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Graph the equation 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)
.
The parabola will have a vertex at left parenthesis 1 comma 0 right parenthesis .

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

1 answer

To analyze the equation \( y = -5x^2 + 1 \), we can identify the key features of the parabola it represents.

  1. Vertex: The equation is in the form \( y = ax^2 + bx + c \), which indicates a parabola. The coefficient \( a = -5 \) shows that it opens downwards. The vertex for this standard form can be found using the formula \( x = -\frac{b}{2a} \). Here, \( b = 0 \) (since there is no \( bx \) term) and \( a = -5 \):

    \[ x = -\frac{0}{2(-5)} = 0 \]

    Plugging \( x = 0 \) back into the equation to find the y-coordinate of the vertex:

    \[ y = -5(0)^2 + 1 = 1 \]

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

  2. Direction: The coefficient of \( x^2 \) is negative (\( -5 \)), indicating that the parabola opens downwards.

Now, we can examine the statements provided:

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

Correct Statement:

  • The correct statement about the graph is: The parabola will open downwards.