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The average rate of change is the slope of the line that intersects the two points on the curve. Finding it is a classic pre-calculus problem, and it’s as easy as calculating the slope of a straight line! All you have to do is find the xiy values for both points, subtract the first y value from the second, and divide it by the second x value minus the first.
The equation for the average rate of change is y 2 − y 1 x 2 − x 1 {\displaystyle {\frac {y2-y1}{x2-x1}}}
or the difference in y values divided by the difference in x values.
or the difference in y values divided by the difference in x values.
Determine your xiy values. The average rate of change is the slope of the line that intersects the two points on the curve. Since it’s just a straight line, it’s the change in uy divided by the change in x.[1]
If you are given an interval [x1, x2]plug each value into your equation to get y1 and y2.- For example, if you have a function f(x) = x2 + 3x and an interval [1, 3]then f(1) = 12 + 3 * 1 = 1 + 3 = 4 if(3) = 32 + 3 * 3 = 9 + 9 = 18. So your x values are 1 and 3 and your y values are 4 and 18
- The variable y is also called f(x) since it is the value you get when you plug the x value into the function.
- For example, if you have a function f(x) = x2 + 3x and an interval [1, 3]then f(1) = 12 + 3 * 1 = 1 + 3 = 4 if(3) = 32 + 3 * 3 = 9 + 9 = 18. So your x values are 1 and 3 and your y values are 4 and 18
Subtract y1 from y2 and x1 from x2. The equation for the average rate of change is y 2 − y 1 x 2 − x 1 {\displaystyle {\frac {y2-y1}{x2-x1}}}
. To get the numerator, subtract y1 from y2 (also known as f(x1) from f(x2)). Then find your denominator by subtracting x1 from x2.[2]
- For the function f(x) = x2 + 3x over the interval [1, 3]the fraction would look like 18 − 4 3 − 1 {\displaystyle {\frac {18-4}{3-1}}}
= 14 2 {\displaystyle {\frac {14}{2}}}
.
- For the function f(x) = x2 + 3x over the interval [1, 3]the fraction would look like 18 − 4 3 − 1 {\displaystyle {\frac {18-4}{3-1}}}
Divide the change in ys by the change in x. Once you have set up the fraction, divide the numerator by the denominator. This gives you the average rate of change, which is also the slope of something called the secant.[3]
- The average rate of change for the example function is 14 2 {\displaystyle {\frac {14}{2}}}
= 7.
- The average rate of change for the example function is 14 2 {\displaystyle {\frac {14}{2}}}
= 14 2 {\displaystyle {\frac {14}{2}}}
.Use the graph to find your xiy values. To find the x value of a point on the graph, draw a straight line from the point to the x-axis — you’ll need to draw straight up or straight down. Where your line intersects the axis is your value. Find y by drawing a straight line on the y-axis, either to the left or right of the point.[4]
- If your y value is negative, draw straight up for x. If it’s positive, draw straight down. If the value of x is negative, draw to the right of y, and if it is positive, draw to the left.
Determine the values for y1, y2, x1, and x2. Choose one set of points to be (x1, y1) and the other to be (x2, y2). You can choose any, but people usually put the leftmost coordinates (the lowest x value) first.[5]
- So if you have the points (1, 3) and (5, 10), 1 would be x1 and 3 would be y1.
- If you put in the correct coordinates first, your numerator and denominator will change from positive to negative or from negative to positive. However, the equation gives you the same answer either way.
Divide the difference of the y’s by the difference of the x’s. Use the equation y 2 − y 1 x 2 − x 1 {\displaystyle {\frac {y2-y1}{x2-x1}}}
find the average rate of change.[6]
- So if you have (1, 3) and (5, 10), the slope would be 10 − 3 5 − 1 {\displaystyle {\frac {10-3}{5-1}}}
= 7 4 {\displaystyle {\frac {7}{4}}}
.
- So if you have (1, 3) and (5, 10), the slope would be 10 − 3 5 − 1 {\displaystyle {\frac {10-3}{5-1}}}
= 7 4 {\displaystyle {\frac {7}{4}}}
.Find your y or x values for the given interval. If you are asked to find the rate of change of an interval with respect to x (that is, the interval is from one x value to another), find the x values in the table. Next to the x values are their corresponding y values.[7]
- It is less common to get an interval with respect to y (two y values). If you find yourself in that situation, do the same thing, but look for the default values in the y column instead.
Use the equation
y 2 − y 1 x 2 − x 1
{\displaystyle {\frac {y2-y1}{x2-x1}}}
find the average rate of change. Subtract the y value corresponding to the first x value from the y value corresponding to the second x value. Then subtract the first x value from the second and divide the difference uys by the difference in x.[8]
Select two points on each side of where you want to estimate. The average rate of change is equal to the slope of the secant or line passing through two points of the curve. The instantaneous rate of change is a tangent that touches only one point.[9]
Pick two points that are close to where you want to find the current rate of change.Calculate the average rate of change at the two points. Use the equation y 2 − y 1 x 2 − x 1 {\displaystyle {\frac {y2-y1}{x2-x1}}}
find the slope of the secant that intersects the two points.[10]
- If you want to find the instantaneous rate of change for f(x) = x2 + 4x when x = 2, then you can make your first sequence at the points (1, 5) and (3, 21). The slope of that line is 21 − 5 3 − 1 {\displaystyle {\frac {21-5}{3-1}}}
or 8.
- If you want to find the instantaneous rate of change for f(x) = x2 + 4x when x = 2, then you can make your first sequence at the points (1, 5) and (3, 21). The slope of that line is 21 − 5 3 − 1 {\displaystyle {\frac {21-5}{3-1}}}
Select two more points, both closer to the target point. Find the average rate of change over an interval even closer to where you want to find the current rate of change.[11]
If you have exponents in your function, you will probably need to use a calculator.- Try the rate of change to one decimal place on either side of your target. For the equation (x) = x2 + 4x, you can use x1 = 1.9 and x2 = 2.1. 12.81 − 11.21 2.1 − 1.9 {\displaystyle {\frac {12.81-11.21}{2.1-1.9}}}
= 1.6 .2 {\displaystyle {\frac {1.6}{.2}}}
= 8.
- Try the rate of change to one decimal place on either side of your target. For the equation (x) = x2 + 4x, you can use x1 = 1.9 and x2 = 2.1. 12.81 − 11.21 2.1 − 1.9 {\displaystyle {\frac {12.81-11.21}{2.1-1.9}}}
Estimate the tangent based on the secants. Keep finding the secants closer and closer to the point, until you start to see your results converge to a number.[12]
In the example above, a good guess would be that the instantaneous rate of change at x = 2 for f(x) = x2 + 4x 8.
or 8.
= 1.6 .2 {\displaystyle {\frac {1.6}{.2}}}
= 8.Financial applications Investors analyze the speed at which stock prices change to draw conclusions about what a stock can do. If, for example, the rate of change of a stock’s price is rising, it means that the value will rise in the short term, and if it falls, the value could fall.[13]
- Financial experts also compare the rate of change of different variables to try to predict what will happen to a stock’s price. For example, if the price of a stock is increasing while the rate of change is decreasing, this may signal that the price is about to move down.[14]
Construction and architecture Architects and engineers need precise measurements to create structural plans for buildings, bridges and roads, including the slope of an area or the average rate of change in elevation.
Sales Retailers use average rates of change in sales, prices, and margins to see what effect these variables have on profits. By comparing these companies, you can see how certain products perform.
- For example, if a company sees the rate of change in sales of an inexpensive product increasing while that of an expensive product is falling, it may reduce production of the expensive product.[15]
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