Answer :
Sure! Let's break down the question step by step:
1. **Graphing the function f(x) = x² + 8x + 18**:
To graph this function, you can start by finding the vertex of the parabola. The vertex form of a parabola is given by \(f(x) = a(x - h)^2 + k\), where (h, k) is the vertex of the parabola. In this case, the function f(x) = x² + 8x + 18 can be rewritten as f(x) = (x + 4)² + 2. So, the vertex of the parabola is (-4, 2).
Now, you know the vertex and the shape of the parabola (opens upwards since the coefficient of x² is positive). You can plot the vertex on the coordinate plane and use the shape of the parabola to sketch the rest of the graph.
2. **Identifying the domain and range of the function**:
- The domain of a function is the set of all possible x-values for which the function is defined. In this case, since f(x) is a polynomial function, it is defined for all real numbers. Therefore, the domain of the function f(x) = x² + 8x + 18 is all real numbers, or (-∞, ∞).
- The range of a function is the set of all possible y-values that the function can output. For a quadratic function in the form f(x) = ax² + bx + c where a > 0 (as in this case), the minimum value of the function occurs at the vertex, and the function is increasing on both sides of the vertex. Therefore, the range of the function f(x) = x² + 8x + 18 is [2, ∞), since the minimum value of the function is 2.
By following these steps, you can graph the function and identify its domain and range accurately.
