Question

find the focus and the directrix of the graph of each equation.
1.) y = 5x^2
2.) y = -1/8y^2

Answer 1

slight mistake on number 2: x = -1/8y^2

Answer 2

you need to solve these?

Answer 3

yes but i'm confused on how and what equation i would use

Answer 4

so um would you like me to slove both of them for you?.

Answer 5

solve*

Answer 6

yes, but I'm feeling a bit confused so maybe you could go through the steps too?

Answer 7

idk if i can help sorry it doesnt make sense

Answer 8

it's okay i understand

Answer 9

These are parabolas whose foci is allocated inside the axis they represent, for instance, a parabola with the form: \(y= kx^2\) has it's foci allocated inside the y-axis, goes through the origin which means that the foci has to be same distant to the vertex (0,0) as the point that defines the intersection between the directrix and the y-axis.
A general form for the parabola whose foci is located in the y-axis, being a foci \(F(0,\frac{ p }{ 2 })\) and the directrix \(d)y=-\frac{ p }{ 2 }\) will yield as a result the following structure for a parabola:
\[y=\frac{ 1 }{ 2p }x^2\]
And look, compare it to \(y=5x^2\) the look exactly the same, and all we have to do is find that distance "p" in order to locate the points that matter, after all, the vertex is already in the origin so no need for complex calculations:
\[y=\frac{ 1 }{ 2p }x^2 \rightarrow y=5x^2 \iff \frac{ 1 }{ 2p }=5\]

Answer 10

so how do we find the distance? do i graph?

Answer 11

Solve for "p"

Answer 12

"p" is the distance from the foci to the directrix where the vertex is the mid-point.

Answer 13

ah okay, sorry

Answer 14

What did you get as result?

Answer 15

i'm sorry i'm still a bit confused on how to go about this

Answer 16

maybe a similar example would help?

Answer 17

Sure, let's for instance consider the equation of the parabola:
\[y=\frac{ 1 }{ 8 }x^2\]
And we desire to find the foci and the directrix of this parabola, well, we know the foci is located in the y-axis, because of the structure responding to \(y=\frac{ 1 }{ 2p }x^2\)
so what we will do is call \(\frac{ 1 }{ 2p }\) as equal to 1/8, therefore:
\[\frac{ 1 }{ 2p }=\frac{ 1 }{ 8 } \iff 2p=8 \iff p=4\]
"p" is the distance from the foci to the directrix, and we know the vertex is the mid point of this parabola, and the VERTEX is located in the origin, meaning (0,0).
So therefore, the foci must be located p/2 distance on the positive axis (because the coefficient of x^2 is positive) so therefore the foci must have coordinates: \(F(0,2)\) and the directrix, must be located in the negative part of the y-axis, and since the directrix is a line, in this case, parallel to the x-axis, we will know that it intersects the y-axis in the negative quadrant at -2, therefore: \(d)y=-2\).
So, now we know the foci and the directrix of the given equation of a parabola, those being \(F(0,2)\) and \(d)y=-2\) respectively.