To analyze the function $f(x) = x^2 + x + 3$, let's explore its key characteristics.

1. Function Definition: The function $f(x) = x^2 + x + 3$ is a quadratic function, which generally has the form $ax^2 + bx + c$.

2. Coefficients: Here, $a = 1$, $b = 1$, and $c = 3$.

3. Vertex: The vertex form of a quadratic function is given by $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. For the given function, the vertex can be found using $h = -\frac{b}{2a}$: $h = -\frac{1}{2 \cdot 1} = -\frac{1}{2}$ To find $k$, substitute $h$ back into the function: $k = f\left(-\frac{1}{2}\right) = \left(-\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right) + 3 = \frac{1}{4} - \frac{1}{2} + 3 = \frac{1}{4} - \frac{2}{4} + \frac{12}{4} = \frac{11}{4} = 2.75$ Thus, the vertex of the parabola is $\left( -\frac{1}{2}, 2.75 \right)$.

4. Axis of Symmetry: The axis of symmetry for the parabola is $x = h$, so: $x = -\frac{1}{2}$

5. Y-intercept: The y-intercept occurs when $x = 0$: $f(0) = 0^2 + 0 + 3 = 3$ Thus, the y-intercept is $(0, 3)$.

6. X-intercepts: To find the x-intercepts, set $f(x) = 0$ and solve for $x$: $x^2 + x + 3 = 0$ This quadratic equation does not have real roots because the discriminant $\Delta = b^2 - 4ac$ is negative: $\Delta = 1^2 - 4 \cdot 1 \cdot 3 = 1 - 12 = -11$ Since the discriminant is negative, there are no real x-intercepts.

7. Graph: The graph of $f(x) = x^2 + x + 3$ is a parabola opening upwards with its vertex at $\left( -\frac{1}{2}, 2.75 \right)$ and no x-intercepts. It intersects the y-axis at $(0, 3)$.

Would you like more details or have any questions about the function analysis?

Here are 8 related questions for further exploration:

1. How do you derive the vertex form of a quadratic function from its standard form?
2. What is the significance of the discriminant in a quadratic equation?
3. How do you complete the square for the function $f(x) = x^2 + x + 3$?
4. How would the graph of $f(x) = -x^2 - x - 3$ compare to the graph of $f(x) = x^2 + x + 3$?
5. How do you find the range of the quadratic function $f(x) = x^2 + x + 3$?
6. What are some real-world applications of quadratic functions?
7. How can you use derivatives to find the critical points of the function $f(x) = x^2 + x + 3$?
8. What transformations can be applied to the function $f(x) = x^2 + x + 3$ to shift or stretch its graph?

Tip: When solving quadratic equations, always check the discriminant first to determine the nature of the roots (real or complex).