SATStudy Topic
SAT Advanced Math Foundations Study Guide
Study quadratics, exponential models, rational expressions, radicals, function composition, and transformations.
Topic Overview
Choose the form that reveals the answer
Equivalent expressions expose different features. Factored form, such as (x - 2)(x - 5), reveals zeros. Vertex form, such as (x - 3)^2 + 4, reveals the smallest output and where it occurs. Expanded form makes coefficients easy to compare. Before calculating, ask which representation matches the requested feature.
Preserve restrictions
Cancel common factors only when the canceled factor is nonzero. A simplified rational expression may still have an excluded input from the original denominator. A real square root requires a nonnegative radicand. Squaring an equation can create candidates that fail in the original equation, so substitution is part of solving rather than an optional final decoration.
Read repeated change correctly
In A(t) = 600(0.8)^t, the initial amount is 600 and each time step retains 80% of the preceding amount. After two steps, the value is 600(0.8)(0.8) = 384, not 600 minus two fixed deductions of 120. For composition, evaluate the inner function first. For horizontal shifts, replacing x with x + h moves input locations left by h; adding h outside the function changes outputs vertically.
- Canceling terms that are added instead of common factors.
- Dropping original denominator restrictions.
- Treating a repeated percentage change as a fixed amount.
- Evaluating function composition in the wrong order.
Checkpoint Quiz
Test your understanding of SAT Advanced Math Foundations
These questions are for study practice only and are not official exam questions.
1. For x not equal to 3, simplify (x^2 - 9)/(x - 3).
2. For which k does x^2 + 6x + k = 0 have exactly one distinct real solution?
3. Which expression is equivalent to (3x^2y)(2xy^3)?
4. A quadratic has zeros -2 and 5 and leading coefficient 1. Which expression represents it?
5. What is the positive solution of 4x^2 = 100?
6. Which expression equals x^(3/2) for x > 0?
7. What is the domain of the real-valued function f(x) = sqrt(9 - x)?
8. The polynomial p(x) = x^3 - 2x + 7 is divided by x - 2. What is the remainder?
9. If f(x) = (x - 1)(x + 6), what is f(1)?
10. What are the solutions of x^2 - 5x + 6 = 0?
Frequently asked questions
What does vertex form show?
For a(x - h)^2 + k, the vertex is (h, k); the sign of a determines whether it is a minimum or maximum.
Why check after squaring?
Squaring can make equations true for values that did not satisfy the original equation.
What does a factor of 0.9 mean?
The model retains 90% of its previous value each time unit, a 10% decline.
How do I evaluate f(g(x))?
Find g(x) first, then use that result as the input to f.
