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

Question

GPT-4o mini · Answer

To analyze the equation $ y = -5(x)^2 + 1 $, let's identify important features of the graph. 1. **Form of the quadratic equation**: The equation is in the standard form $ y = ax^2 + bx + c $, where in this case, $ a = -5 $, $ b = 0 $, and $ c = 1 $. 2. **Vertex**: The vertex of a quadratic equation $ y = ax^2 + bx + c $ can be found using the formula: $$ x = -\frac{b}{2a} $$ Substituting $ b = 0 $ and $ a = -5 $: $$ x = -\frac{0}{2 \times -5} = 0 $$ 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 $, which is negative, the parabola opens downwards. Now, let's summarize: - The vertex is at $ (0, 1) $. - The parabola opens downwards. Based on the response options given, the correct statements are: - **None of the vertex options are correct**, since they all suggest a vertex at $ (0, 0) $, $ (1, 0) $, or $ (-1, 0) $. - The correct statement about the direction of the parabola is: **The parabola will open downwards.** So, the final answer is that the parabola will open downwards.