Respuesta :

Check the picture below.  So the parabola looks more or less like so.

[tex]\bf \textit{horizontal parabola vertex form with focus point distance} \\\\ 4p(x- h)=(y- k)^2 \qquad \begin{cases} \stackrel{vertex}{(h,k)}\qquad \stackrel{focus~point}{(h+p,k)}\qquad \stackrel{directrix}{x=h-p}\\\\ p=\textit{distance from vertex to }\\ \qquad \textit{ focus or directrix}\\\\ \stackrel{"p"~is~negative}{op ens~\supset}\qquad \stackrel{"p"~is~positive}{op ens~\subset} \end{cases} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\bf \begin{cases} h=-5\\ k=2\\ p=4 \end{cases}\implies 4(4)[x-(-5)]=[y-2]^2\implies 16(x+5)=(y-2)^2 \\\\\\ x+5=\cfrac{1}{16}(y-2)^2\implies x = \cfrac{1}{16}(y-2)^2-5[/tex]

Ver imagen jdoe0001

The standard form of the equation of the parabola with vertex (-5, 2) and focus (-1, 2) is

[tex](y - 2) ^{2} = - (x + 5)[/tex]

The vertex of the parabola (h, k) = (-5, 2)

That is, h = -5, k = 2

The focus (4p, k) = (-1, 2)

From the relationship above:

4p  = -1

p = -1/4

The equation of a parabola is given as:

[tex](y - k) ^{2} = 4p(x - h)[/tex]

Substitute h = -5, k = 2, p = -1/4 into the parabola equation above

[tex](y - 2)^{2} = 4( \frac{ - 1}{4} )(x - ( - 5))[/tex]

Simplifying the equation above

[tex](y - 2) ^{2} = - 1(x - ( - 5))[/tex]

Simplifying further, we have:

[tex](y - 2) ^{2} = - (x + 5)[/tex]

Therefore, the equation of the parabola with vertex (-5, 2) and focus (-1, 2) is

[tex](y - 2) ^{2} = - (x + 5)[/tex]

Learn more here on standard form of a parabola here: https://brainly.com/question/13493333

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