SAT Math — Algebra and Functions

Chapter Overview & Weightage

Algebra and Functions is the single largest content domain on the SAT Math section. The College Board’s “Heart of Algebra” and “Passport to Advanced Math” together account for approximately 50–55% of all SAT Math questions (roughly 22–24 out of 44 questions across both Math modules).

The digital SAT (2024 onwards) has two Math modules of 22 questions each (44 total). Module 2 adapts based on your Module 1 performance — performing well in Module 1 gets you a harder Module 2 with a higher score ceiling. Algebra questions appear in both modules, but harder function questions dominate the adaptive Module 2.

Topic	Approx. questions	Difficulty
Linear equations (1 variable)	3–4	Easy–Medium
Systems of linear equations	3–4	Easy–Medium
Linear inequalities	2–3	Easy–Medium
Quadratic equations	3–4	Medium–Hard
Functions (notation, composition, transformations)	3–4	Medium–Hard
Exponential functions	2–3	Medium–Hard
Polynomial manipulation	2–3	Hard

Key Concepts You Must Know

Linear equations in one variable: Solve by isolating $x$ . Watch for equations with no solution ( $3x + 1 = 3x + 5$ ) or infinite solutions ( $2(x+1) = 2x + 2$ ).

Systems of two linear equations: Use substitution or elimination. The SAT also asks about the number of solutions — one solution (lines intersect), no solution (parallel lines, same slope, different intercept), infinitely many (same line).

Function notation: $f(3)$ means substitute $x = 3$ . Composite functions: $f(g(x))$ means plug $g(x)$ into $f$ . Inverse functions: $f^{-1}(x)$ swaps inputs and outputs.

Quadratic functions: $f(x) = ax^2 + bx + c$ . Know vertex form $a(x-h)^2 + k$ , factored form $a(x-r_1)(x-r_2)$ . The vertex is at $x = -b/(2a)$ , $y = c - b^2/(4a)$ .

Transformations of functions:

$f(x) + k$ : shifts graph up by $k$
$f(x) - k$ : shifts graph down by $k$
$f(x + k)$ : shifts graph left by $k$
$f(x - k)$ : shifts graph right by $k$
$-f(x)$ : reflects over $x$ -axis
$f(-x)$ : reflects over $y$ -axis

Exponential functions: $f(x) = a \cdot b^x$ where $b > 0$ , $b \neq 1$ . If $b > 1$ , exponential growth. If $0 < b < 1$ , exponential decay.

Important Formulas

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Discriminant $= b^2 - 4ac$ :

> 0: two real roots
= 0: one real root (repeated)
< 0: no real roots (complex roots, SAT won’t ask this)

Vertex $x$ -coordinate: $x = -\frac{b}{2a}$

Vertex form: $f(x) = a(x - h)^2 + k$ where $(h, k)$ is the vertex.

Slope-intercept: $y = mx + b$ (slope $m$ , $y$ -intercept $b$ )

Standard form: $Ax + By = C$ (slope $= -A/B$ )

Two lines are parallel if same slope; perpendicular if slopes multiply to $-1$ .

Solved Previous Year Questions

PYQ 1 — Systems of equations (Medium)

Q: The system $2x + 3y = 7$ and $4x + ky = 14$ has infinitely many solutions. Find $k$ .

Solution: For infinitely many solutions, the second equation must be a multiple of the first.

$4x + ky = 14$ is $2 \times (2x + 3y = 7)$ — so $k = 2 \times 3 = 6$ .

Check: $4x + 6y = 14$ is indeed $2(2x + 3y) = 2(7)$ ✓. Answer: $k = \mathbf{6}$ .

PYQ 2 — Function composition (Hard)

Q: If $f(x) = 2x - 1$ and $g(x) = x^2 + 3$ , find $f(g(2))$ .

Solution: First find $g(2) = (2)^2 + 3 = 7$ .

Then find $f(7) = 2(7) - 1 = 13$ .

Answer: $f(g(2)) = \mathbf{13}$ .

PYQ 3 — Quadratic with no solution (Hard)

Q: For which value of $k$ does $x^2 - kx + 9 = 0$ have exactly one real solution?

Solution: Exactly one real solution means discriminant $= 0$ .

$b^2 - 4ac = (-k)^2 - 4(1)(9) = k^2 - 36 = 0$

$k^2 = 36 \implies k = \pm 6$ .

Both $k = 6$ and $k = -6$ give exactly one real solution.

Difficulty Distribution

Level	What to expect	Strategy
Easy (800–550)	Direct linear equation solving, simple function evaluation, reading graphs	Apply formula directly, no tricks
Medium (550–650)	Systems of equations, quadratic factoring, transformations	Eliminate answer choices, use substitution
Hard (650–800)	No-solution/infinite-solution conditions, polynomial division, complex function composition	Algebraic manipulation + conceptual understanding

Expert Strategy

Work backwards from answer choices on any “which value of k” type question — it’s usually faster than algebra. Plug each answer choice into the condition and check.

For function problems, translate function notation into plain language: $f(g(x))$ means “the output of $g$ becomes the input of $f$ .” Draw an arrow diagram if it helps.

The SAT loves “how many solutions” questions for systems and quadratics. For two linear equations: same slope + different intercept = no solution; same slope + same intercept = infinitely many solutions; different slopes = exactly one solution. Memorise this pattern — it appears multiple times per test.

When you see $f(x+2)$ in a transformation question, always ask: “Is this a horizontal shift, and which direction?” The counterintuitive answer is that $+2$ inside shifts LEFT by 2 (because you need $x = -2$ to make the input zero).

Common Traps

Trap 1: Confusing $f(g(x))$ and $g(f(x))$ . These are different! $f(g(x))$ is “apply $g$ first, then $f$ .” Work from the inside out — always evaluate the innermost function first.

Trap 2: For the equation $3x + c = 3x + 7$ , concluding $c = 7$ means “one solution.” No — if $c = 7$ , the equation becomes $3x + 7 = 3x + 7$ , which is true for ALL $x$ (infinitely many solutions). For NO solution, $c \neq 7$ (e.g., $c = 5$ gives $3x + 5 = 3x + 7$ , which is $5 = 7$ — impossible).

Trap 3: Forgetting both roots from the quadratic formula. If $x^2 = 25$ , then $x = \pm 5$ . The SAT sometimes gives answer choices with only $x = 5$ as one option — check both roots against any given constraints.

Trap 4: Applying transformations in the wrong order for combined shifts. $f(2x - 6)$ is NOT the same as $f(2x) - 6$ . Always rewrite in the form $f(a(x - h)) + k$ before identifying transformations.