1. Basic Algebraic Operations
1.1 Addition and Subtraction of Like Terms
- Combine coefficients of like terms.
- Example: \( 3x + 4x = (3 + 4)x = 7x \)
1.2 Multiplication and Division of Monomials
- Multiplication: \( a^m \cdot a^n = a^{m+n} \)
- Division: \( \frac{a^m}{a^n} = a^{m-n} \)
1.3 Distributive Property
- \( a(b + c) = ab + ac \)
- Example: \( 2(x + 3) = 2x + 6 \)
2. Solving Linear Equations
2.1 Standard Form of a Linear Equation
- The standard form is expressed as: \( Ax + By = C \)
- Where \( A \), \( B \), and \( C \) are integers, and \( A \) should be non-negative.
2.2 Slope-Intercept Form
- The slope-intercept form is given by: \( y = mx + b \)
- Where \( m \) is the slope and \( b \) is the y-intercept.
2.3 Point-Slope Form
- The point-slope form is written as: \( y - y_1 = m(x - x_1) \)
- Where \( (x_1, y_1) \) is a point on the line, and \( m \) is the slope.
3. Solving Inequalities
3.1 Properties of Inequalities
- If \( a < b \), then \( a + c < b + c \)
- If \( a < b \), then \( ac < bc \) (if \( c > 0 \)) and \( ac > bc \) (if \( c < 0 \))
3.2 Graphing Inequalities
- Use a number line for one-variable inequalities.
- For two-variable inequalities, shade above the line for \( y > mx + b \) and below for \( y < mx + b \).
4. Functions and Relations
4.1 Definition of a Function
- A relation where each input \( x \) has exactly one output \( y \).
4.2 Function Notation
- Functions are often represented as \( f(x) \).
- Example: If \( f(x) = 2x + 3 \), then \( f(2) = 2(2) + 3 = 7 \).
4.3 Evaluating Functions
- Substitute the input value into the function.
- Example: To evaluate \( f(x) = x^2 - 1 \) at \( x = 4 \):
\[
f(4) = 4^2 - 1 = 16 - 1 = 15
\]
5. Systems of Equations
5.1 Solving Systems by Substitution
- Solve one equation for one variable and substitute into the other.
5.2 Solving Systems by Elimination
- Align equations and eliminate one variable by adding or subtracting.
5.3 Graphical Method
- Graph both equations on the same coordinate plane.
- The intersection point(s) represent the solution(s).
6. Quadratic Equations
6.1 Standard Form of a Quadratic Equation
- A quadratic equation is expressed as: \( ax^2 + bx + c = 0 \)
6.2 Factoring Quadratics
- To factor \( ax^2 + bx + c \), look for two numbers that multiply to \( ac \) and add to \( b \).
6.3 Quadratic Formula
- The solutions to \( ax^2 + bx + c = 0 \) can be found using:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
6.4 Vertex Form of a Quadratic Equation
- The vertex form is expressed as: \( y = a(x - h)^2 + k \)
- Where \( (h, k) \) is the vertex of the parabola.
7. Polynomials
7.1 Polynomial Definition
- A polynomial is an expression composed of variables and coefficients, involving only non-negative integer exponents.
7.2 Adding and Subtracting Polynomials
- Combine like terms.
7.3 Multiplying Polynomials
- Use the distributive property or FOIL (First, Outside, Inside, Last) for binomials.
7.4 Polynomial Long Division
- Divide the leading term of the dividend by the leading term of the divisor.
8. Exponents and Radicals
8.1 Laws of Exponents
- Product of Powers: \( a^m \cdot a^n = a^{m+n} \)
- Power of a Power: \( (a^m)^n = a^{mn} \)
- Power of a Product: \( (ab)^n = a^n \cdot b^n \)
8.2 Simplifying Radicals
- Example: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \)
8.3 Rationalizing Denominators
- To eliminate radicals from the denominator: \( \frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a} \)
9. Statistics and Probability
9.1 Mean, Median, and Mode
- Mean: \( \text{Mean} = \frac{\sum x}{n} \)
- Median: The middle value when data is ordered.
- Mode: The value that appears most frequently.
9.2 Simple Probability
- Probability of an event: \( P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \)
9.3 Counting Principles
- Addition Rule: \( P(A \text{ or } B) = P(A) + P(B) \)
- Multiplication Rule: \( P(A \text{ and } B) = P(A) \cdot P(B) \)
Conclusion
In conclusion, the formulas in algebra 1 provide a comprehensive toolkit for students to solve various mathematical problems. Understanding these formulas is crucial for success in algebra and serves as a stepping stone for more advanced mathematics. By practicing the application of these formulas, students can build confidence and proficiency in their mathematical abilities, paving the way for future academic success.
Frequently Asked Questions
What is the formula for the slope of a line in slope-intercept form?
The slope-intercept form of a linear equation is given by the formula y = mx + b, where m represents the slope and b represents the y-intercept.
How do you calculate the area of a triangle using algebra?
The area of a triangle can be calculated using the formula A = 1/2 base height, where 'base' is the length of the base of the triangle and 'height' is the perpendicular height from the base to the opposite vertex.
What is the quadratic formula and when is it used?
The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a) and is used to find the solutions (roots) of a quadratic equation in the standard form ax² + bx + c = 0.
What is the formula for the distance between two points in a coordinate plane?
The distance d between two points (x1, y1) and (x2, y2) is calculated using the formula d = √((x2 - x1)² + (y2 - y1)²).
What is the formula for the perimeter of a rectangle?
The perimeter P of a rectangle can be calculated using the formula P = 2(length + width), where 'length' is the length of the rectangle and 'width' is the width.
How do you find the midpoint between two points?
The midpoint M between two points (x1, y1) and (x2, y2) is found using the formula M = ((x1 + x2)/2, (y1 + y2)/2).
What is the formula for solving systems of equations using substitution?
To solve a system of equations using substitution, solve one equation for one variable and substitute that expression into the other equation, then solve for the remaining variable.
