Rate of Change Calculator
The slope of a line is a rate of change:
( x1 , y1 ) ( x2 , y2 )
Rate of change (Slope or m) =
The rate of change calculator is a free online tool that gives the change in slope for the given input coordinate points. BYJU’S online rate of change calculator tool makes the calculations faster and easier where it displays the result in a fraction of seconds.
How to Use the Rate of Change Calculator?
The procedure to use the rate of change calculator is as follows:
Step 1: Enter the X and Y coordinate points in the given input field. I.e., (x1, y1) and (x2, y2)
Step 2: Now click the button “calculate Rate of Change” to get the output
Step 3: The result will be displayed in the output field
What is the Rate of Change?
A rate of change defines how one quantity changes in relation to another quantity. The rate of change can be either positive or negative. Since the slope of a line is the ratio of vertical and horizontal change between two points on the plane or a line, then the slope equals the ratio of the rise and the run.
Where,
Rise is the vertical change between two points
Run is the horizontal change between two points
Standard Formula
The standard form for the rate of change of slope, m is given by
Rate of change = Rise/ Run
= Δy / Δx
Therefore, the rate of change of slope =(y2 – y1) / (x2 – x1)
Where
Δ represents the rate of change
(x1, x2) is the X coordinates
(y1, y2) is the Y coordinates
Frequently Asked Questions on Rate of Change Calculator
The slope represents the gradient of the line. It is
denoted by the symbol “m”. It defines the rate of change. A slope can be either positive or negative. If a line has a positive slope (i.e., m > 0), then y always increases when x increases and y always decreases when x decreases. Also, if a line
has a negative slope (i.e., m<0), then y always increases when x decreases and y always decreases when x increases. The formula to find the slope is m = (y2 – y1) / (x2 – x1) Where (x1, x2) is the X coordinates (y1, y2) is the Y coordinatesWhat is meant by slope?
Can a slope be negative?
What is the formula for the slope?
A rate of change is a rate that describes how one quantity changes in relation to another quantity. If x is the independent variable and y is the dependent variable, then
rate of change = change in y change in x
Rates of change can be positive or negative. This corresponds to an increase or decrease in the y -value between the two data points. When a quantity does not change over time, it is called zero rate of change.
Positive rate of change
When the value of x increases, the value of y increases and the graph slants upward.
Negative rate of change
When the value of x increases, the value of y decreases and the graph slants downward.
Zero rate of change
When the value of x increases, the value of y remains constant. That is, there is no change in y value and the graph is a horizontal line .
Example:
Use the table to find the rate of change. Then graph it.
Time Driving ( h ) x Distance Travelled ( mi ) y 2 80 4 160 6 240
A rate of change is a rate that describes how one quantity changes in relation to another quantity.
rate of change = change in y change in x = change in distance change in time = 160 − 80 4 − 2 = 80 2 = 40 1
The rate of change is 40 1 or 40 . This means a vehicle is traveling at a rate of 40 miles per hour.
Learning Outcomes
- Find the average rate of change of a function.
Gasoline costs have experienced some wild fluctuations over the last several decades. The table below[1] lists the average cost, in dollars, of a gallon of gasoline for the years 2005–2012. The cost of gasoline can be considered as a function of year.
| [latex]y[/latex] | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 |
| [latex]C\left(y\right)[/latex] | 2.31 | 2.62 | 2.84 | 3.30 | 2.41 | 2.84 | 3.58 | 3.68 |
If we were interested only in how the gasoline prices changed between 2005 and 2012, we could compute that the cost per gallon had increased from $2.31 to $3.68, an increase of $1.37. While this is interesting, it might be more useful to look at how much the price changed per year. In this section, we will investigate changes such as these.
The price change per year is a rate of change because it describes how an output quantity changes relative to the change in the input quantity. We can see that the price of gasoline in the table above did not change by the same amount each year, so the rate of change was not constant. If we use only the beginning and ending data, we would be finding the average rate of change over the specified period of time. To find the average rate of change, we divide the change in the output value by the change in the input value.
[latex]\begin{align}\text{Average rate of change} &=\frac{\text{Change in output}}{\text{Change in input}}\\[2mm] &=\frac{\Delta y}{\Delta x}\\[2mm] &=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\\[2mm] &=\frac{f\left({x}_{2}\right)-f\left({x}_{1}\right)}{{x}_{2}-{x}_{1}}\end{align}[/latex]
The Greek letter [latex]\Delta [/latex] (delta) signifies the change in a quantity; we read the ratio as “delta-[latex]y[/latex] over delta-[latex]x[/latex]” or “the change in [latex]y[/latex] divided by the change in [latex]x[/latex].” Occasionally we write [latex]\Delta f[/latex] instead of [latex]\Delta y[/latex], which still represents the change in the function’s output value resulting from a change to its input value.
