Question

1. y = \frac{1}{12}(x - 3)^2+5

vertex: (3,5)
c = 5
focus=(3,10)
directrix: y = - 3

Explanation:

Step1: Identify vertex form

The equation $y=\frac{1}{12}(x - 3)^{2}+5$ is in vertex - form $y=a(x - h)^{2}+k$, where $(h,k)$ is the vertex. Here $h = 3$ and $k = 5$, so the vertex is $(3,5)$.

Step2: Find the value of $c$

For a parabola of the form $y=a(x - h)^{2}+k$, the relationship between $a$ and $c$ is $a=\frac{1}{4c}$. Given $a=\frac{1}{12}$, we solve $\frac{1}{12}=\frac{1}{4c}$ for $c$. Cross - multiplying gives $4c = 12$, so $c = 3$.

Step3: Determine the focus

Since the parabola opens upwards (because $a=\frac{1}{12}>0$) and the vertex is $(h,k)=(3,5)$, the focus of the parabola is $(h,k + c)$. Substituting $h = 3$, $k = 5$, and $c = 3$, we get the focus $(3,5 + 3)=(3,8)$.

Step4: Find the directrix

The directrix of a parabola opening upwards with vertex $(h,k)$ is the line $y=k - c$. Substituting $k = 5$ and $c = 3$, we get $y=5-3 = 2$.

Answer:

Vertex: $(3,5)$
$c = 3$
Focus: $(3,8)$
Directrix: $y = 2$