Consider the following quadratic function. \[ g(x)=-3 x^{2}-18 x-25 \]

The given quadratic function is \[ g(x) = -3x^2 - 18x - 25. \] To analyze this function, we can find several key features, such as the vertex, the axis of symmetry, the x-intercepts, and the y-intercept. ### 1. Vertex The vertex of a quadratic function in the form \( ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex: \[ x = -\frac{b}{2a}. \] Here, \( a = -3 \) and \( b = -18 \). Plugging in these values: \[ x = -\frac{-18}{2 \cdot -3} = \frac{18}{-6} = -3. \] Now, we can find the y-coordinate of the vertex by substituting \( x = -3 \) back into the function: \[ g(-3) = -3(-3)^2 - 18(-3) - 25. \] \[ = -3(9) + 54 - 25. \] \[ = -27 + 54 - 25 = 2. \] Thus, the vertex is at the point \((-3, 2)\). ### 2. Axis of Symmetry The axis of symmetry is the vertical line that passes through the vertex. Therefore, the equation of the axis of symmetry is: \[ x = -3. \] ### 3. X-Intercepts To find the x-intercepts, we set \( g(x) = 0 \): \[ -3x^2 - 18x - 25 = 0. \] We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = -3 \), \( b = -18 \), and \( c = -25 \): First, we calculate the discriminant: \[ b^2 - 4ac = (-18)^2 - 4(-3)(-25) = 324 - 300 = 24. \] Now, applying the quadratic formula: \[ x = \frac{-(-18) \pm \sqrt{24}}{2(-3)} = \frac{18 \pm 2\sqrt{6}}{-6}. \] This simplifies to: \[ x = -3 \mp \frac{\sqrt{6}}{3}. \] Thus, the x-intercepts are: \[ x = -3 + \frac{\sqrt{6}}{3} \quad \text{and} \quad x = -3 - \frac{\sqrt{6}}{3}. \] ### 4. Y-Intercept To find the y-intercept, we evaluate \( g(0) \): \[ g(0) = -3(0)^2 - 18(0) - 25 = -25. \] Thus, the y-intercept is at the point \((0, -25)\). ### Summary - **Vertex**: \((-3, 2)\) - **Axis of Symmetry**: \(x = -3\) - **X-Intercepts**: \(x = -3 + \frac{\sqrt{6}}{3}\) and \(x = -3 - \frac{\sqrt{6}}{3}\) - **Y-Intercept**: \((0, -25)\) If you have any further questions or need additional information, feel free to ask!