Study guide
Algebra and Advanced Math are the two biggest Math domains on the digital SAT, together supplying roughly 70 percent of the 44 Math questions. Algebra covers linear equations, inequalities, systems, and linear functions, while Advanced Math covers quadratics, exponentials, and general function fluency. A calculator is allowed on the entire Math section, and the built-in Desmos graphing calculator in the testing app is a legitimate solving tool, not just a checking tool.
Linear Equations, Inequalities, and Systems
A linear equation in one variable has a single solution unless something special happens, and the SAT loves the special cases. When solving leaves you with a true statement containing no variable, such as 6 equals 6, the equation has infinitely many solutions; when it leaves a false statement, such as 6 equals 4, there is no solution. Questions often exploit this in reverse: for what value of k does the equation 3(2x + 5) = kx + 15 have infinitely many solutions? Expand the left side to 6x + 15 and match coefficients, so k must be 6. Systems of two linear equations follow the same logic geometrically. Two lines with different slopes cross exactly once, one solution. Two lines with the same slope and different y-intercepts are parallel and never cross, no solution. Identical lines overlap everywhere, infinitely many solutions. So when a problem asks for the constant that makes a system have no solution, force the slopes equal while keeping the intercepts different. To actually solve a system, use substitution when one equation isolates a variable, elimination when coefficients line up, or Desmos: graph both equations and read the intersection point directly. Inequalities behave like equations with one crucial exception: multiplying or dividing both sides by a negative number flips the inequality sign, so dividing negative 3x is less than 12 by negative 3 gives x is greater than negative 4. For systems of inequalities, the solution set is the region where the shaded areas overlap, and a point is a solution only if it satisfies every inequality simultaneously.
Linear Functions in Context
Many Algebra questions dress a line in a story and ask what its parts mean. Every linear model has the form y = mx + b, and the two constants have fixed real-world jobs: the slope m is the rate of change, how much y changes for each one-unit increase in x, and the y-intercept b is the starting value, the value of y when x is zero. Suppose a gym charges a one-time fee plus a monthly rate, modeled by C(t) = 45 + 30t, where t is months. The 45 is the joining fee, paid even at t equals zero, and the 30 means the total cost grows by 30 dollars per month. When a question asks for the meaning of a constant, substitute strategically: plug in x equals zero to interpret b, and compare consecutive x-values to interpret m. Read units obsessively, because answer choices will offer the right number with the wrong units or the right units attached to the wrong constant, such as calling the monthly rate a one-time fee. You should also be able to build the line from data: given two points, the slope is the change in y divided by the change in x, so a delivery service charging 14 dollars for 2 miles and 26 dollars for 6 miles has slope (26 minus 14) over (6 minus 2), which is 3 dollars per mile, and back-substituting gives the base fee of 8 dollars. Finally, know the quick facts: parallel lines share a slope, perpendicular lines have slopes that multiply to negative 1, and x-intercepts are found by setting y to zero.
Quadratics: Roots, Vertex, and the Discriminant
A quadratic function y = ax squared + bx + c graphs as a parabola, opening upward when a is positive and downward when a is negative, and every quadratic question is asking about one of three features: the roots, the vertex, or the number of solutions. Roots, also called zeros or x-intercepts, are the x-values where y equals zero. Find them by factoring when possible, using the quadratic formula when not, or graphing in Desmos and reading the intercepts. In factored form y = a(x minus p)(x minus q), the roots are p and q, and two useful shortcuts follow for a monic quadratic, one where a equals 1: the roots sum to negative b and multiply to c. The vertex is the maximum or minimum point, the turning point of the parabola. In vertex form y = a(x minus h) squared + k, the vertex is (h, k); otherwise the vertex's x-coordinate is negative b over 2a, and by symmetry it sits exactly midway between the roots. Word problems about maximum height or minimum cost are vertex questions in disguise. The discriminant, b squared minus 4ac, counts real solutions without solving: positive means two distinct real solutions, zero means exactly one, negative means none. The SAT frequently asks for the value of a constant that gives an equation exactly one solution, which means set the discriminant to zero and solve. The same tool handles line-parabola intersections: substitute the line into the quadratic, and the discriminant of the resulting equation tells you whether they cross twice, touch once, or miss entirely.
