Algebra
Algebra is a fundamental component of the SAT Math section. Familiarity with algebraic expressions, equations, and inequalities is crucial for solving various problems.
Basic Algebraic Formulas
1. Linear Equations:
The standard form of a linear equation is:
\[
Ax + By = C
\]
where \(A\), \(B\), and \(C\) are constants.
2. Slope-Intercept Form:
This form is expressed as:
\[
y = mx + b
\]
where \(m\) is the slope and \(b\) is the y-intercept.
3. Quadratic Equations:
The standard form of a quadratic equation is:
\[
ax^2 + bx + c = 0
\]
The solutions can be found using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
4. Factoring:
If \(x^2 + bx + c = 0\) can be factored, it can be expressed as:
\[
(x - p)(x - q) = 0
\]
where \(p\) and \(q\) are the roots of the equation.
5. Exponential Functions:
The general formula for exponential growth or decay is:
\[
A = A_0 e^{kt}
\]
where \(A_0\) is the initial amount, \(k\) is the rate, and \(t\) is time.
Systems of Equations
- Substitution Method: Solve one equation for one variable and substitute it into the other equation.
- Elimination Method: Add or subtract equations to eliminate one variable, making it easier to solve for the other.
Geometry
Geometry is another significant aspect of the SAT Math section, covering various shapes, angles, and properties.
Basic Geometric Formulas
1. Area Formulas:
- Rectangle:
\[
A = l \times w
\]
- Triangle:
\[
A = \frac{1}{2} \times b \times h
\]
- Circle:
\[
A = \pi r^2
\]
2. Perimeter Formulas:
- Rectangle:
\[
P = 2l + 2w
\]
- Triangle:
\[
P = a + b + c
\]
- Circle (Circumference):
\[
C = 2\pi r
\]
3. Volume Formulas:
- Rectangular Prism:
\[
V = l \times w \times h
\]
- Cylinder:
\[
V = \pi r^2 h
\]
- Sphere:
\[
V = \frac{4}{3}\pi r^3
\]
4. Pythagorean Theorem:
For any right triangle:
\[
a^2 + b^2 = c^2
\]
where \(c\) is the hypotenuse.
Special Right Triangles
- 45-45-90 Triangle:
\[
\text{Leg} = x, \quad \text{Hypotenuse} = x\sqrt{2}
\]
- 30-60-90 Triangle:
\[
\text{Short leg} = x, \quad \text{Long leg} = x\sqrt{3}, \quad \text{Hypotenuse} = 2x
\]
Statistics and Probability
Understanding statistics and probability can help you interpret data sets and make informed predictions.
Basic Statistical Formulas
1. Mean (Average):
\[
\text{Mean} = \frac{\sum x}{n}
\]
where \(x\) represents each value, and \(n\) is the number of values.
2. Median:
The middle value when a data set is ordered from least to greatest. If there are two middle numbers, the median is their average.
3. Mode:
The value that appears most frequently in a data set.
4. Range:
\[
\text{Range} = \text{Maximum} - \text{Minimum}
\]
Probability Basics
1. Probability Formula:
\[
P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}
\]
2. Combined Probability:
- For independent events \(A\) and \(B\):
\[
P(A \text{ and } B) = P(A) \times P(B)
\]
- For mutually exclusive events:
\[
P(A \text{ or } B) = P(A) + P(B)
\]
Functions
Functions are integral to the SAT Math section, as they involve relationships between variables.
Function Basics
1. Function Notation:
A function \(f\) of \(x\) is written as:
\[
f(x) = mx + b
\]
2. Evaluating Functions:
To find \(f(a)\), substitute \(a\) into the function:
\[
f(a) = ma + b
\]
3. Inverse Functions:
If \(y = f(x)\), then the inverse function \(f^{-1}(y)\) can be found by solving for \(x\) in terms of \(y\).
Tips for Using Formulas Effectively
1. Memorization: Create flashcards to memorize key formulas.
2. Practice Problems: Apply formulas in practice problems to reinforce understanding.
3. Understand Context: Don’t just memorize; understand how and when to use each formula.
4. Time Management: Familiarize yourself with the formulas to save time during the test.
Conclusion
Mastering the formulas for SAT Math is critical for achieving a high score on the mathematics section of the SAT. By categorizing formulas into algebra, geometry, statistics, and functions, you can create a structured approach to study and review these essential tools. As you prepare, remember to practice applying these formulas in various contexts, and familiarize yourself with the types of questions you may encounter. With diligence and the right strategies, you’ll be well-equipped to tackle the SAT Math section with confidence.
Frequently Asked Questions
What are the key formulas for geometry on the SAT Math section?
Key geometry formulas include the area of a triangle (1/2 base height), the area of a rectangle (length width), the circumference of a circle (2 π radius), and the Pythagorean theorem (a² + b² = c²).
How do you calculate the slope of a line in the SAT Math section?
The slope of a line is calculated using the formula (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.
What is the formula for the quadratic equation and how is it used on the SAT?
The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a). It is used to find the roots of a quadratic equation ax² + bx + c = 0.
Which formulas are important for solving systems of equations on the SAT?
Important formulas include substitution (solving one equation for a variable and substituting it into another) and elimination (adding or subtracting equations to eliminate a variable).
What is the formula for the area of a circle and how can it be applied?
The area of a circle is given by the formula A = π r², where r is the radius. This can be applied to word problems involving circular objects or shapes.
How do you find the distance between two points on the SAT?
The distance formula is d = √((x2 - x1)² + (y2 - y1)²), which is used to calculate the distance between two points (x1, y1) and (x2, y2) on a coordinate plane.
What is the formula for compound interest and why is it relevant for the SAT?
The formula for compound interest is A = P(1 + r/n)^(nt), where A is the amount of money accumulated after n years, P is the principal amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years. It is relevant for financial word problems.
What is the formula for the volume of a cylinder and how is it used?
The volume of a cylinder is given by the formula V = π r² h, where r is the radius and h is the height. This can be used in problems involving three-dimensional objects.