In our example, the gasoline price increased by $1.37 from 2005 to 2012. Over 7 years, the average rate of change was
[latex]\dfrac{\Delta y}{\Delta x}=\dfrac{{1.37}}{\text{7 years}}\approx 0.196\text{ dollars per year}[/latex]
On average, the price of gas increased by about 19.6¢ each year.
Other examples of rates of change include:
- A population of rats increasing by 40 rats per week
- A car traveling 68 miles per hour (distance traveled changes by 68 miles each hour as time passes)
- A car driving 27 miles per gallon of gasoline (distance traveled changes by 27 miles for each gallon)
- The current through an electrical circuit increasing by 0.125 amperes for every volt of increased voltage
- The amount of money in a college account decreasing by $4,000 per quarter
A General Note: Rate of Change
A rate of change describes how an output quantity changes relative to the change in the input quantity. The units on a rate of change are “output units per input units.”
The average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values.
[latex]\dfrac{\Delta y}{\Delta x}=\dfrac{f\left({x}_{2}\right)-f\left({x}_{1}\right)}{{x}_{2}-{x}_{1}}[/latex]
How To: Given the value of a function at different points, calculate the average rate of change of a function for the interval between two values [latex]{x}_{1}[/latex] and [latex]{x}_{2}[/latex].
- Calculate the difference [latex]{y}_{2}-{y}_{1}=\Delta y[/latex].
- Calculate the difference [latex]{x}_{2}-{x}_{1}=\Delta x[/latex].
- Find the ratio [latex]\dfrac{\Delta y}{\Delta x}[/latex].
Example: Computing an Average Rate of Change
Using the data in the table below, find the average rate of change of the price of gasoline between 2007 and 2009.
| [latex]y[/latex] | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 |
| [latex]C\left(y\right)[/latex] | 2.31 | 2.62 | 2.84 | 3.30 | 2.41 | 2.84 | 3.58 | 3.68 |
The following video provides another example of how to find the average rate of change between two points from a table of values.
Try It
Using the data in the table below, find the average rate of change between 2005 and 2010.
| [latex]y[/latex] | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 |
| [latex]C\left(y\right)[/latex] | 2.31 | 2.62 | 2.84 | 3.30 | 2.41 | 2.84 | 3.58 | 3.68 |
Example: Computing Average Rate of Change from a Table
After picking up a friend who lives 10 miles away, Anna records her distance from home over time. The values are shown in the table below. Find her average speed over the first 6 hours.
| t (hours) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| D(t) (miles) | 10 | 55 | 90 | 153 | 214 | 240 | 282 | 300 |
Example: Computing Average Rate of Change from a Graph
Given the function [latex]g\left(t\right)[/latex], find the average rate of change on the interval [latex]\left[-1,2\right][/latex].
Example: Computing Average Rate of Change for a Function Expressed as a Formula
Compute the average rate of change of [latex]f\left(x\right)={x}^{2}-\frac{1}{x}[/latex] on the interval [latex]\text{[2,}\text{4].}[/latex]
The following video provides another example of finding the average rate of change of a function given a formula and an interval.
Try It
Find the average rate of change of [latex]f\left(x\right)=x - 2\sqrt{x}[/latex] on the interval [latex]\left[1,9\right][/latex].
Example: Finding the Average Rate of Change of a Force
The electrostatic force [latex]F[/latex], measured in newtons, between two charged particles can be related to the distance between the particles [latex]d[/latex], in centimeters, by the formula [latex]F\left(d\right)=\frac{2}{{d}^{2}}[/latex]. Find the average rate of change of force if the distance between the particles is increased from 2 cm to 6 cm.
Example: Finding an Average Rate of Change as an Expression
Find the average rate of change of [latex]g\left(t\right)={t}^{2}+3t+1[/latex] on the interval [latex]\left[0,a\right][/latex]. The answer will be an expression involving [latex]a[/latex].
Try It
Find the average rate of change of [latex]f\left(x\right)={x}^{2}+2x - 8[/latex] on the interval [latex]\left[5,a\right][/latex].
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