Question

What is an equation of a parabola with the given vertex and focus? vertex: (-2, 5)focus: (-2, 6)show each step

Answer 1

`y=(1)/(4)(x+2)^2+5`

Step-by-step explanation

the equation of a parabola in vertex form is give by:

`\begin{gathered} y=a(x-h)^2+k \\ \text{where} \\ (h,k)\text{ is the vertex} \\ and\text{ the focus is( h,k}+(1)/(4a)) \end{gathered}`

Step 1

so

let

a) vertex

`\begin{gathered} vertex\colon(h.k)\text{ }\rightarrow(-2,5) \\ h=-2 \\ k=5 \end{gathered}`

and

b) focus

`\begin{gathered} \text{( h,k}+(1)/(4a))\rightarrow(-2,6) \\ so \\ h=-2 \\ \text{k}+(1)/(4a)=6 \\ \end{gathered}`

replace the k value and solve for a,

`\begin{gathered} \text{k}+(1)/(4a)=6 \\ 5+(1)/(4a)=6 \\ \text{subtract 5 in both sides} \\ 5+(1)/(4a)-5=6-5 \\ (1)/(4a)=1 \\ \text{cross multiply } \\ 1=1\cdot4a \\ 1=4a \\ \text{divide both sides by }4 \\ (1)/(4)=(4a)/(4)=a \\ a=\text{ }(1)/(4) \end{gathered}`

Step 2

finally, replace in the formula

`\begin{gathered} y=a(x-h)^2+k \\ y=(1)/(4)(x-(-2))^2+5 \\ y=(1)/(4)(x+2)^2+5 \\ \end{gathered}`

therefore, the answer is

`y=(1)/(4)(x+2)^2+5`

I hope this helps you