Lecture Notes: Linear Functions and Slope

Linear vs Non-linear Functions

Topic from transcript: identifying which functions among labeled options (C, D, E, G, etc.) are linear and which are not. There is some confusion in the dialogue (e.g., statements like “G is not linear” followed by “G is linear”). The core idea is to distinguish linear functions from non-linear ones based on a constant rate of change.
Key idea: a function is linear if it can be written in the form y = mx + b where:
- m is the slope (rate of change)
- b is the intercept (starting value when the input is zero)
In the context of the discussion, the value m represents the rate of change of the dependent variable with respect to the independent variable (e.g., time). The transcript explicitly notes that m represents the rate of change.
Non-linear functions do not have a constant rate of change; their graphs are not straight lines. The dialogue’s back-and-forth about which option is linear highlights the practical exercise students do in class to identify linearity.

The speaker states: "this m value represents the rate of change."
In a real-world context (population growth example mentioned in the transcript):
- If a population increases by 2000 people over 5 years, the rate of change (slope) is:
  m = rac{2000}{5} = 400 ext{ people per year}
- The corresponding linear model for population P as a function of time t (in years) would be:
  P(t) = P{0} + m t = P{0} + 400 t
- Here, 400 is the constant rate of change per year, illustrating a linear relationship with a constant slope.
General note: for a linear function y = mx + b, m = rac{dy}{dx} in continuous terms or m = rac{ riangle y}{ riangle x} for discrete changes.

Criteria from the content: a function is linear if it exhibits a constant rate of change and can be expressed in the form y = mx + b.
Practical checks:
- Try to fit the function to the form y = mx + b. If you can identify a constant slope m across different intervals, the function is linear.
- If the relation shows changing slope or curvature, it is not linear.
The dialogue shows a classroom exercise where students must determine which labeled function is not linear; this reinforces the idea of constant rate of change as the hallmark of linearity.

The lecturer invites a student to come to the board to solve problems, signaling active participation and practice with identifying linear vs non-linear relationships.
The line: "I will call anyone to come to the board and take the responsibility of the board." reflects a collaborative, problem-solving classroom culture.
Expectation set in the transcript: students will encounter many problems in the class focused on linear relationships and slope concepts.
The teacher also emphasizes repetition for clarity: "Repeat and everybody would be able to…" indicating a pedagogy of ensuring everyone follows.

There are moments of ambiguity in the transcript, such as:
- The statement "G is not linear" followed by "G is linear". This suggests a correction or uncertainty during the discussion.
- Phrases like "One is correct. One is transfer" appear to be transcription errors or miscommunications and are not mathematically definitive.
Note for study: in such exams, be prepared to verify linearity by checking the form and slope consistency, and be aware that spoken transcripts can contain inconsistencies that require applying the standard definitions.

Given a linear model with a constant rate of change m, the population P as a function of time t can be written as:
P(t) = P_{0} + m t
If the scenario states an increase of 2000 residents over 5 years, then:
m = rac{2000}{5} = 400
P(t) = P_{0} + 400 t
Interpretation: for each year, the population increases by 400 people under this linear model.

Linear function form: y = mx + b with constant slope m and intercept b.
Slope/m (rate of change) interpretation: how much y changes per unit change in x.
Example of rate-of-change calculation: m = rac{ riangle y}{ riangle x} or m = rac{dy}{dx} in calculus terms.
To identify linearity from data: check if the average rate of change is constant across intervals; if yes, likely linear; if not, non-linear.
Classroom dynamics: expect practice problems, board-work, and participation to reinforce linearity concepts.