Question

If a parabola's focus is at (−1, −1) and the directrix is at y = 7, what is the equation representing this parabola?

Answer 1

Use the definition of Parabola, more formally speaking:
"The set of all points equidistant to a fixed point called focus and a fixed line called 'directrix'".
This of course, implies that, say for instance we have a fixed point \(F( \alpha, \beta )\) and a line"d" \((d)y=k \rightarrow y-k=0\) we can then say that point \(A(x_a,-k)\) is then a point belonging to the line (d) and we can then generalize a point \(P(x_p,y_p)\) that has de following conditions:
\[dist(F,P)=dist(P,A)\]
Which applying the distance formula we obtain:
\[\sqrt{(x_p-\alpha)^2+(y_p-\beta)^2}=\sqrt{(x_a-x_p)^2+(-k-y_p)^2}\]
This is what will represent the very definition of parabola.
of course if you narrow this down you will eventually obtain the equation for the parabola, but for now, I have just stated how you can do it with the very definition of parabola using generalized points.

Answer 2

Thank you

Answer 3

if you have any further questions, ask away :)

Answer 4

If the cubic function P(x) includes the points (−4, 0), (0, 0), and (2, 0), which of the following represents this function?

Answer 5

You can find any function where the roots are given, take notice that the points you were given are "roots" of the function since they have their y-coordinate equal to zero.
Any function has it's factorial form as the following:
\[f(x)=(x-a_1)(x-a_2)...( x-a _{n-1})(x-a_n)\]
Where \(a_1,a_2 ... a_{n-1},a_n \in \mathbb{R}\) and they represent the root or zeroes of the function, all you have to do is plot the points in this structure and then distribute the parenthesis.