Graph the equation y=−5(x)2+1

Question

. Which of the following statements about the graph is correct?(1 point)
Responses

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

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

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

Answer 1

To analyze the equation \( y = -5x^2 + 1 \), let's determine the vertex and the direction of the parabola.

1. Form of the Equation: The equation is in the standard form of a quadratic equation \( y = ax^2 + bx + c \). Here, \( a = -5 \), \( b = 0 \), and \( c = 1 \).

2. Vertex Calculation: For a parabola given by \( y = ax^2 + bx + c \), the x-coordinate of the vertex can be found using the formula: \[ x = -\frac{b}{2a} \] Substituting \( b = 0 \) and \( a = -5 \): \[ x = -\frac{0}{2 \cdot -5} = 0 \]

   To find the y-coordinate of the vertex, substitute \( x = 0 \) back into the equation: \[ y = -5(0)^2 + 1 = 1 \] Thus, the vertex is at \( (0, 1) \).

3. Direction of the Parabola: Since \( a = -5 < 0 \), the parabola opens downwards.

From this analysis, we can determine:

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

Given these findings, the correct statements from your options would be:

• The parabola will have a vertex at (0, 0): Incorrect (the vertex is at \( (0, 1) \)).
• The parabola will have a vertex at (1, 0): Incorrect.
• The parabola will have a vertex at (−1, 0): Incorrect.
• The parabola will open downwards: Correct.

Therefore, the correct statement is:

The parabola will open downwards.