To analyze the equation \( y = -5(x)^2 + 1 \), let's identify its key features and then describe the graph.
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Type of Graph: This equation is a quadratic function, characterized by the \( x^2 \) term. Since the coefficient of \( x^2 \) is negative (-5), the graph is a downward-opening parabola.
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Vertex: The vertex form of a parabola is given by \( y = a(x-h)^2 + k \), where \( (h, k) \) is the vertex of the parabola. In this case, the equation can be rewritten as \( y = -5(x - 0)^2 + 1 \). Thus, the vertex is at the point \( (0, 1) \).
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Axis of Symmetry: The axis of symmetry is the vertical line that passes through the vertex. Since the vertex is at \( (0, 1) \), the axis of symmetry is x = 0 (the y-axis).
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Y-Intercept: The y-intercept occurs when \( x = 0 \). Substituting \( x = 0 \) into the equation gives \( y = -5(0)^2 + 1 = 1 \). Therefore, the y-intercept is at the point \( (0, 1) \).
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X-Intercepts: To find the x-intercepts, we set \( y = 0 \): \[ 0 = -5(x)^2 + 1 \ 5(x)^2 = 1 \ (x)^2 = \frac{1}{5} \ x = \pm \frac{1}{\sqrt{5}} \approx \pm 0.447 \] So, the x-intercepts are approximately at \( \left(\frac{1}{\sqrt{5}}, 0\right) \) and \( \left(-\frac{1}{\sqrt{5}}, 0\right) \).
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Graph Shape: The graph will be a downward-opening parabola, with the vertex at the top.
Summary of Key Points:
- Vertex: (0, 1)
- Opens downward.
- Y-intercept: (0, 1)
- X-intercepts: (±0.447, 0)
- Axis of symmetry: x = 0
Statement Analysis:
A correct statement about the graph would likely be something along the lines of:
- The graph has a maximum point at (0, 1).
- The parabola opens downwards.
- The vertex represents the highest point on the graph.
If you provide specific statements, I will help you determine which one is correct.
3 answers