Question

Using the whole numbers 0 through 9 only once, make two points in the format of `(x,y)` that will create the steepest slope possible. Then do the exercise again to achieve the shallowest slope possible. What are the points?

Answer 1

See below:

Explanation:

We have to find 2 points with the format `(x,y)` and inserting in any whole number, 1 through 9 (using a number only once) to find the steepest slope, and then again for the flattest slope.

Let's first talk about slope :

The symbol often used for slope is `m` and is found by dividing the "rise" (or the change in `y` values between 2 points) by the "run" (or the change in `x` values between those same 2 points). We can express the equation this way:

`m=(y_2-y_1)/(x_2-x_1)`

Steepest possible slope

We're looking for the greatest possible slope and so we want `y_2-y_1` to be as big as possible and `x_2-x_1` to be as small as possible.

The biggest `y_2-y_1` we can create is `9-0=9`

The smallest possible `x_2-x_1` we can create will be any two consecutive numbers (and so creating a difference of 1), so let's use `5-4=1`. This gives us:

`m=9/1=9`

And this gives us for the two points:

`(4,0),(5,9)`

(Note: if we reverse the order of the two points, we'll end up with `m=-9`, which is just as steep.)

Flattest possible slope

We can use the same logic to find the flattest possible slope. This time we want `x_2-x_1` to be as big as possible and `y_2-y_1` to be as small as possible. And so we can use the same 4 numbers:

`m=(5-4)/(9-0)=1/9`

which gives us points:

`(0,4),(9,5)`