Question
Graph the equation y=−5(x)2+1 . Which of the following statements about the graph is correct?(1 point)
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GPT-4o mini
To graph the equation \( y = -5x^2 + 1 \), we can start by identifying key features of the quadratic function.
1. **Shape**: This is a downward-opening parabola because the coefficient of \( x^2 \) is negative (-5).
2. **Vertex**: The vertex of the parabola can be found using the vertex form or by completing the square. Here, since it’s already in the form \( y = ax^2 + bx + c \), the vertex occurs at the maximum point. In standard form \( y = ax^2 + c \) (with \( b = 0 \)), the vertex is at the point \( (0, c) \). Thus, the vertex for this equation is at \( (0, 1) \).
3. **Y-intercept**: The y-intercept occurs where \( x = 0 \), which gives us \( y = -5(0)^2 + 1 = 1 \). So, the y-intercept is \( (0, 1) \).
4. **X-intercepts**: To find the x-intercepts, set \( y = 0 \):
\[
0 = -5x^2 + 1 \\
5x^2 = 1 \\
x^2 = \frac{1}{5} \\
x = \pm \frac{1}{\sqrt{5}} \approx \pm 0.447
\]
5. **Direction**: The parabola opens downward due to the negative coefficient of \( x^2 \).
To summarize about the graph:
- The vertex is at \( (0, 1) \).
- The parabola opens downwards.
- The y-intercept is at \( (0, 1) \).
- The x-intercepts are approximately \( (0.447, 0) \) and \( (-0.447, 0) \).
If you have specific statements to evaluate, please provide them, and I'll help determine which is correct based on the analysis above!
1. **Shape**: This is a downward-opening parabola because the coefficient of \( x^2 \) is negative (-5).
2. **Vertex**: The vertex of the parabola can be found using the vertex form or by completing the square. Here, since it’s already in the form \( y = ax^2 + bx + c \), the vertex occurs at the maximum point. In standard form \( y = ax^2 + c \) (with \( b = 0 \)), the vertex is at the point \( (0, c) \). Thus, the vertex for this equation is at \( (0, 1) \).
3. **Y-intercept**: The y-intercept occurs where \( x = 0 \), which gives us \( y = -5(0)^2 + 1 = 1 \). So, the y-intercept is \( (0, 1) \).
4. **X-intercepts**: To find the x-intercepts, set \( y = 0 \):
\[
0 = -5x^2 + 1 \\
5x^2 = 1 \\
x^2 = \frac{1}{5} \\
x = \pm \frac{1}{\sqrt{5}} \approx \pm 0.447
\]
5. **Direction**: The parabola opens downward due to the negative coefficient of \( x^2 \).
To summarize about the graph:
- The vertex is at \( (0, 1) \).
- The parabola opens downwards.
- The y-intercept is at \( (0, 1) \).
- The x-intercepts are approximately \( (0.447, 0) \) and \( (-0.447, 0) \).
If you have specific statements to evaluate, please provide them, and I'll help determine which is correct based on the analysis above!
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