SAT Advanced Math – Important Points and SAT Advanced Math Practice Test
There are 44 SAT Math questions in the SAT Exam. 13 to 15 of these 44 questions come from the SAT Advanced Math content domain. Approximately 30-35% of the SAT Math questions come from the SAT Advanced Math. Advanced Math is a little bit harder than SAT Algebra. However, having a solid background and solving as many SAT Advanced Math practices as possible will improve your SAT Advanced Math scores.
In this article
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We’ve listed the most important and frequently occurring concepts in this SAT Advanced Math post. You will see SAT Advanced Math practice test questions and exercises, totaling 27 SAT Advanced Math questions with rationales for FREE.
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SAT Advanced Math Content Domain
Advanced Math focuses on the math you’ll need to pursue further study in disciplines such as science or economics and for career opportunities in the STEM fields of science, technology, engineering, and math. SAT Advanced Math area measures skills and knowledge central for progression to more advanced math courses, including demonstrating an understanding of absolute value, quadratic, exponential, polynomial, rational, radical, and other nonlinear equations.
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SAT Advanced Math Skills and Knowledge Testing Points
The SAT exam will have around 13 to 15 questions from the Advanced Math content domain. There are 3 skills and knowledge testing points in the SAT Advanced Math content domain:
- Equivalent expressions
- Nonlinear equations in one variable and systems of equations in two variables
- Nonlinear functions
Let’s review each skill and knowledge point and see some SAT Advanced Math Exercises for each.
1. Equivalent Expressions
Equivalent expressions are algebraic expressions that, despite having different forms, produce the same result for any value of the variable(s).
Importance: Understanding equivalent expressions is crucial for simplifying algebraic SAT Advanced Math problems, solving equations, and performing algebraic manipulations.
Basic Principles of Equivalent Expressions
Commutative Property:
Addition: a+b = b+a
Multiplication: a×b = b×a
Associative Property:
Addition: (a+b)+c=a+(b+c)
Multiplication: (a×b)×c=a×(b×c)
Distributive Property:
a(b+c) = ab + ac
Recognizing Equivalent Expressions
Simplification: Combine like terms and use properties to simplify expressions.
Example 1: Simplify 2x + 3x −4 + x.
Solution: Combine like terms: 2x+3x+x−4=6x−4.
Equivalent Expression: 6x−4
Techniques for Finding Equivalent Expressions
Factoring and Expanding: Use the distributive property to factor and expand expressions.
Example 2: Find an equivalent expression for 3(x+2)+4x.
Solution: Expand 3(x+2)=3x+6, then add 4x:
3x+6+4x = 7x+6
Equivalent Expression: 7x+6
Equivalent Expressions Practice Problems
Example 3: Identify if the expressions 4(x+1)−2 and 4x+2 are equivalent.
Solution: Expand 4(x+1)−2=4x+4−2 = 4x+2
Equivalent: Yes, both expressions simplify to 4x+2.
Example 4: Determine the equivalent expression for 2(a+b)−3(b−a).
Solution:
Expand: 2a+2b−3b+3a = 5a−b
Equivalent Expression: 5a−b.
Application in SAT Advanced Math Problems
Strategy: Recognize and use equivalent expressions to simplify complex problems and solve equations efficiently.
Example 5: Solve for x if 3x+4=2x+8.
Solution: Rearrange to form an equivalent expression: 3x−2x = 8−4.
Solve: x = 4.
Common Mistakes to Avoid
Ignoring Parentheses: Remember to apply the distributive property correctly when parentheses are involved.
Forgetting to Combine Like Terms: Always combine like terms to simplify the expression fully.
Summary and Key Takeaways
- Equivalent expressions represent the same quantity and are essential for algebraic manipulation.
- Mastery of properties (commutative, associative, distributive) is crucial in recognizing and forming equivalent expressions.
- Practice identifying and creating equivalent expressions to build confidence for the SAT Advanced Math section.
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SAT Advanced Math Exercises for Equivalent Expressions
To improve your math skills, we do not recommend using a calculator when solving these SAT Advanced Math Exercises.
Exercise I. 
The given equation relates the positive numbers p, x, and y. Write the p-value in terms of x and y.
Exercise II. 
What is the value of 
?
Exercise III. 
What is the value of x?
Exercise IV. 
Simplify the given expression.
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. This equation yields;
. We need to isolate p to write the p-value in terms of x and y. Divide both sides of the equation by 221x;
. This yields;
.
;
. This yields;
. If we substitute the x value;
.
, and;
. With the help of these two, we can rewrite the left side of the equation as follows;
. This is the left side of the equation.
. This yields;
. This yields;
. This is the right side of the equation.
. The bases of the right and left sides of the equation are the same now. Therefore, powers must be the same as well.
. If multiply both sides by 3;
. This yields;
. If we subtract 5x+3 from both sides;
. This yields;
.
. Therefore, the first fraction can be rewritten as;
. There are (x-2) both in the numerator and denominator, so we can eliminate them, and the first fraction yields;
. This is the simplified expression for 1st fraction.
