How do you find the rate of change?

The rate of change measures how a quantity changes over time or across space. It's often represented as the slope of a line connecting two points on a graph.

Rate of change generally refers to how something changes with respect to something else. For example, velocity is the rate of change of distance with respect to time, and acceleration is the rate of change of velocity with respect to time.

The average rate of change represents the average rate at which something changes from one point to another.

So you need to set up the equation like this:

Rate of change of {x} = (change in x) divided by time taken for the change to happen.

E.g. Rate of change of distance is acceleration.

a = d2 - d1 / t

This is an easy thing to find out. If you have a straight line on a X-Y graph, the formula you would use is y=mx+c. The rate of change is m in this formula is is how steep the line is . To calculate m for this formula you choose two separate points on the line (taking note of their X-Y coordinates). Then you take the difference between the y coordinates then take the difference between the x coordinates. Then you take these two numbers and divide the difference between the y coordinates by the difference between the x coordinates. This will give you the steepness of the line which is m in the formula of a straight line.

The rate of change of a quantity is found using calculus, specifically by finding the derivative of the function that describes the quantity with respect to the variable of interest.

For a function y=f(x):

• The rate of change of y with respect to x is given by the derivative dy/dx
• This derivative represents how y changes as x changes.

Take the variable that is changing over a period of time, making note of the first time point and second time point. Subtract this variable at the first time point from the variable at the second time point. Then, take this value and divide it by the difference in time between the two points to get the rate of change.

Rate of change = change in y/change in x

When we discuss how objects move, we start with displacement, which indicates an object's position over time. However, just knowing the position isn't always enough—we need to understand how quickly this position changes. This is where velocity comes in. Velocity is the rate of change of displacement with respect to time, telling us how fast an object moves and in what direction. On a graph of position versus time, the slope of the curve at any given point represents the object's velocity. A steeper slope means a higher velocity, while a flatter slope indicates a lower velocity.

Acceleration takes this concept a step further by measuring how velocity changes over time. If an object speeds up, slows down, or changes direction, its velocity is changing, meaning it experiences acceleration. On a graph of velocity versus time, the slope of the curve at any point represents the object's acceleration. A steeper slope here means a greater change in velocity, while a flatter slope means a smaller change. Just as velocity is the rate of change of displacement, acceleration is the rate of change of velocity.

Differentiation is the mathematical tool we use to find these rates of change. It allows us to determine the slope of a curve at any point on a graph. For a displacement-time graph, differentiation helps us find the velocity, the rate at which displacement changes. For a velocity-time graph, differentiation helps us find the acceleration, the rate at which velocity changes. When approaching problems about rates of change, it's crucial to identify the variables, interpret the graph, determine what rate of change you need, and analyze the slope accordingly. 

How much one quantity change in relation to another

Referring to a straight line graph, the rate of change can be found using the formula: change in y/change in x, practically speaking, When dealing with a straight line, it's best to "make a right angled triangle" choosing points appropriately. Your result is the "gradient" or "slope" of the line which tells you the rate of change. It can either be positive, negative or 0 if the straight line is parallel to the x axis.

As far as GCSE Maths is concerned, you take the two pairs of values that are given, and with them:

i) you calculate a difference between respective valued

ii) you then take a ratio of these teo differences (if the two quantities are concerned are Y and X, or some other quantity and time, then the difference of Y/some other quantity goes in the numerator while the difference of X/time goes in the denominator, in the ratio).

iii) the value you get from this ratio is the rate.

For example, if the two pairs are (2,10) and (6,14). Then:

-the difference of the Y values is 14-10=4

-the difference of the X values is 6-2=4

-the ratio is simply 4/4=1

-the rate is therefore 1

It is simply a total concerned amount divided by the total time. For example, if you want to find out rate of change of distance. You consider total distance covered by the vehicle and divide it by the total time it took for that change.

Rate of change of position = total covered distance / total taken time.

This depends on what type of rate of change you are looking for, however either it be speed, temperature or mass, you are usually able to plot this variable against time on a graph, to find the rate of change you have to find the gradient of this graph.

If looking at a straight line graph, you are able to simply identify two points on the line and note down their coordinates. You are then able to do (Y2-Y1)/(X2-X1) to find the rate of change.

If it is a curved graph, draw a tangent and use the (Y2-Y1)/(X2-X1) formula on the tangent. The rate of change is not the same all across and you have to draw a tangent on a specific point on the graph to find the rate of change at that time specifically.

You would also be able to use differentiation to be able to find the gradient at a specific point.

I hope this helps :)

Use the formula (new-old)/old and multiply by 100 to get a percentage

To find the rate of change of something you calculate how much the “something” has changed by and then divide it by the number of seconds that have passed.