How do you find the rate of change?

divide the change in y-values by the change in x-values

Rate of change of what?

The rate of change of anything is usually given by its end value, minus its start value, divided by time

for example momentum=0 at t=0s, and momentum=5 at t=10s, rate of change = (5-0)/10 = 5/10 = 1/2

rate of change = (change in quantity 1)/ (change in quantity 2)

Find the change in y, and divide it by the change in x!

For example...

On a graph, we may find two points (5,4) and (6,8).

We can calculate rate of change by taking the x values (5 and 6), and finding the difference between them (6-5=1). We do the same for the y values (4 and 8), with the difference being 4.

Therefore, the rate of change = 4 / 1 = 4!

The rate of change can be found on a straight-line graph by selecting two coordinates and dividing the change in the y-values by the change in the x-values. This can be shown in the formula : m = y2-y1/ x2-x1 with m representing the gradient of the straight-line. Additionally, x and y can be two different variables and the formula is used to calculate average speed or average velocity.The rate of change of a straight line is equivalent to the gradient of a straight line graph. However, we use differentiation with non-linear graphs to find the rate of change at a particular point. This is another topic in itself.

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.

Divide the change of one variable with the change of the other variable

For a linear set of values the rate of change will be the change in y-values divided by the change in x-values in other words the gradient of a straight line. In the set of values relating to a curved line the rate of change on each point can be found by differentiating the equation of that curve and using the values of x of the point you need to find the rate of change at.

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.

Rate of Change = (Change in Output) / (Change in Input)

• Change in Output: This refers to the difference between the final and initial values of the quantity you're measuring (often denoted by y).
• Change in Input: This refers to the difference between the final and initial values of the independent variable (often denoted by x).

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.

### 1. Rate of Change in a Graph:

The rate of change can be understood as the slope of a line on a graph. To find the rate of change between two points on a straight line:

- Identify the Two Points: Let’s say you have two points on the graph: ( (x_1, y_1) ) and ( (x_2, y_2) ).

- Use the Formula for the Slope: The rate of change (slope) is calculated using the formula:

[

\text{Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1}

]

- Interpretation: This formula gives you the change in the (y)-value (output) per unit change in the (x)-value (input). A positive result indicates that the function is increasing, while a negative result indicates it is decreasing.

### 2. Rate of Change in a Function:

For a more general approach with functions, especially when dealing with curves:

- Find the Derivative: The rate of change at any point on a curve can be found using calculus, specifically by taking the derivative of the function. For example, if ( f(x) ) is a function, the derivative ( f'(x) ) gives you the rate of change of ( f ) with respect to ( x ).

- Evaluate the Derivative at a Point: To find the rate of change at a specific point ( x = a ), substitute ( a ) into the derivative:

[

\text{Rate of Change} = f'(a)

]

### 3. Rate of Change in Real-Life Situations:

In practical scenarios, the rate of change can be interpreted as how one quantity changes in relation to another. For example:

- Speed: If a car travels a distance of 150 km in 3 hours, the rate of change (speed) can be calculated as:

[

\text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{150 \text{ km}}{3 \text{ hours}} = 50 \text{ km/h}

]

### Example Problem:

Find the rate of change of the function ( f(x) = 2x^2 + 3x ) at the point ( x = 2 ):

1. Find the derivative:

  [

  f'(x) = \frac{d}{dx}(2x^2 + 3x) = 4x + 3

  ]

2. Evaluate the derivative at ( x = 2 ):

  [

  f'(2) = 4(2) + 3 = 8 + 3 = 11

  ]

Thus, the rate of change of the function ( f(x) ) at ( x = 2 ) is 11.

The rate of change tells you how fast one quantity changes compared to another. For example, for a straight line, you find the rate of change by using the formula:

Rate of change=Change in y / Change in x. This is often called the gradient or slope. If y goes from 3 to 7 while x goes from 2 to 4, then the rate of change is:

(7−3)/(4−2)= 4/2 = 2. This means y increases by 2 for every 1 unit increase in x.

The Rate of change (ROC) is the increase or decrease of one value related to another value.

The speed of a car for instance is the distance covered divided by the time taken.

In mathematics this can be the gradient of a graph in general terms.

We can find the rate of change by calculating the gradient of the curve or line. We identify two points on the line. Next we find the difference in the y-axis points and divide by the difference in the x-axis points. E.g. (Y2 - Y1)/(X2 - X1)