Answer:
[tex]\displaystyle{\sf{slope} = -\dfrac{16}{3}}[/tex]
Step-by-step explanation:
Let's understand this notation:
[tex]\displaystyle{f(a) = b}[/tex] means that at x = a, there exists value y = b. We can write [tex]\displaystyle{f(a) = b}[/tex] in the coordinate form of [tex]\displaystyle{(a,b)}[/tex].
So according to the problem, we can rewrite the notation in form of coordinate as:
[tex]\displaystyle{\left(\dfrac{1}{6}, -3 \right)}[/tex] and [tex]\displaystyle{\left(-\dfrac{1}{3}, -\dfrac{1}{3}\right)}[/tex]
Finding slope, we can use the slope formula of:
[tex]\displaystyle{m = \dfrac{f(x_2)-f(x_1)}{x_2-x_1}}[/tex]
Since y = f(x) then:
[tex]\displaystyle{m = \dfrac{y_2-y_1}{x_2-x_1}}[/tex]
Determine that:
- [tex]\displaystyle{\left(-\dfrac{1}{3}, -\dfrac{1}{3}\right) = (x_2,y_2)}[/tex]
- [tex]\displaystyle{\left(\dfrac{1}{6}, -3 \right) = (x_1,y_1)}[/tex]
Substitute in the formula:
[tex]\displaystyle{=\dfrac{-\dfrac{1}{3}-(-3)}{-\dfrac{1}{3}-\dfrac{1}{6}}}\\\\\\\displaystyle{=\dfrac{-\dfrac{1}{3}+3}{-\dfrac{1}{3}-\dfrac{1}{6}}}\\\\\\\displaystyle{=\dfrac{-\dfrac{1}{3}+\dfrac{9}{3}}{-\dfrac{2}{6}-\dfrac{1}{6}}}\\\\\\\displaystyle{=\dfrac{-\dfrac{8}{3}}{-\dfrac{3}{6}}}\\\\\\\displaystyle{=-\dfrac{8}{3} \cdot \left(-\dfrac{6}{3}\right)}\\\\\\\displaystyle{=-8 \cdot \left(-\dfrac{2}{3}\right)}\\\\\\\displaystyle{=-\dfrac{16}{3}}[/tex]
Therefore, the slope is -16/3
