Proving Equal Differences in Linear Functions over Equal Intervals

The video tutorial explains that linear functions grow by equal differences over equal intervals. It begins by introducing linear functions as equations of the form y = mx + b, with graphs as straight lines. The tutorial demonstrates that for any interval length, the difference in y-coordinates is consistent, depending only on the slope and interval length. A general proof is provided, showing that the difference in heights is independent of specific x-coordinates. An example with a specific line equation further illustrates this concept, confirming that the difference in y-coordinates remains constant for equal intervals.

What is the general form of a linear function?

• y = ax^3 + bx^2 + cx + d

• y = a/x + b

• y = ax^2 + bx + c

• y = mx + b

In the context of linear functions, what does the term 'equal intervals' refer to?

• Intervals with the same starting point

• Intervals with the same slope

• Intervals with the same height on the y-axis

• Intervals with the same length on the x-axis

What is the significance of the variable 'k' in the algebraic proof?

• It represents the length of the interval on the x-axis

• It represents the slope of the line

• It represents the y-intercept of the line

• It represents the x-coordinate of a point

In the algebraic proof, what does the expression 'mk' represent?

• The difference in y-coordinates over an interval

• The slope of the line

• The x-coordinate of a point

• The y-intercept of the line

Why does the difference in y-coordinates not depend on the specific x-coordinate 'w'?

• Because it depends on the x-intercept

• Because it depends on the y-intercept

• Because it depends on the curvature of the line

• Because it only depends on the slope and interval length

In the example provided, what is the difference in y-coordinates when the interval length is 5?

• 15

• 10

• 5

• 20

What does the example with the line equation demonstrate about linear functions?

• They grow by equal differences over equal intervals

• They are always increasing

• They have a constant y-intercept

• They have a variable slope