Question

write the equation of the function \\( f(x) \\) represented by the following graph. write the equation in vertex form.
identify the vertex (lowest point) on the graph and write its coordinates as \\( (h, k) \\).

Explanation:

Step1: Identify vertex (h,k)

The vertex (lowest point) is $(-4, -1)$. So $h=-4$, $k=-1$.

Step2: Use vertex form template

Vertex form: $f(x)=a(x-h)^2+k$. Substitute $h,k$:
$f(x)=a(x-(-4))^2+(-1)=a(x+4)^2-1$

Step3: Solve for a using a point

Use the y-intercept $(0,14)$:
$14=a(0+4)^2-1$
$14=16a-1$
$16a=15$
$a=\frac{15}{16}$

Step4: Substitute a into vertex form

$f(x)=\frac{15}{16}(x+4)^2-1$

Answer:

1. Vertex coordinates: $(-4, -1)$
2. Function equation: $f(x)=\frac{15}{16}(x+4)^2 - 1$