Question

Explaining Linear Functions
Given a graph of a function, explain how to find the rate of change and how to determine whether it is a linear or nonlinear function.

Answer 1

Short answer: The rate of change is the slope of the graph: pick two points (x1,y1) and (x2,y2) and compute (y2−y1)/(x2−x1). The function is linear exactly when that slope is the same for every choice of two points (equivalently the graph is a straight line).

How to do it step by step from a given graph
1. Pick two clear points on the graph and read their coordinates: (x1,y1) and (x2,y2).
2. Compute the rate of change (slope): m = (y2 − y1) / (x2 − x1). Interpret as “change in y per change in x.”
3. To decide linear vs. nonlinear:
- If the graph is a straight line (visually), it is linear.
- Or check by choosing several different pairs of points; if every pair gives the same m, the function is linear.
- If m changes with different pairs (or the graph curves), the function is nonlinear.

Quick examples
- Example 1 (linear): Points (1,2) and (4,8) give m = (8−2)/(4−1) = 6/3 = 2. If a third point fits that same slope (e.g., (0,0) also gives slope 2 to (1,2)), the graph is a straight line → linear.
- Example 2 (nonlinear): Points (1,1), (2,4), (3,9). Slopes: (4−1)/(2−1)=3, (9−4)/(3−2)=5. Slopes differ → nonlinear (this is a curve).

Notes and edge cases
- Horizontal line: slope 0 (linear).
- Vertical line: slope undefined; vertical lines are not functions of x.
- For discrete data or a table, compute Δy/Δx for each interval; constant Δy/Δx → linear.
- For curves you can still compute an average rate of change on an interval (a secant slope); instantaneous rate of change needs a tangent.

If you have a specific graph or image, send it and I’ll compute the rate(s) of change and tell you whether it’s linear.