. Select every visible x-intercept and apply any domain condition from the question.

The x-intercepts are (2, 0) and (5, 0), so the equation has solutions x = 2 and x = 5.

For y = x² − 7x + 10, the y-intercept is 10. The minimum is −2.25 at x = 3.5, which follows from y = (x − 3.5)² − 2.25. Select the turning point to read it; the minimum is the y-value, not the x-value.

Watch out: A graphing window can hide a root. Zoom out or set the x-axis bounds when the problem suggests large values.

Use this when two equations share x and y, especially when elimination would create awkward arithmetic.

1. Enter both equations exactly as written. You do not need to solve each one for y first.
2. Select the intersection point.
3. Substitute the coordinates mentally into the simpler equation as a quick check.

The lines meet at (5, 3). If the problem asks for x + y, calculate 5 + 3 rather than entering 5.

Watch out: Parallel lines have no solution, while the same line entered twice represents infinitely many solutions. Do not invent an intersection.

Use this for feasible regions, boundary questions, and answer choices that give ordered pairs.

1. Enter each inequality with its original strict or inclusive symbol.
2. Locate the region where all shading overlaps.
3. Test a choice by plotting its point or substituting its coordinates into the inequalities.

A solid boundary means equality is included. A dotted boundary means points on that line are excluded.

Watch out: A point can satisfy one inequality and fail the system. Check every condition, including whether the boundary is strict.

Another worked example in Desmos

Use this when time, length, population, or another variable has a stated domain and the unrestricted graph creates extra answers.

1. Graph the model normally.
2. Add the allowed input interval in curly braces at the end.
3. Read intersections only from the visible, context-valid segment.

Only the portion from x = 0 through x = 12 belongs to the situation. Desmos hides values outside that interval.

Domain means allowed inputs; range means resulting outputs. For this rental, 0 ≤ x ≤ 12 produces 18 ≤ y ≤ 48. Curly braces restrict the display; they do not turn continuous values into whole numbers.

Watch out: A mathematically valid negative time or fractional person is still invalid in context. The calculator does not interpret units for you.

Use this when one formula must be evaluated at several inputs or compared with a target value.

1. Define the rule with function notation.
2. Evaluate a single input with f(value), or use a list for several inputs.
3. For a target output, graph y = f(x) and the horizontal target line.

The single evaluation answers one substitution. The list creates four outputs at once without retyping the formula.

Watch out: Keep parentheses around negative inputs. f(-3) and -f(3) are not generally the same.

Use this to understand vertex form, exponential bases, line slopes, and questions asking how a parameter changes a graph.

1. Enter a model containing an undefined letter such as a, h, or k.
2. Create the slider when Desmos offers it.
3. Move one parameter at a time, then use algebra for the exact final value if the slider only gives an estimate.

The vertex stays at (3, 2) while a changes the opening direction and width. The intersections with y = 10 move accordingly.

Watch out: A slider is excellent for structure and estimation, but dragging is not an exact solution method. Verify a final parameter algebraically or by direct substitution.

Use this when a table is paired with a line, quadratic, or exponential model and the question asks for a best-fit coefficient or prediction.

1. Enter the x and y data in a table so the columns are x₁ and y₁.
2. On a new line, type the model with a tilde.
3. Read the fitted parameters, then substitute only the value the question requests.

Use + → Table first and enter the paired data. The headings x₁ and y₁ supply the lists for the regression below.

Desmos estimates m and b for the linear model, or a, b, and c for the quadratic model. Choose the family named or justified by the problem.

Watch out: Regression produces an estimate. Do not round a parameter before using it in a later calculation unless the question tells you to.

Use this for mean, median, total, spread, and the effect of changing or adding a data value.

1. Put the data in square brackets and store it as a list if you will reuse it.
2. Apply the needed function to the list.
3. For a changed data set, edit the list once and compare the updated result.

The list has mean 8, median 7, and total 40. These outputs stay linked if any entry changes.

Watch out: The median requires an ordered position, but Desmos handles the ordering internally. Your interpretation still has to match the question.

Another worked example in Desmos

Use this for circle equations, line-circle intersections, tangency, and coordinate-geometry questions.

1. Enter the circle equation in standard or expanded form exactly as given.
2. Enter the line or second circle on another expression row.
3. Select each intersection and apply the requested coordinate condition.

The circle has center (2, -1) and radius 5. The line y = 2 meets it at (-2, 2) and (6, 2).

Watch out: In (x - h)² + (y - k)² = r², the center signs are opposite what appears inside each parenthesis.

Another worked example in Desmos

A 20% increase multiplies by 1.20. A subsequent 20% decrease multiplies the new amount by 0.80, not the original amount.

The result is 96: a net 4% decrease. For repeated growth, use a power, such as 100(1.05)^3 for three successive 5% increases.

Watch out: Identify the base for each percentage. Equal increases and decreases do not cancel when their bases differ.