Question

How is average rate of change related to slope?

Answer 1

Hint: Slope is just a way of measuring the inclination of a line or the steepness of the line. It is defined by, change in $y$ due to the change in x. The average rate of change about a point as the interval over which the average is being taken is reduced to zero.
Average rate of change $$={\displaystyle \frac{y_2-y_1}{x_2-x_1}}$$
$(x_1,y_1)$ - coordinates of first point in the line
$$(x_2,y_2)$$ - coordinates of second point in the line

Complete step-by-step solution:
Let’s assume the coordinates of those points as follows:
$x_1$ coordinate is $-1$ and $x_2^{}$ coordinate is $3$
$y_1$ coordinate is $-6$ and $y_2$ coordinate is $6$
Let us consider the two points $\text{ (}-1,-6\text{) and }(3,6)$
Average rate of change $$={\displaystyle \frac{y_2-y_1}{x_2-x_1}}$$
$x_1=-1$ and $x_2=3$
$y_1=-6$ and $y_2=6$
Now, substitute the two points in the formula of average rate of change
Average rate of change $={\displaystyle \frac{6-(-6)}{3-(-1)}}$
Adding the two terms,
$={\displaystyle \frac{6+6}{3+1}}={\displaystyle \frac{12}{4}}$
Simplifying the terms,
$={\displaystyle \frac{12}{4}}=3$
Average rate of change $=3$
Slope is calculated by finding the ratio of the vertical change to horizontal change between two distinct points on a line. The coordinates points to point a graph. And then to draw a straight line through the two points. Slope is something also referred to as the rate of change.

When we moved $1$ in the $x$ , we moved up $3$ in the $y$ . If you move 2 in the $x-$ direction, you’re going to move $6$ in the $y$. $6/2$ is the same thing as $3$ . This is the average rate of change between two points.

Note: The rate of change between two points on either side of x becomes closer to the slope of x as the distance between the two points is reduced. When we work with functions, the average rate of change is expressed using function notation.
For the function is,
$$\mathrm{y}=f(x)$$ , where $\mathrm{x}=a$ and $\mathrm{x}=b$
Average rate of change $={\displaystyle \frac{\text{change in }y}{\text{change in }x}}={\displaystyle \frac{f(b)-f(a)}{b-a}}$