Lesson

Welcome to Algebra II: A Review of Algebra I Concepts

Before we dive into the complexities of Algebra II, let's take some time to refresh our understanding of the fundamental concepts learned in Algebra I. This review will ensure a solid foundation upon which we can build more advanced algebraic skills. We will revisit linear equations, inequalities, and functions – the building blocks of algebra.

Linear Equations: The Basics

Linear equations are equations that can be written in the form \( ax+b=c \) , where \( a \) , \( b \) , and \( c \) are constants and \( x \) is the variable. Solving a linear equation involves isolating the variable to find its value. Remember the key principle: whatever operation you perform on one side of the equation, you must perform on the other side to maintain equality.

Solving Linear Equations: A Step-by-Step Approach

Here's a general strategy for solving linear equations:

1. Simplify: Combine like terms on both sides of the equation.
2. Isolate the variable term: Use addition or subtraction to get the term with the variable alone on one side.
3. Solve for the variable: Use multiplication or division to isolate the variable and find its value.

For example, let's solve the equation \( 2x+3=7 \) :

1. Subtract 3 from both sides: \( 2x=4 \)
2. Divide both sides by 2: \( x=2 \)

Linear Inequalities: Beyond Equality

Linear inequalities are similar to linear equations, but instead of an equals sign, they use inequality symbols such as \( < \) (less than), \( > \) (greater than), \( \leq \) (less than or equal to), or \( \geq \) (greater than or equal to). Solving inequalities is very similar to solving equations, with one important exception: when you multiply or divide both sides by a negative number, you must reverse the inequality sign.

Solving Linear Inequalities: Key Differences

Consider the inequality \( -3x<9 \) . To solve for \( x \) , we divide both sides by -3. Because we are dividing by a negative number, we must flip the inequality sign, resulting in \( x>-3 \) .

The solution to an inequality is a range of values, not just a single value. We can represent this range on a number line. A closed circle indicates that the endpoint is included in the solution (for \( \leq \) or \( \geq \) ), while an open circle indicates that the endpoint is not included (for \( < \) or \( > \) ).

Graphing Linear Equations

Linear equations can be represented graphically as straight lines on the coordinate plane. The standard form of a linear equation is \( y=mx+b \) , where \( m \) represents the slope of the line and \( b \) represents the y-intercept (the point where the line crosses the y-axis). The slope \( m \) describes the steepness and direction of the line. It is calculated as the "rise over run," or the change in \( y \) divided by the change in \( x \) .

Slope-Intercept Form

The slope-intercept form, \( y=mx+b \) , makes it easy to graph a linear equation. Start by plotting the y-intercept \( (0,b) \) . Then, use the slope \( m \) to find another point on the line. For example, if \( m=2/3 \) , move 2 units up and 3 units to the right from the y-intercept. Draw a line through these two points to graph the equation.

Functions: Relationships Between Variables

A function is a relationship between two sets of elements, called the domain and the range. For every element in the domain, there is exactly one corresponding element in the range. We often represent functions using the notation \( f(x) \) , where \( x \) is the input (an element from the domain) and \( f(x) \) is the output (the corresponding element in the range). \( f(x) \) is read as "f of x".

Function Notation

For the function \( f(x)=3x+2 \) , if we want to find the value of the function when \( x=4 \) , we substitute 4 for \( x \) in the equation: \( f(4)=3(4)+2=14 \) . Therefore, \( f(4)=14 \) . This means that when the input is 4, the output is 14.

Domain and Range

The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values). For example, for the function \( f(x)=\sqrt{x} \) , the domain is all non-negative real numbers (since we can't take the square root of a negative number), and the range is also all non-negative real numbers.

Linear Functions

A linear function is a function whose graph is a straight line. It can be written in the form \( f(x)=mx+b \) , where \( m \) is the slope and \( b \) is the y-intercept. Linear functions have a constant rate of change, meaning that the slope is the same between any two points on the line.

Identifying Linear Functions

To determine if a function is linear, check if the rate of change between any two points is constant. If the rate of change is constant, the function is linear. If the rate of change varies, then the function is non-linear.