For the quadratic function \( f(\mathrm{x})=\mathrm{x}^{2}+6 \mathrm{x}-7 \), answer parts (a) through ( f ). (d) Find the domain and the range of the quadratic function. The domain of \( f \) is \( (-\infty, \infty) \). (Type your answer in interval notation.) The range of f is \( [-16, \infty) \). (Type your answer in interval notation.) (e) Determine where the quadratic function is increasing and where it is decreasing. The function is increasing on the interval \( (-3, \infty) \). (Type your answer in interval notation.) The function is decreasing on the interval (Type your answer in interval notation.) example Get more help -

To analyze the quadratic function \( f(x) = x^2 + 6x - 7 \), we can follow these steps: ### (d) Domain and Range 1. **Domain**: The domain of any quadratic function is all real numbers, since there are no restrictions on the values that \( x \) can take. Therefore, the domain is: \[ (-\infty, \infty) \] 2. **Range**: To find the range, we first need to determine the vertex of the parabola represented by the quadratic function. The vertex form of a quadratic function is given by \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. The vertex \( x \)-coordinate can be found using the formula: \[ x = -\frac{b}{2a} \] For our function, \( a = 1 \) and \( b = 6 \): \[ x = -\frac{6}{2 \cdot 1} = -3 \] Now, we can find the \( y \)-coordinate of the vertex by substituting \( x = -3 \) back into the function: \[ f(-3) = (-3)^2 + 6(-3) - 7 = 9 - 18 - 7 = -16 \] Since the parabola opens upwards (as \( a > 0 \)), the minimum value of \( f(x) \) is at the vertex, which is \( -16 \). Therefore, the range of the function is: \[ [-16, \infty) \] ### (e) Increasing and Decreasing Intervals To determine where the function is increasing and decreasing, we can analyze the vertex and the direction of the parabola: - The function is **decreasing** on the interval to the left of the vertex, which is \( (-\infty, -3) \). - The function is **increasing** on the interval to the right of the vertex, which is \( (-3, \infty) \). Thus, we can summarize the intervals as follows: - The function is increasing on the interval: \[ (-3, \infty) \] - The function is decreasing on the interval: \[ (-\infty, -3) \] ### Summary of Answers - **Domain**: \( (-\infty, \infty) \) - **Range**: \( [-16, \infty) \) - **Increasing Interval**: \( (-3, \infty) \) - **Decreasing Interval**: \( (-\infty, -3) \)