To analyze the function \( g(x) = x^2 - 3x - 10 \), we can look at various aspects such as the vertex, roots, or the nature of the function (e.g., whether it is increasing or decreasing).
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Finding the Roots: We can find the roots of the function by setting \( g(x) = 0 \):
\[ x^2 - 3x - 10 = 0 \]
We can factor this quadratic:
\[ (x - 5)(x + 2) = 0 \]
Thus, the roots are \( x = 5 \) and \( x = -2 \).
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Vertex: The vertex of a parabola in the form \( ax^2 + bx + c \) can be determined using the formula:
\[ x = -\frac{b}{2a} \]
In this case, \( a = 1 \) and \( b = -3 \):
\[ x = -\frac{-3}{2 \cdot 1} = \frac{3}{2} \]
To find the y-coordinate of the vertex, substitute \( x = \frac{3}{2} \) back into \( g(x) \):
\[ g\left(\frac{3}{2}\right) = \left(\frac{3}{2}\right)^2 - 3\left(\frac{3}{2}\right) - 10 \] \[ = \frac{9}{4} - \frac{9}{2} - 10 = \frac{9}{4} - \frac{18}{4} - \frac{40}{4} = \frac{9 - 18 - 40}{4} = \frac{-49}{4} \]
So the vertex is at \( \left(\frac{3}{2}, -\frac{49}{4}\right) \).
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Nature of the Function: Since the leading coefficient \( a = 1 \) is positive, the parabola opens upwards. This means:
- The minimum value of the function occurs at the vertex.
- The function decreases until it reaches \( x = \frac{3}{2} \) and then increases thereafter.
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Conclusion: Depending on the statements you have, here is what you can conclude:
- The function has roots at \( x = 5 \) and \( x = -2 \).
- The vertex of the parabola is at \( \left(\frac{3}{2}, -\frac{49}{4}\right) \), which represents a minimum point.
- The function is decreasing on the interval \( (-\infty, \frac{3}{2}) \) and increasing on \( (\frac{3}{2}, \infty) \).
If you provide specific statements, I can help determine which one is true based on this analysis.
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