Question

How do you find the slope given (3,3) and (4,0)?

Answer 1

The slope in the function f(x)=ax+b, where f(x) goes through (3, 3) and (4, 0) is `a=-3`

Explanation:

We need to start by making a graph. It makes it easier to see what we have:

(By the way, it is important that you don't write `(3,3)` and `(4,0)` as these can be read as the decimal numbers `3.3` and `4.0`, which at least some non English speaking languages use. To ensure no misunderstanding, therefore, please add a space after the comma: `(3, 3)`, `(4, 0)`)

As we can see, this is a linear function on the form
`y=ax+b`, where the slope is the constant `a`.
(Alternatively we can write it as `f(x)=ax+b`, but it is slightly easier to handle the function if we use `y`.)

From the graph we can read directly that `a=-3`, since if x increases with 1 from 3 to 4, y falls from 3 to 0 in value.

We can show this the following way: The two value pairs `x=3, f(x)=3` and `x=4, y=0` both fulfill the function
`y=ax+b` which we are interested in.

If we plug these two value pairs into the function, we get the following pair of equations:
(1) `3=3a+b => 3a+b=3`
(2) `0=4a+b => 4a+b=0`

As we have `+b` in both, we subtract (1) from (2) to get rid of it:

`(4a+b)-(3a+b)= 0-3`
`=> a=-3`
Our slope is `-3`

(We are not asked about this, but as
`4a+b=0` it follows that `b=-4a=(-4)*(-3)=12`
so that the function is #f(x)=-3x+12)