To find the equation of the parabola with a given vertex at [tex]\((8, -1)\)[/tex] and a [tex]\(y\)[/tex]-intercept at [tex]\((0, -17)\)[/tex], we should use the vertex form of a parabola equation, which is given by:
[tex]\[ y = a(x - h)^2 + k \][/tex]
Here, [tex]\((h, k)\)[/tex] is the vertex of the parabola. Substituting the vertex [tex]\((8, -1)\)[/tex] into the equation, we get:
[tex]\[ y = a(x - 8)^2 - 1 \][/tex]
Next, we need to use the [tex]\(y\)[/tex]-intercept to find the value of [tex]\(a\)[/tex]. The [tex]\(y\)[/tex]-intercept is the point where [tex]\(x = 0\)[/tex] and [tex]\(y = -17\)[/tex]. Substituting [tex]\(x = 0\)[/tex] and [tex]\(y = -17\)[/tex] into the equation, we obtain:
[tex]\[ -17 = a(0 - 8)^2 - 1 \][/tex]
Simplifying the equation:
[tex]\[ -17 = a(64) - 1 \][/tex]
Add 1 to both sides:
[tex]\[ -16 = 64a \][/tex]
Solving for [tex]\(a\)[/tex]:
[tex]\[ a = \frac{-16}{64} = -\frac{1}{4} \][/tex]
Now that we have the value of [tex]\(a\)[/tex], we can write the final equation of the parabola:
[tex]\[ y = -\frac{1}{4}(x - 8)^2 - 1 \][/tex]
Thus, the equation of the parabola is:
[tex]\[ y = -\frac{1}{4}(x - 8)^2 - 1 \][/tex]
This corresponds to option:
A) [tex]\( y = -\frac{1}{4}(x - 8)^2 - 1 \)[/tex]
