To find the equation of a parabola given its vertex and orientation, we use the vertex form of the equation for a parabola. The general vertex form is:
[tex]\[ y = a(x-h)^2 + k \][/tex]
Here, [tex]\((h, k)\)[/tex] is the vertex of the parabola and [tex]\(a\)[/tex] determines the direction (upward or downward) and the width of the parabola.
### Given Information:
1. The vertex of the parabola is at [tex]\((-1, 0)\)[/tex].
2. The parabola opens down.
### Step-by-Step Solution:
1. Identify [tex]\(h\)[/tex] and [tex]\(k\)[/tex]:
Since the vertex is at [tex]\((-1,0)\)[/tex], we have:
[tex]\[
h = -1 \quad \text{and} \quad k = 0
\][/tex]
2. Determine the sign of [tex]\(a\)[/tex]:
The parabola opens downwards, meaning the coefficient [tex]\(a\)[/tex] should be negative.
3. Substitute the values into the vertex form:
Replace [tex]\(h\)[/tex] and [tex]\(k\)[/tex] in the vertex form equation:
[tex]\[
y = a(x - h)^2 + k
\][/tex]
Substituting [tex]\(h = -1\)[/tex] and [tex]\(k = 0\)[/tex]:
[tex]\[
y = a(x - (-1))^2 + 0
\][/tex]
Simplify inside the parentheses:
[tex]\[
y = a(x + 1)^2
\][/tex]
4. Set [tex]\(a\)[/tex] to -1 (since the parabola opens downwards):
[tex]\[
y = -(x + 1)^2
\][/tex]
Therefore, the equation of the parabola is:
[tex]\[
y = -(x + 1)^2
\][/tex]
### Comparison with Given Options:
- [tex]\(y=-(x+1)^2\)[/tex] matches our determined equation perfectly.
- [tex]\(y=-x^2-1\)[/tex] and [tex]\(y=-(x-1)^2\)[/tex] do not match the vertex form derived.
Thus, the correct equation of the parabola is:
[tex]\[ y = -(x+1)^2 \][/tex]