Exponential Versus Linear Growth
The test repeatedly checks one deep distinction: linear change adds a constant amount per step, while exponential change multiplies by a constant factor per step. A population gaining 200 residents every year is linear; a population growing 4 percent every year is exponential. Exponential models take the form y = a times b to the power t, where a is the initial value at t equals zero and b is the growth factor per time unit. For percent growth at rate r, the factor is 1 plus r, and for decay it is 1 minus r, so an account growing 4 percent annually from 500 dollars is modeled by 500 times 1.04 to the power t, and a car losing 12 percent of its value yearly is 30000 times 0.88 to the power t. Given a table, diagnose the model by checking consecutive outputs: constant differences mean linear, constant ratios mean exponential. A table reading 6, 12, 24, 48 doubles each step, so it is exponential with factor 2. Interpretation questions ask what a and b mean in context, and the answers are always the same: a is the starting amount, and b describes the per-period multiplier, with b of 1.04 meaning 4 percent growth per period. Watch the time units: if a quantity doubles every 3 months and t is measured in months, the exponent must be t over 3. Also know the qualitative fact the test rewards: any exponentially growing quantity eventually overtakes any linear one, no matter how steep the line, which is why increasing exponential graphs start shallow and end steep.
Function Notation and Interpreting Constants
Function notation is a naming convention, and fluency with it unlocks easy points. The statement f(x) = 3x + 2 defines a machine named f that takes an input x and returns 3x + 2, so f(4) means the output at input 4, which is 14, and the statement f(4) = 14 is exactly the same information as the point (4, 14) on the graph of f. When a table defines a function, f(a) = b means the row where the input is a shows output b. Composite evaluations work inside out: to find g(f(2)), compute f(2) first, then feed the result to g. Solving f(x) = g(x) means finding inputs where the two functions produce equal outputs, which is geometrically the x-coordinates of the intersection points of their graphs, a task Desmos performs instantly: type both functions, click the intersections, and read the coordinates. Many Advanced Math questions blend notation with interpretation, giving a model like P(d) = 120 times 0.95 to the power d for the milligrams of a medication remaining d hours after a dose, then asking for the meaning of P(6) or of the number 120. Translate mechanically: P(6) is the amount remaining after 6 hours, 120 is the initial dose because it is the output when d is zero, and 0.95 means the amount shrinks by 5 percent each hour. The reliable technique for any interpret-the-constant question is substitution: set the variable to zero to expose the initial value, or increase it by one unit to expose the rate or factor. Constants never change meaning midstream; find their job once and the answer follows.
Key terms
- Slope
- — The rate of change of a line, the amount y changes when x increases by one unit.
- Y-intercept
- — The value of y when x equals zero, representing the starting value in a linear model.
- System of equations
- — Two or more equations solved together, with solutions at the intersection points of their graphs.
- No-solution system
- — A linear system whose equations describe parallel lines with equal slopes but different intercepts.
- Root (zero)
- — An input value that makes a function equal zero, appearing on the graph as an x-intercept.
- Vertex
- — The maximum or minimum point of a parabola, located at x equals negative b over 2a.
- Discriminant
- — The quantity b squared minus 4ac, whose sign tells how many real solutions a quadratic equation has.
- Exponential growth
- — Change by a constant multiplicative factor each period, modeled by y equals a times b to the power t.
- Growth factor
- — The base b in an exponential model, equal to 1 plus the percent growth rate expressed as a decimal.
- Function notation
- — The convention where f(a) = b means the function f gives output b at input a, matching the point (a, b).
- Vertex form
- — The expression y = a(x minus h) squared + k, which displays the parabola's vertex (h, k) directly.
- Elimination method
- — Solving a system by adding or subtracting scaled equations to cancel a variable.
Exam tips
- For no-solution and infinite-solution questions, match coefficients: same slope with different intercepts gives no solution, and identical equations give infinitely many.
- Use the built-in Desmos graphing calculator as a solver, not just a checker: graph both sides of an equation and click the intersection to read the answer.
- When a question asks how many solutions a quadratic has, or asks for the constant giving exactly one solution, go straight to the discriminant and set it to zero if needed.
- To interpret a constant in context, substitute t equals zero to reveal the initial value and increase the variable by one unit to reveal the rate or growth factor.
- Check time units in exponential models: a quantity that doubles every 3 years needs exponent t over 3 when t is in years, and mismatched exponents are a favorite trap.