To determine how many real solutions the function \( f(x) = -2x^2 - 1 \) has based on its graph, we can analyze the function's properties.
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Identify the function type: This is a quadratic function, and it can be rewritten in standard form as \( f(x) = -2(x^2) - 1 \).
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Direction of the parabola: Since the coefficient of \( x^2 \) is negative (\(-2\)), the parabola opens downward.
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Vertex of the parabola: To find the vertex, we can use the formula \( x = -\frac{b}{2a} \) for a quadratic equation in the form \( ax^2 + bx + c \). Here, \( a = -2 \), \( b = 0 \), and \( c = -1 \).
- \( x = -\frac{0}{2 \cdot -2} = 0 \)
- Now, we can find the function value at this vertex: \[ f(0) = -2(0)^2 - 1 = -1 \] Thus, the vertex is at point \( (0, -1) \).
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Intersections with the x-axis: The graph of the function will intersect the x-axis (where \( f(x) = 0 \)) based on the vertex being below the x-axis. Since the vertex is at \( y = -1 \) (which is below the x-axis) and the parabola opens downwards, it does not cross the x-axis.
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Conclusion: Since the parabola does not intersect the x-axis, the number of real solutions to the equation \( -2x^2 - 1 = 0 \) is 0.
Therefore, the function \( f(x) = -2x^2 - 1 \) has no real solutions.
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