Graph the equation y=−5(x)2+1

Question

𝑦
=
−
5
(
𝑥
)
2
+
1
. Which of the following statements about the graph is correct?(1 point)
Responses

The parabola will have a vertex at (1,0)
(
1
,
0
)
.
The parabola will have a vertex at left parenthesis 1 comma 0 right parenthesis .

The parabola will have a vertex at (0,0)
(
0
,
0
)
.
The parabola will have a vertex at left parenthesis 0 comma 0 right parenthesis .

The parabola will open downwards.
The parabola will open downwards.

The parabola will have a vertex at (−1,0)
(
−
1
,
0
)
.

Answer 1

To analyze the quadratic equation \( y = -5(x)^2 + 1 \), let's identify its key features.

1. Determine the vertex: The given equation is in the standard form of a parabola \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. In this case, we can rewrite the equation to match the standard vertex form.

   The equation can be rewritten as: \[ y = -5(x - 0)^2 + 1 \]

   From this, we see that:

   • \(h = 0\)
   • \(k = 1\)

   Therefore, the vertex of the parabola is at \((0, 1)\).

2. Determine the direction of opening: The coefficient \(a = -5\) is negative, which indicates that the parabola opens downwards.

Based on this analysis, we can summarize the findings:

• The vertex of the parabola is at \((0, 1)\).
• The parabola opens downwards.

Now, let's review the responses provided:

1. The parabola will have a vertex at \((1, 0)\).
2. The parabola will have a vertex at \((0, 0)\).
3. The parabola will open downwards.
4. The parabola will have a vertex at \((-1, 0)\).

The correct statements about the graph are:

• "The parabola will open downwards."

And, the vertex is actually \((0, 1)\), which is not included in the options you provided. Thus, only the third statement is correct.