Question

Given `g\left(x\right)=x^{2}-3x-10`, which statement is true?Given `g\left(x\right)=x^{2}-3x-10`, which statement is true?

Answer 1

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).

1. **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 \( x = -2 \).

2. **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 \( \left(\frac{3}{2}, -\frac{49}{4}\right) \).

3. **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.

4. **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.