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.

To find the rate of change from the graph of a function and determine whether the function is linear or nonlinear, follow these steps:

Finding the Rate of Change from a Graph

1. Select two points on the graph : Choose two points with clearly identifiable coordinates (x1,y1)(x_1,y_1)(x1​,y1​) and (x2,y2)(x_2,y_2)(x2​,y2​), ensuring they have different xxx-values.
2. Calculate the change in yyy: Compute the difference in the yyy-coordinates:

Δy=y2−y1\Delta y=y_2-y_1Δy=y2​−y1​

3. Calculate the change in xxx: Compute the difference in the xxx-coordinates:

Δx=x2−x1\Delta x=x_2-x_1Δx=x2​−x1​

4. Compute the rate of change (slope) :

Rate of change=ΔyΔx\text{Rate of change}=\frac{\Delta y}{\Delta x}Rate of change=ΔxΔy​

This value represents how much yyy changes for each unit change in xxx over the interval between the two points.

Determining if the Function is Linear or Nonlinear

• Linear function : If the graph is a straight line and the rate of change is constant between any two points on the graph, the function is linear. This means the slope ΔyΔx\frac{\Delta y}{\Delta x}ΔxΔy​ remains the same no matter which two points you select. Linear functions can be expressed in the form y=mx+by=mx+by=mx+b, where mmm is the constant slope and bbb is the yyy-intercept

• Nonlinear function : If the graph is curved or the rate of change varies between different intervals (i.e., the slope changes as you move along the graph), the function is nonlinear. Nonlinear graphs do not form a straight line, and their slopes are not constant

Summary

Aspect| Linear Function| Nonlinear Function
---|---|---
Graph shape| Straight line| Curved or non-straight line
Rate of change (slope)| Constant for all intervals| Changes depending on the interval
Equation form| y=mx+by=mx+by=mx+b (slope-intercept form)| Does not fit the linear form

By calculating the rate of change between multiple pairs of points on the graph and checking if it remains constant, you can confidently classify the function as linear or nonlinear

. This approach provides a clear, step-by-step method to analyze any function graphically.