Question

Find the slope of the line containing the given point

f(1/6)= -3 and f(-1/3)= -1/3

the slope is?___

Answer 1

`\displaystyle{\sf{slope} = -(16)/(3)}`

Explanation:

Let's understand this notation:

`\displaystyle{f(a) = b}` means that at x = a, there exists value y = b. We can write
`\displaystyle{f(a) = b}` in the coordinate form of
`\displaystyle{(a,b)}`.

So according to the problem, we can rewrite the notation in form of coordinate as:

`\displaystyle{\left((1)/(6), -3 \right)}` and
`\displaystyle{\left(-(1)/(3), -(1)/(3)\right)}`

Finding slope, we can use the slope formula of:

`\displaystyle{m = (f(x_2)-f(x_1))/(x_2-x_1)}`

Since y = f(x) then:

`\displaystyle{m = (y_2-y_1)/(x_2-x_1)}`

Determine that:

• 
  `\displaystyle{\left(-(1)/(3), -(1)/(3)\right) = (x_2,y_2)}`
• 
  `\displaystyle{\left((1)/(6), -3 \right) = (x_1,y_1)}`

Substitute in the formula:

`\displaystyle{=(-(1)/(3)-(-3))/(-(1)/(3)-(1)/(6))}\\\\\\\displaystyle{=(-(1)/(3)+3)/(-(1)/(3)-(1)/(6))}\\\\\\\displaystyle{=(-(1)/(3)+(9)/(3))/(-(2)/(6)-(1)/(6))}\\\\\\\displaystyle{=(-(8)/(3))/(-(3)/(6))}\\\\\\\displaystyle{=-(8)/(3) \cdot \left(-(6)/(3)\right)}\\\\\\\displaystyle{=-8 \cdot \left(-(2)/(3)\right)}\\\\\\\displaystyle{=-(16)/(3)}`

Therefore, the slope is -16/3