Average Rate of Change of a Function
Determine your x and y values. The average rate of change is the slope of a line that intersects two points on a curve. Since it’s just a straight line, it’s the change in y divided by the change in x. If you’re given the interval [x1, x2], plug each value into your equation to get y1 and y2. For example, if you had the function f(x) = x + 3x and the interval [1, 3], then f(1) = 1 + 3 * 1 = 1 + 3 = 4 and f(3) = 3 + 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 referred to as f(x) since it’s the value you get when you plug an x value into the function.
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}}} {\displaystyle {\frac {y2-y1}{x2-x1}}}. To get your numerator, subtract y1 from y2 (AKA f(x1) from f(x2)). Then find your denominator by subtracting x1 from x2. For the function f(x) = x + 3x over the interval [1, 3], the fraction would look like 18 − 4 3 − 1 {\displaystyle {\frac {18-4}{3-1}}} {\displaystyle {\frac {18-4}{3-1}}} = 14 2 {\displaystyle {\frac {14}{2}}} {\displaystyle {\frac {14}{2}}}.
Divide the change in y by the change in x. Once you’ve set up your fraction, divide the numerator by the denominator. This gives you the average rate of change, which is also the slope of something called a secant line. The average rate of change for the example function is 14 2 {\displaystyle {\frac {14}{2}}} {\displaystyle {\frac {14}{2}}} = 7.
Average Rate of Change From a Graph
Use the graph to find your x and y values. To find the x value of a point on a graph, draw a straight line from the point to the x-axis—you will either have to draw straight up or straight down. Where your line intersects the axis is your value. Find y by drawing a straight line to the y-axis, either to the left or right of the point. If your y value is negative, draw straight up for x. If it’s positive, draw straight down. If the x value is negative, draw right for y, and if it’s positive draw 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 either, but usually, people put the leftmost coordinates first (lowest value of x). So if you had the points (1, 3) and (5, 10), 1 would be x1 and 3 would be y1. If you put the right coordinates first your numerator and denominator will switch 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}}} {\displaystyle {\frac {y2-y1}{x2-x1}}} to find the average rate of change. So if you had (1, 3) and (5, 10), the slope would be 10 − 3 5 − 1 {\displaystyle {\frac {10-3}{5-1}}} {\displaystyle {\frac {10-3}{5-1}}} = 7 4 {\displaystyle {\frac {7}{4}}} {\frac {7}{4}}.
Average Rate of Change From a Table
Find your y or x values, given a certain interval. If you’re asked to find the rate of change of an interval with respect to x (AKA the interval is from one x value to another), find the x values on the table. Next to the x values are their corresponding y values. It’s less common to be given an interval with respect to y (two y values). If you find yourself in that situation, do the same thing, but look for the given values in the y column, instead.
Use the equation y 2 − y 1 x 2 − x 1 {\displaystyle {\frac {y2-y1}{x2-x1}}} {\displaystyle {\frac {y2-y1}{x2-x1}}} to find the average rate of change. Subtract the y value that corresponds with the first x value from the y value that corresponds to the second x value. Then, subtract the first x value from the second, and divide the difference in y by the difference in x.
Estimating Instantaneous Rate of Change
Choose two points on either side of where you want to estimate. The average rate of change is the same as the slope of a secant line, or a line that goes through two points of a curve. The instant rate of change is a tangent line, which only touches a single point. Choose two points that are close to where you want to find the instantaneous rate of change.
Calculate the average rate of change over the two points. Use the equation y 2 − y 1 x 2 − x 1 {\displaystyle {\frac {y2-y1}{x2-x1}}} {\displaystyle {\frac {y2-y1}{x2-x1}}} to find the slope of the secant line that intersects the two points. If you wanted to find the instantaneous rate of change for f(x) = x + 4x when x = 2, then you could make your first secant line at the points (1, 5) and (3, 21) The slope of that line is 21 − 5 3 − 1 {\displaystyle {\frac {21-5}{3-1}}} {\displaystyle {\frac {21-5}{3-1}}}, or 8.
Choose 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 instantaneous rate of change. If you have exponents in your function you will likely need to use a calculator. Try the rate of change at one decimal place on either side of your target. For the equation (x) = x + 4x, you could 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}}} {\displaystyle {\frac {12.81-11.21}{2.1-1.9}}} = 1.6 .2 {\displaystyle {\frac {1.6}{.2}}} {\displaystyle {\frac {1.6}{.2}}} = 8.
Estimate the tangent line based on the secant lines. Keep finding secant lines closer and closer to the point, until you start to see your results converge on a number. In the example above, a good guess would be that the instantaneous rate of change at x = 2 for f(x) = x + 4x is 8.
When to Use Average Rate of Change
Financial applications Investors analyze the speed at which stock prices change to draw conclusions about what a stock might do. If the rate of change of a stock’s price is rising, for example, it means the value will go up in the short term, and if it goes down the value may decline. Financial professionals also compare the rate of change of different variables to try and predict what will happen to the price of a stock. For example, if the price of a stock is rising while the rate of change is lowering, that can signal that the price is about to trend down.
Construction and architecture Architects and engineers need precise measurements to make 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 effects those variables have on profits. By comparing these businesses can see how certain products are performing. For example, if a company sees the rate of change of a low-cost product’s sales go up while a high-cost product’s goes down, they may lower production of the high-cost product.
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