. Therefore the numerator can be rewritten as follows;
.
as 
. Therefore, we can rewrite the denominator of the 2nd fraction as follows;
. If we rewrite the 2nd fraction with the expressions we found;
. We see that (x-2)(x+2) are present both in the numerator and denominator of the 2nd fraction. Therefore, we can eliminate them.
. This is the simplified expression for 2nd fraction.
. The result yields;
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2. Nonlinear Equations in One Variable and Systems of Equations in Two Variables
Nonlinear Equations in One Variable
A nonlinear equation in one variable is an equation in which the variable is raised to a power other than one, appears in the denominator, or is part of a function like a square root or absolute value. Common examples include quadratic equations, cubic equations, and equations involving square roots or absolute values.
We’ve listed types of nonlinear equations and how to solve these SAT Advanced Math questions below.
Types of Nonlinear Equations
Quadratic Equations: 
Cubic Equations: 
Equations Involving Square Roots: 
Equations with Absolute Values: |x-3| = 5
Solving Nonlinear Equations
Quadratic Equations:
Factoring: Find two numbers that multiply to ac (coefficient of x2 times constant term) and add to b (coefficient of x).
Quadratic Formula: 
Completing the Square: Rewriting the equation in the form 
💡Tip: Quadratic Equations and Formula is a frequently occurring SAT Advanced Math concept in SAT exam.
Example 1: Solve 
Solution: 
Therefore, x = 2 or x = 3.
Equations with Square Roots:
Isolate the square root on one side of the equation and then square both sides.
Example 2: Solve 
Solution: Square both sides: x + 2 = 9
x = 7
Equations with Absolute Values:
Split into two cases, one where the expression inside the absolute value is positive and one where it is negative.
Example 3: Solve ∣x−3∣ = 5.
Solution: x – 3 = 5 or x – 3 = – 5
x = 8 or x = -2
Systems of Equations in Two Variables
A system of equations consists of two or more equations with the same set of variables. Solving systems of equations means finding the set of values for the variables that satisfy all equations in the system.
Types of Systems
Linear-Linear Systems: Both equations are linear.
Linear-Nonlinear Systems: One equation is linear, and the other is nonlinear (like a quadratic).
Solving Systems of Equations in SAT Advanced Math
Substitution Method:
Solve one equation for one variable and substitute that expression into the other equation.
Example 4: Solve the system:
y=2x+3
Solution: Substitute y = 2x + 3 into the second equation:
Expand and simplify:
Solve the quadratic equation using the quadratic formula or factoring.
Elimination Method:
Add or subtract the equations to eliminate one of the variables.
Example 5: Solve the system:
x+y=7
x−y=1
Solution: Add the two equations to eliminate y:
2x = 8 ⇒ x = 4
Substitute x=4 back into x+y=7
4 + y = 7 ⇒ y = 3
The solution is x = 4, y = 3
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SAT Advanced Math Exercises for Nonlinear Equations in One Variable and Systems of Equations in Two Variables
To improve your math skills, we do not recommend using a calculator when solving these SAT Advanced Math Exercises.
Exercise I. Two variables, x, and y are related such that for each increase of 1 in the value of x, the value of y increases by a factor of 5. When x=0, y=10. Write the y in terms of x.
Exercise II. 
.
One solution to the given equation can be written as 
, where k is a constant. What is the value of k?
Exercise III. 
.
In the given equation, p is a constant. The equation has exactly one solution. What is the value of p?
Exercise IV. 
.
A solution to the given system of equations is (x, y), where x>0. What is the value of x?
. a is a constant. Let’s test this.
5 times 5a equals to 25a. Therefore, the equation is correct.
. The zero power of a number is equal to 1. Therefore;
. This yields;
, the solution to the equation is as follows;
.
and 
values.
. This yields;
. This is in the form of 
. This yields;
. This yields;
. This yields;
± 
. Therefore;
and 
.
is called discriminant. If the discriminant of an equation;
. There are two 
. There is exactly one solution x.
. There are no real solutions.
. This yields;
. If we add 144 on both sides;
. If we apply the square root on both sides;
. This yields;
. This yields;
. If we substitute this in the second equation for y;
. If we subtract 
from both sides;
. This yields;
.
as 
. Therefore, we can rewrite the equation as;
.
, then 
. This yields;
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3. Nonlinear Functions
Nonlinear functions are those whose graphs are not straight lines. These functions have at least one variable raised to a power other than one, multiplied by itself, or in some other configuration that creates a curve rather than a line. In SAT Advanced Math, understanding nonlinear functions involves recognizing their different forms and how they behave on a graph.
Key Characteristics of Nonlinear Functions:
- Nonlinear Graphs: The graphs of nonlinear functions can be parabolas, circles, ellipses, hyperbolas, or any other shape that is not a straight line.
- Variable Powers: At least one variable is raised to a power other than one (e.g., x², x³).
