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).

Explore the characteristics of the quadratic function f(x) = x^2 + x + 3, including its vertex, axis of symmetry, y-intercept, and graph properties. Learn how to determine the vertex form, discriminant, and absence of real x-intercepts. Suitable for high school students studying quadratic functions.

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