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Let's solve the quadratic function \( f(x) = 2x^2 - 10x - 14 \) by finding the required points step by step:

### A) Finding the Vertex:
The vertex of a parabola given by \( f(x) = ax^2 + bx + c \) is found using the formula:
\[
x_{\text{vertex}} = -\frac{b}{2a}
\]
For \( f(x) = 2x^2 - 10x - 14 \), we have:
- \( a = 2 \)
- \( b = -10 \)

Using the vertex formula:
\[
x_{\text{vertex}} = -\frac{-10}{2(2)} = \frac{10}{4} = 2.5
\]

To find the \( y \)-coordinate of the vertex, plug \( x = 2.5 \) back into the equation:
\[
f(2.5) = 2(2.5)^2 - 10(2.5) - 14 = 2(6.25) - 25 - 14 = 12.5 - 25 - 14 = -26.5
\]

Thus, the vertex is \( (2.5, -26.5) \).

### B) Vertical Intercept (y-intercept):
The vertical intercept is the point where \( x = 0 \). Substituting \( x = 0 \) into the equation:
\[
f(0) = 2(0)^2 - 10(0) - 14 = -14
\]
Thus, the vertical intercept is \( (0, -14) \).

### C) Finding the x-intercepts:
To find the x-intercepts, solve \( f(x) = 0 \):
\[
2x^2 - 10x - 14 = 0
\]
We can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For \( a = 2 \), \( b = -10 \), and \( c = -14 \):
\[
x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(2)(-14)}}{2(2)} = \frac{10 \pm \sqrt{100 + 112}}{4} = \frac{10 \pm \sqrt{212}}{4}
\]
\[
\sqrt{212} \approx 14.56
\]
Thus, the x-intercepts are:
\[
x = \frac{10 + 14.56}{4} = \frac{24.56}{4} = 6.14
\]
\[
x = \frac{10 - 14.56}{4} = \frac{-4.56}{4} = -1.14
\]

So, the x-intercepts are approximately \( (6.14, 0) \) and \( (-1.14, 0) \).

### Summary:
- A) Vertex: \( (2.5, -26.5) \)
- B) Vertical intercept: \( (0, -14) \)
- C) x-intercepts: \( (6.14, 0), (-1.14, 0) \)

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you derive the quadratic formula?
2. What is the significance of the vertex in a quadratic function?
3. Can a parabola have more than two x-intercepts?
4. How does the value of 'a' affect the shape of the parabola?
5. What is the axis of symmetry for this quadratic function?

**Tip**: The discriminant \( b^2 - 4ac \) in the quadratic formula tells you the nature of the roots: if positive, there are two real roots; if zero, one real root; if negative, no real roots.

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

Learn how to solve the quadratic function f(x) = 2x^2 - 10x - 14 step by step. Find the vertex, vertical intercept, and x-intercepts using key algebraic concepts. Ideal for Grades 9-11 students.

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