Standard Form of Linear Equations

This page focuses on the standard form of linear equations and provides examples of graphing linear functions.

Definition: The standard form of a linear equation is Ax + By = C, where A, B, and C are real numbers, and A and B are not both zero.

The lesson provides an example of graphing the linear equation 5x + y = 10:

1. Solve for y: y = 10 - 5x
2. Create a table of values
3. Plot points on a coordinate plane

Example: To determine if a point (2,5) is on the graph of 5x + y = 10, substitute the values: 5(2) + 5 = 15 ≠ 10, so (2,5) is not on the graph.

Highlight: When graphing linear functions, it's important to note that A and B in the standard form cannot be zero simultaneously.

Real-World Applications and Graphing Techniques

This page covers real-world applications of linear functions and introduces graphing techniques, including using intercepts.

Example: Sal's video store pricing model is represented by the function F$x$ = 2x, where x is the number of DVDs purchased. This results in a discrete graph.

The lesson explains how to determine the domain and range of this function:

• Domain: 1, 2, 3, 4, ... (number of DVDs purchased)
• Range: 2, 4, 6, 8, ... (cost in dollars)

Highlight: When graphing discrete functions, use individual points rather than a continuous line.

The lesson introduces a problem about miners ascending in an elevator, which will be used to demonstrate graphing linear functions using intercepts in the next lesson.

Vocabulary: Intercepts are the points where a graph crosses the x-axis (x-intercept) or y-axis (y-intercept).

This comprehensive guide provides students with a solid foundation in understanding and working with linear functions, preparing them for more advanced mathematical concepts.

Understanding Linear Functions

Linear functions are characterized by a constant rate of change between variables. This lesson introduces the concept of linear functions, their properties, and how to distinguish them from non-linear functions.

Definition: A linear function is a mathematical relationship where a constant change in one variable corresponds to a constant change in another variable.

The lesson emphasizes that all linear functions behave similarly and can be represented by linear equations with two variables. Examples of linear equations include:

• y = 3x + 2
• y = x
• 5x + y = 10

Example: The equation y = x² is not a linear function because it does not have a constant change in y for a constant change in x.

Highlight: In linear equations, the highest exponent for x and y is always 1.

The lesson also introduces non-linear equations, such as:

• y = x²
• y² = 3x + 5
• y = 4x⁵ - 2x⁴ + √x

Vocabulary: Non-linear functions are mathematical relationships where the rate of change between variables is not constant.