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Approximating Rate of Change From Graphs & Tables

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  • 0:04 Rate of Change & Graphs
  • 0:55 How to Calculate Rate…
  • 2:57 Real-Life Example
  • 4:43 Lesson Summary
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Lesson Transcript
Instructor: Cameron Smith

Cameron has a Master's Degree in education and has taught HS Math for over 25 years.

This lesson will help you approximate the rate of change of a function from a graph or a table. You will look at some real-life examples and approximate the rate of change related to a specific situation.

Rate of Change & Graphs

You have taken some data about a falling ball. You want to approximate the rate of change of the height of the ball with respect to the time at t = 2 seconds. You can approximate this rate of change using information from the data you collected.

The rate of change of a function is the slope of the graph of the equation at a given point on the graph. The tangent line to the graph has the same slope as the graph at that point.


The tangent line to the graph has the same slope as the graph at that point.
Tangent Line


It can be hard to find the slope of the tangent line, so sometimes you can approximate it by looking at the slope that connects two points that are around the point you're interested in. You can also use the point of interest and another point near it. This line is called a secant line. The slopes of secant lines can be used to approximate the slope of the tangent line.


The secant line
Secant Line


Sometimes, the information about the two points you'll use can be found in a table, and sometimes it will come from a graph. Let's look at both!

How to Calculate Rate of Change

When you know the coordinates of two points on a graph you can calculate the slope of the line segment that connects them. Remember that the slope formula is given by:


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Now here's how we approximate the rate of change while using a table. Let's look at a table and approximate the rate of change between x = 3 and x = 6.


Table 1


We can then substitute the values from the table, using the points (3, 5.5) and (6, 7), into the slope formula which, as you can see, is:


Example 1


We can use this slope to approximate the rate of change of the graph at x-values in between x = 3 and x = 6. The approximate rate of change of the function is about 0.5. The closer the two x-values are to each other, the more accurate your approximation.

Now let's approximate the rate of change using a graph. If the information is from a graph, then you use the coordinates of two points from the graph to approximate the rate of change, or slope of the graph between the two points. You will need to estimate the coordinates of the two points as part of this process.

Let's approximate the slope of the graph at x = 2 in the following graph, appearing here:


Function Graph


You can start by looking between x = 1 and x = 2. It looks like the coordinates of the two points are (1, 1) and (2, 1.4). Now you can compare that calculation with the slope between the points when x = 2 and x = 4, which are the points (2, 1.4) and (4, 2).

Example 3

Notice with these formulas that the slopes are a little different. You can say that the approximate slope of the graph at x = 2 is between 0.4 and 0.3. It turns out the actual slope of the graph at x = 2 is 0.35, so the approximation was pretty accurate.

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