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

1 Answer
May 13, 2018

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

Explanation:

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

(By the way, it is important that you don't write (3,3)(3,3) and (4,0)(4,0) as these can be read as the decimal numbers 3.33.3 and 4.04.0, which at least some non English speaking languages use. To ensure no misunderstanding, therefore, please add a space after the comma: (3, 3)(3,3), (4, 0)(4,0))

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

From the graph we can read directly that a=-3a=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)=3x=3,f(x)=3 and x=4, y=0x=4,y=0 both fulfill the function
y=ax+by=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=33=3a+b3a+b=3
(2) 0=4a+b => 4a+b=00=4a+b4a+b=0

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

(4a+b)-(3a+b)= 0-3(4a+b)(3a+b)=03
=> a=-3a=3
Our slope is -33

(We are not asked about this, but as
4a+b=04a+b=0 it follows that b=-4a=(-4)*(-3)=12b=4a=(4)(3)=12
so that the function is #f(x)=-3x+12)