- Multiple Solutions: Nonlinear functions can have multiple x-intercepts, y-intercepts, or roots.
- Changes in Direction: Nonlinear graphs can change direction, unlike linear graphs that are consistently increasing or decreasing.
Common Types of Nonlinear Functions in SAT Advanced Math
1. Quadratic Functions
Form: f(x) = ax² + bx + c
Graph: Parabola (U-shaped curve)
Vertex: The highest or lowest point of the parabola.
Examples:
f(x) = x² – 4x + 3
Graph this function: The parabola opens upwards because the coefficient of x² is positive. The roots are where the function crosses the x-axis.
2. Cubic Functions
Form: f(x) = ax³ + bx² + cx + d
Graph: S-shaped curve with one or more turns.
Examples:
f(x) = x³ − 3x² + 2x
Graph this function: The curve starts from the lower left, turns upward, turns again, and moves downward or upward depending on the coefficients.
3. Exponential Functions
Form: f(x) = 
where b > 0 and b≠1
Graph: Exponential growth or decay curve.
Examples:
f(x)=
shows exponential growth.
f(x)=
shows exponential decay.
These functions rapidly increase or decrease and never touch the x-axis.
4. Absolute Value Functions
Form: f(x) = ∣ax+b∣
Graph: V-shaped graph.
Examples:
f(x) = ∣x−2∣
Graph this function: The graph has a vertex at x=2 and opens upwards.
5. Rational Functions
Form: f(x) = 
where p(x) and q(x) are polynomials, and q(x) ≠ 0
Graph: Can have asymptotes and undefined points.
Examples:
Graph this function: The function has a vertical asymptote at x=1 and a horizontal asymptote at y=0.
Examples and Practice Problems
Example 1: Quadratic Function
Problem: Graph the function 
Solution:
Identify the coefficients: a=1, b=−4, c=3.
Find the roots using the quadratic formula: x=1 and x=3.
The vertex is at (2,−1).
The parabola opens upwards.
Example 2: Exponential Function
Problem: Solve for x in the equation 
Solution: Rewrite 16 as 
, so 
, thus x = 4.
Example 3: Rational Function
Problem: Determine the domain of the function 
Solution: The function is undefined when the denominator is zero, so x ≠ 3. The domain is all real numbers except x = 3.
Graphing Nonlinear Functions
Graphing nonlinear functions involves plotting points and understanding the shape of the function. Here are the steps to graph a nonlinear function:
- Identify the function type (quadratic, cubic, etc.)
- Determine key features: roots, intercepts, asymptotes, vertex, etc.
- Plot critical points and sketch the graph based on these points.
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SAT Advanced Math Exercises for Nonlinear Functions
To improve your math skills, we do not recommend using a calculator when solving these SAT Advanced Math Exercises.
Exercise I. 
The given equation defines the function f. What is the minimum value of f(x)?
Exercise II. The function f is defined by 
. What is the value of f(6)?
Exercise III. The function 
gives a spring’s length, in feet, when an object of w kilograms is hung, where 
. What is the best interpretation of 15 in this context?
, if a > 0, then the graph is upward. In the given equation, a = 3. So, the minimum value of the f(x) will be its vertex.
where a, h, and k are constants, (h,k) is the vertex point.
. We can rewrite as 
. Therefore;
. If we expand the parenthesis;
. This is in the form of 
. This yields x = 2. If we substitute x = 2 in the function equation;
. This yields;
. This yields;
.
. So, when there are no objects, the length of the spring is 15 feet. We can conclude that the initial length of the spring when no object is hung is 15 feet.Free Digital SAT Math Prep Course
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SAT Advanced Math Practice Test
We’ve listed 3 hard SAT Advanced Math practice test questions below. Note that this test does not resemble the typical question difficulty distribution on an SAT Advanced Math domain. Instead, we wanted to show you the hardest SAT Advanced Math questions you may see on the SAT.
Besides, since these are the hardest questions for the SAT Advanced Math, it is very normal that you will spend longer than usual time to solve each question. It is also super normal that you may score lower than your previous SAT Advanced Math Practice tests in this one. Because a typical Digital SAT Math Practice Test covers easy, medium, and hard questions. However, this one contains only the hardest questions.
Question 1
In the expression above, b and c are positive integers. If the expression is equivalent to x+b and x ≠ b, which of the following could be the value of c ?
A. 4
B. 6
C. 8
D. 10
Skill and Knowledge Testing Point: Equivalent expressions
Question 2
If 
and 
are the two solutions to the system of equations above, what is the value of 
?
A. -3
B. -2
C. -1
D. 1
Skill and Knowledge Testing Point: Nonlinear equations in one variable and systems of equations in two variables
Question 3
| x |  |
| 1 |  |
| 2 |  |
| 3 |  |
For the exponential function 
, the table above shows several values of 
and their corresponding values of , where is a constant greater than 1. If 
is a constant and 
, what is the value of ?
Skill and Knowledge Testing Point: Nonlinear functions
SAT Advanced Math Practice Test Answers and Rationales
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