Given a graph of a function, you find the rate of change by looking at how much the yyy-values change when the xxx-values increase, and you decide if it is linear or nonlinear by checking whether the graph is a straight line or a curve.

Quick Scoop

Imagine you’re watching a car move along a road on a map.
The rate of change tells you how fast its position is changing, and the graph’s shape tells you whether that speed stays the same (linear) or keeps changing (nonlinear).

Step 1: Finding the Rate of Change from a Graph

For a graph, “rate of change” is the same idea as slope : how steep the graph is.

  1. Pick two clear points on the graph
    • Choose points where the graph exactly crosses grid intersections, like (x1,y1)(x_1,y_1)(x1​,y1​) and (x2,y2)(x_2,y_2)(x2​,y2​).
  1. Find the change in xxx
    • Compute Δx=x2−x1\Delta x=x_2-x_1Δx=x2​−x1​.
    • This is how far you move horizontally.
  1. Find the change in yyy
    • Compute Δy=y2−y1\Delta y=y_2-y_1Δy=y2​−y1​.
    • This is how far you move vertically.
  1. Use the slope formula (rate of change)
    • Rate of change =ΔyΔx=y2−y1x2−x1=\dfrac{\Delta y}{\Delta x}=\dfrac{y_2-y_1}{x_2-x_1}=ΔxΔy​=x2​−x1​y2​−y1​​.
    • If this is positive, the graph is going up as xxx increases; if negative, it’s going down.

Mini story:
Think of climbing stairs. Each step right (change in xxx) comes with some height change (change in yyy). The ratio “rise per step” is your rate of change.

If the graph is linear , this slope will be the same no matter which two points you pick on the line.

Step 2: Checking if the Function Is Linear or Nonlinear (from the Graph)

Use the shape of the graph and the behavior of the slope.

A. Linear Function (straight line)

A function is linear if:

  • Its graph is a perfect straight line.
  • The rate of change (slope) is constant everywhere on the graph.

On the graph, that means:

  • If you pick different pairs of points on the graph and compute ΔyΔx\dfrac{\Delta y}{\Delta x}ΔxΔy​, you always get the same number.
  • The line doesn’t bend, curve, or change steepness.

This matches the typical equation form y=mx+by=mx+by=mx+b, where mmm is the constant rate of change.

B. Nonlinear Function (curved or changing)

A function is nonlinear if:

  • Its graph is not a straight line — it might curve, bend, or change direction.
  • The rate of change is not constant ; the steepness changes as you move along the graph.

On the graph, that looks like:

  • Arcs, parabolas (U-shapes), waves, or any graph that clearly bends.
  • If you calculate ΔyΔx\dfrac{\Delta y}{\Delta x}ΔxΔy​ between different pairs of points, you get different values.

Side-by-Side View: Linear vs Nonlinear on a Graph

Here’s a quick comparison.

html

<table>
  <tr>
    <th>Feature</th>
    <th>Linear Function</th>
    <th>Nonlinear Function</th>
  </tr>
  <tr>
    <td>Shape on graph</td>
    <td>Straight line [web:3][web:5][web:7]</td>
    <td>Curve, bend, or changing shape [web:3][web:5][web:7]</td>
  </tr>
  <tr>
    <td>Rate of change (slope)</td>
    <td>Constant everywhere [web:3][web:7][web:9]</td>
    <td>Changes from point to point [web:3][web:7][web:6]</td>
  </tr>
  <tr>
    <td>Example behavior</td>
    <td>“Every hour, you earn 15 dollars” – same amount each hour [web:5][web:7]</td>
    <td>“The speed of a falling object increases over time” – gets steeper as time goes on [web:6][web:3]</td>
  </tr>
  <tr>
    <td>Typical equation form</td>
    <td>Looks like y = mx + b [web:3][web:5]</td>
    <td>Includes exponents, roots, or curves (like y = x², y = 3x³ + x² − 7) [web:3][web:5]</td>
  </tr>
</table>

Putting It All Together (Quick Checklist)

When you are given a graph of a function , do this:

  1. Find the rate of change
    • Pick two points on the graph.
    • Compute y2−y1x2−x1\dfrac{y_2-y_1}{x_2-x_1}x2​−x1​y2​−y1​​.
    • That value is the rate of change between those two points.
  1. Check if it’s linear or nonlinear
    • Look at the overall shape :
      • Straight line → linear.
      • Curved or bending → nonlinear.
 * Optionally, test with multiple point pairs:
   * Same slope each time → **linear**.
   * Different slopes → **nonlinear**.

Tiny Story Wrap-Up + TL;DR

Think of a jogger on a treadmill:

  • If the treadmill stays at exactly 6 mph the entire time, their distance vs. time graph is a straight line with constant rate of change → linear.
  • If the treadmill keeps speeding up and slowing down, the graph bends and the steepness changes → nonlinear.

TL;DR:

  • Rate of change from a graph: pick two points, use ΔyΔx\dfrac{\Delta y}{\Delta x}ΔxΔy​.
  • Linear if the graph is a straight line with a constant rate of change.
  • Nonlinear if the graph curves and the rate of change is not constant.

Information gathered from public forums or data available on the internet and portrayed here.