# SAT Advanced Math: Quadratics, Exponentials, Tips | topin

SAT guidesAdvanced Math

Section

Math, about 35% of questions (13–15 scored)

Skills

Equivalent expressions, nonlinear equations and systems, nonlinear functions

Functions

Quadratic, exponential, polynomial, rational, radical and absolute value

Formats

Multiple choice and typed answers

Calculator

Allowed; graphing finds vertices, zeros and intersections

## How it works

Advanced Math questions ask you to rewrite expressions, solve quadratic and other nonlinear equations, interpret the parts of a function, or match a function to its graph. Harder versions add a constant and ask for the value that gives one solution, compare exponential models written in different forms, or test function transformations and composition.

## Worked example

f(x) = x² − 6x + c In the function above, c is a constant. The graph of y = f(x) in the xy-plane touches the x-axis at exactly one point. What is the value of c?

1. A3
2. B6
3. C9
4. D36
Show the answer

C: 9

A parabola touches the x-axis at exactly one point when the equation x² − 6x + c = 0 has exactly one real solution, which happens when the discriminant b² − 4ac is zero. Here a = 1, b = −6 and the constant term is c: 36 − 4c = 0, so c = 9\. Check with factoring: x² − 6x + 9 = (x − 3)², which is zero only at x = 3\. Choosing 36 comes from stopping at b²; choosing 3 or 6 comes from confusing c with the root or with b.

## A method for Advanced Math

1. 1  
### Know the three quadratic forms  
Standard form ax² + bx + c gives the y-intercept, factored form a(x − p)(x − q) gives the x-intercepts, and vertex form a(x − h)² + k gives the vertex (h, k).
2. 2  
### Use the discriminant for number of solutions  
b² − 4ac positive: two real solutions. Zero: exactly one. Negative: none. It is not on the reference sheet, so learn it.
3. 3  
### Read exponential models  
In a(b)^t, a is the starting value and b the growth factor: 1.04 means 4% growth, 0.85 means 15% decay. Check what one unit of t is, months or years.
4. 4  
### Match forms by testing a value  
For equivalent-expression questions, substitute an easy number such as x = 2 into the original and each choice. The one that gives the same value is equivalent.
5. 5  
### Graph when the algebra is long  
Graphing a function shows its zeros, vertex and intersections directly. If the question has a constant, try typical values to see the pattern, then confirm with algebra.

## Traps to avoid

* Losing a negative solution when taking a square root.
* Reading the vertex of a(x − h)² + k as (−h, k) instead of (h, k).
* Treating an exponential rate per month as a rate per year, or the reverse.
* Answering the number of solutions when the question asks for a solution, or the reverse.

## Advanced Math FAQs

What is on SAT Advanced Math?

Equivalent expressions; quadratic, exponential, polynomial, rational, radical and absolute value functions; and nonlinear equations and systems. It makes up about 35% of the Math section.

Is the quadratic formula on the SAT reference sheet?

No. The reference sheet covers geometry formulas only, so learn the quadratic formula, the discriminant and vertex form.

Can the graphing calculator solve SAT quadratics?

Often, yes: graphing shows the zeros and vertex directly. Questions with an unknown constant are harder to graph, so know the algebra too.

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