𝒙(𝒕) = −𝟔 − 𝟏𝟓𝒕 + 𝟒𝒕𝟐

To analyze the function \( x(t) = -6 - 15t + 4t^2 \), we can follow several steps to understand its properties, such as finding its critical points, behavior, and roots.

1. Determine the Type of Function

This is a quadratic function in standard form \( ax^2 + bx + c \), where:

• \( a = 4 \)
• \( b = -15 \)
• \( c = -6 \)

2. Vertex of the Parabola

The vertex of a parabola given by \( ax^2 + bx + c \) can be found using the formula: \[ t = -\frac{b}{2a} \] Substituting \( a \) and \( b \): \[ t = -\frac{-15}{2 \cdot 4} = \frac{15}{8} = 1.875 \]

3. Value at the Vertex

To find the value of the function at the vertex: \[ x\left(\frac{15}{8}\right) = -6 - 15\left(\frac{15}{8}\right) + 4\left(\frac{15}{8}\right)^2 \] Calculating each term: \[ -6 = -6, \] \[ -15\left(\frac{15}{8}\right) = -\frac{225}{8}, \] \[ 4\left(\frac{15}{8}\right)^2 = 4 \cdot \frac{225}{64} = \frac{900}{64} = \frac{225}{16}. \] Putting it all together: \[ x\left(\frac{15}{8}\right) = -6 - \frac{225}{8} + \frac{225}{16}. \] Converting \(-6\) to fractions with a common denominator: \[ -6 = -\frac{48}{8}, \] \[ -\frac{48}{8} - \frac{225}{8} = -\frac{273}{8}. \] Now, convert \(-\frac{273}{8}\) to the common denominator of \(16\): \[ -\frac{273 \cdot 2}{16} = -\frac{546}{16}, \] So, we have: \[ x\left(\frac{15}{8}\right) = -\frac{546}{16} + \frac{225}{16} = -\frac{321}{16}. \]

4. Roots of the Quadratic

To find the roots, set \( x(t) = 0 \): \[ 4t^2 - 15t - 6 = 0. \] Using the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): \[ t = \frac{15 \pm \sqrt{(-15)^2 - 4 \cdot 4 \cdot (-6)}}{2 \cdot 4}, \] \[ t = \frac{15 \pm \sqrt{225 + 96}}{8} = \frac{15 \pm \sqrt{321}}{8}. \]

5. Summary

• The vertex occurs at \( t = \frac{15}{8} \) with the value \( x\left(\frac{15}{8}\right) = -\frac{321}{16} \).
• The roots are at \( t = \frac{15 \pm \sqrt{321}}{8} \).

If you need further analysis or have specific questions about this function, feel free to ask!