SAT · Domain 2

Advanced Math

35% of the SAT math section

Quadratics, polynomials and non-linear models — the other third of the section. Translating between the forms of a parabola, and reading what the parts of an exponential function mean.

Advanced Math · Topic 1 of 4

Equivalent Expressions

Rewriting without changing the value: a radical as a fractional exponent, a fraction of polynomials with a factor cancelled, a product expanded. “Which of the following is equivalent to” is the most common opening in this domain, and the fastest check is usually a number — put $x = 2$ into the original and into the choice you like.

The problem

For

x > 3

, which of the following is equivalent to

\dfrac{x^2 - 9}{x - 3}

?

A $x - 3$
B $x + 3$
C $x - 6$
D $x^2 - 3$

Working1 / 4

The top is a difference of squares $x^2 - 9 = (x - 3)(x + 3)$

Advanced Math · Topic 2 of 4

Quadratics & Polynomials

Standard form shows the $y$ -intercept, factored form shows the roots, vertex form shows the vertex. Each hides what the others show, and the SAT tests whether you can move between them — most often from standard to factored.

The problem

Which of the following is equivalent to the expression

x^2 - 8x + 15

?

A $(x - 3)(x - 5)$
B $(x + 3)(x + 5)$
C $(x - 2)(x - 6)$
D $(x + 2)(x + 6)$

Working1 / 5

Two numbers that multiply to $+15$ and add to $-8$ $\text{factor pairs of } 15:\; (1, 15),\ (3, 5)$

Advanced Math · Topic 3 of 4

Nonlinear Equations & Systems

A line and a parabola meet where their $y$ -values agree, so set the two expressions equal and you have a quadratic. How many solutions it has is how many times the graphs cross — and the discriminant $b^2 - 4ac$ answers that without solving anything: positive means two, zero means one, negative means none.

The problem

In the

xy

-plane, the graphs of

y = x^2 + 4x + 5

and

y = 2x + k

intersect at exactly one point. What is the value of

k

?

A $1$
B $4$
C $5$
D $6$

Working1 / 4

Set the two expressions for $y$ equal $x^2 + 4x + 5 = 2x + k$

Advanced Math · Topic 4 of 4

Exponential Functions

An exponential model is a starting amount multiplied by the same factor over and over. The SAT rarely asks you to compute one; it asks what a number in the model represents, or which model fits a sentence about bacteria or a bank balance.

The problem

A population of bacteria is modeled by the function

P(t) = 500(1.08)^t

, where

t

is the number of days. What does the value

1.08

represent in this context?

AThe initial population
BThe population after one day
CThe factor the population grows by each day — an 8% daily increase
DThe number of days the population takes to double

Working1 / 4

The general form of exponential growth $A = P(1 + r)^t$