Answers:
Equation in standard form: [tex]y = \frac{1}{8}(x+1)^{2} -4[/tex]
Axis of Symmetry: x = -1
Vertex: (-1, -4)
Focus: (-1, -2)
Directrix: y = -6
Step-by-step explanation:
First, let's clean up some of the mess with the equation that we have been given in this question and put it in standard form so that we can find the features we've been asked to give.
Our initial equation is: [tex]8(y+4)=(x+1)^{2}[/tex]
In order to put our equation in standard form, we must isolate y, so that it is in terms of y.
First, we need to divide both sides by 8. When we do this, we get [tex]y+4=\frac{(x+1)^{2} }{8}[/tex] or [tex]y+4=\frac{1}{8}(x+1)^{2}[/tex]. Then we need to subtract 4 from both sides to get: [tex]y = \frac{1}{8}(x+1)^{2} -4[/tex].
Now, we can start to find the features of the parabola. Remember, the standard form of equations involving parabolas is [tex]y=a(x-h)+k[/tex] where [tex]a =\frac{1}{4p}[/tex].
The axis of symmetry is at [tex]x=h[/tex], in this case [tex]h=-1[/tex] so the axis of symmetry is [tex]x=-1[/tex].
The vertex is located at [tex](h, k)[/tex]. As we previously stated [tex]h=-1[/tex], but additionally, [tex]k=-4[/tex] making the vertex located at [tex](-1,-4)[/tex].
The focus is located at [tex](h, k+p)[/tex] and the directrix is located at [tex]y=k-p[/tex]. For this, we need to find p which we can find with the equation [tex]a =\frac{1}{4p}[/tex]. Substituting the value of “a” into the equation, we get [tex]\frac{1}{8} = \frac{1}{4p}[/tex]. We can cross-multiply to get the equation [tex]4p=8[/tex]. Dividing by 4 on both sides, we get [tex]p=2[/tex]. Now we can find the focus and directrix.
The focus is located at [tex](-1, [-4+2])[/tex] or [tex](-1,-2)[/tex] and the directrix is represented by the equation [tex]y=(-4-2)[/tex] or [tex]y=-6[/tex].
Hopefully this answered your question this time, and with this, I hope you are able to solve it without help the next time
