c) \( K(x)=-x^{2}-2 x-2 \)

The function you've provided is a quadratic function given by: \[ K(x) = -x^2 - 2x - 2 \] To analyze this function, we can identify several key features: 1. **Standard Form**: The function is in standard form \( K(x) = ax^2 + bx + c \) where \( a = -1 \), \( b = -2 \), and \( c = -2 \). 2. **Direction of the Parabola**: Since the coefficient of \( x^2 \) (which is \( a \)) is negative, the parabola opens downwards. 3. **Vertex**: The vertex of a quadratic function in standard form can be found using the formula for the x-coordinate of the vertex: \[ x = -\frac{b}{2a} \] Substituting the values of \( a \) and \( b \): \[ x = -\frac{-2}{2 \cdot -1} = \frac{2}{-2} = -1 \] Now, we can find the y-coordinate of the vertex by substituting \( x = -1 \) back into the function: \[ K(-1) = -(-1)^2 - 2(-1) - 2 = -1 + 2 - 2 = -1 \] Thus, the vertex of the parabola is at the point \( (-1, -1) \). 4. **Y-intercept**: The y-intercept occurs when \( x = 0 \): \[ K(0) = -0^2 - 2(0) - 2 = -2 \] So, the y-intercept is at the point \( (0, -2) \). 5. **X-intercepts**: To find the x-intercepts, we set \( K(x) = 0 \): \[ -x^2 - 2x - 2 = 0 \] Multiplying through by -1 gives: \[ x^2 + 2x + 2 = 0 \] Now we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): Here, \( a = 1 \), \( b = 2 \), and \( c = 2 \): \[ x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1} = \frac{-2 \pm \sqrt{4 - 8}}{2} = \frac{-2 \pm \sqrt{-4}}{2} = \frac{-2 \pm 2i}{2} = -1 \pm i \] Thus, the x-intercepts are complex: \( -1 + i \) and \( -1 - i \). ### Summary of Key Features: - **Vertex**: \( (-1, -1) \) - **Y-intercept**: \( (0, -2) \) - **X-intercepts**: \( -1 + i \) and \( -1 - i \) (complex) - **Direction**: Opens downwards If you have any specific questions or need further analysis, feel free to ask!