The function you've provided is \( f(x) = 7x^2 + 7 \). This is a quadratic function, which has the general form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants.

In this case:
- \( a = 7 \)
- \( b = 0 \) (since there is no linear term \( x \))
- \( c = 7 \)

### Key Properties of the Function:
1. **Vertex:** The vertex of the parabola (the graph of the quadratic function) can be found using the formula for the x-coordinate of the vertex, \( x = -\frac{b}{2a} \). Since \( b = 0 \), the x-coordinate of the vertex is \( x = 0 \). Substituting \( x = 0 \) into the function, we get the y-coordinate of the vertex:
   \[
   f(0) = 7(0)^2 + 7 = 7
   \]
   So, the vertex is at \( (0, 7) \).

2. **Axis of Symmetry:** The axis of symmetry is a vertical line that passes through the vertex, which in this case is \( x = 0 \).

3. **Direction of Opening:** Since the coefficient of \( x^2 \) (which is \( 7 \)) is positive, the parabola opens upwards.

4. **Y-intercept:** The y-intercept occurs when \( x = 0 \). From the function, \( f(0) = 7 \), so the y-intercept is \( (0, 7) \).

5. **X-intercepts:** To find the x-intercepts (if any), set \( f(x) = 0 \) and solve for \( x \):
   \[
   7x^2 + 7 = 0
   \]
   \[
   7(x^2 + 1) = 0
   \]
   \[
   x^2 + 1 = 0
   \]
   \[
   x^2 = -1
   \]
   Since \( x^2 = -1 \) has no real solutions (because the square of a real number cannot be negative), there are no x-intercepts.

Would you like to go into more detail on any of these points or ask about something else related to this function?

### Follow-up Questions:
1. How do you determine the vertex of any quadratic function?
2. What is the significance of the axis of symmetry in a quadratic function?
3. Can you explain why this particular quadratic function has no real x-intercepts?
4. What does it mean when a parabola opens upwards or downwards?
5. How do you find the y-intercept of a quadratic function?
6. What would change in the graph if the coefficient of \( x^2 \) was negative?
7. How can you graph this function by hand?
8. What are some real-world applications of quadratic functions?

**Tip:** When analyzing a quadratic function, always start by identifying the coefficients \( a \), \( b \), and \( c \), as they give you crucial information about the shape and position of the parabola.

Explore the quadratic function f(x) = 7x^2 + 7. Learn about its vertex, axis of symmetry, y-intercept, and why it has no real x-intercepts. Suitable for high school students studying quadratic functions.

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