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Let's analyze the given problem. The function provided is:

\[
f(x) = 2x^2 - 10x - 14
\]

You are asked to find:
A) The vertex of the parabola.
B) The vertical intercept.
C) The x-intercepts.

### A) Finding the Vertex:
The vertex form of a parabola is given by \( f(x) = a(x-h)^2 + k \), where \( (h, k) \) is the vertex. But when you have a quadratic function in standard form \( f(x) = ax^2 + bx + c \), you can calculate the vertex using the following formula:

- \( h = -\frac{b}{2a} \)
- \( k = f(h) \)

For the function \( f(x) = 2x^2 - 10x - 14 \):
- \( a = 2 \)
- \( b = -10 \)
- \( c = -14 \)

Let's calculate the vertex. 

We will first calculate \( h \) and then substitute it back into the function to find \( k \).The vertex of the parabola is \( (2.5, -26.5) \).

It seems you entered \( \left(\frac{5}{2}, \frac{53}{2}\right) \), which is incorrect because the \( y \)-coordinate should be \(-26.5\), not \(\frac{53}{2}\).

### B) The Vertical Intercept:
You already correctly identified the vertical intercept as \( (0, -14) \). This is found by substituting \( x = 0 \) into the function \( f(x) = 2x^2 - 10x - 14 \).

### C) The x-intercepts:
The x-intercepts are found by solving \( f(x) = 0 \), which gives us the quadratic equation:

\[
2x^2 - 10x - 14 = 0
\]

We will solve this quadratic equation using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Let's compute the x-intercepts.The x-intercepts of the parabola are approximately \( (-1.14, 0) \) and \( (6.14, 0) \), which you entered correctly.

### Summary of Answers:
- A) The vertex is \( (2.5, -26.5) \).
- B) The vertical intercept is \( (0, -14) \).
- C) The x-intercepts are \( (-1.14, 0) \) and \( (6.14, 0) \).

Would you like more details on any part of the solution? Here are some related questions you might explore:

1. How do you derive the quadratic formula?
2. What is the significance of the discriminant in a quadratic equation?
3. How does the shape of the parabola change when the coefficient \( a \) changes?
4. How do you complete the square to find the vertex form of a quadratic equation?
5. What are the general steps to graphing a parabola?

**Tip:** Always double-check your vertex calculations by substituting the \( x \)-coordinate of the vertex back into the original equation to ensure accuracy.

Consider the parabola given by the equation: f(x) = 2x^2 - 10x - 14. Find the following for this parabola: A) The vertex. B) The vertical intercept. C) The x-intercepts.

Learn how to solve the parabola equation f(x) = 2x^2 - 10x - 14 by finding its vertex, vertical intercept, and x-intercepts. This step-by-step solution covers key algebraic concepts such as using the vertex formula and the quadratic formula for x-intercepts. Suitable for students in Grades 9-12.

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