Consider the quadratic function:  
f
(
x
)
=
4
x
2
+
14
x
−
6
   
  .  Find the following for this parabola.

A) The line of symmetry: 
  

B)  The vertex: 
  

C) The vertical intercept is the point 
  

D) Give the coordinates of the two  
x
  intercepts of the parabola as ordered pairs.
  
Round your values to two decimal places for this part, if the roots are irrational.

We are given the quadratic function:

\[
f(x) = 4x^2 + 14x - 6
\]

Let's find the requested information step by step.

### A) Line of Symmetry
The line of symmetry for a quadratic function \(f(x) = ax^2 + bx + c\) is given by the formula:

\[
x = -\frac{b}{2a}
\]

In this case, \(a = 4\) and \(b = 14\). Substituting into the formula:

\[
x = -\frac{14}{2(4)} = -\frac{14}{8} = -1.75
\]

So, the line of symmetry is:

\[
x = -1.75
\]

---

### B) Vertex
The vertex of a parabola is found by using the line of symmetry (which gives the x-coordinate), and then plugging that value back into the quadratic function to find the y-coordinate.

We already know \(x = -1.75\). Now, substitute this value into the original equation to find the y-coordinate:

\[
f(-1.75) = 4(-1.75)^2 + 14(-1.75) - 6
\]

\[
f(-1.75) = 4(3.0625) - 24.5 - 6 = 12.25 - 24.5 - 6 = -18.25
\]

So, the vertex is:

\[
(-1.75, -18.25)
\]

---

### C) Vertical Intercept
The vertical intercept (y-intercept) occurs when \(x = 0\). To find this, simply substitute \(x = 0\) into the quadratic equation:

\[
f(0) = 4(0)^2 + 14(0) - 6 = -6
\]

So, the vertical intercept is:

\[
(0, -6)
\]

---

### D) x-Intercepts (Roots)
To find the x-intercepts, we need to solve the equation \(f(x) = 0\), or:

\[
4x^2 + 14x - 6 = 0
\]

We can solve this quadratic equation using the quadratic formula:

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

Where \(a = 4\), \(b = 14\), and \(c = -6\).

First, calculate the discriminant:

\[
\Delta = b^2 - 4ac = 14^2 - 4(4)(-6) = 196 + 96 = 292
\]

Now, substitute the values into the quadratic formula:

\[
x = \frac{-14 \pm \sqrt{292}}{2(4)} = \frac{-14 \pm \sqrt{292}}{8}
\]

Approximating \(\sqrt{292} \approx 17.09\):

\[
x = \frac{-14 \pm 17.09}{8}
\]

Now, calculate the two possible solutions:

1. \(x_1 = \frac{-14 + 17.09}{8} = \frac{3.09}{8} \approx 0.39\)
2. \(x_2 = \frac{-14 - 17.09}{8} = \frac{-31.09}{8} \approx -3.89\)

So, the x-intercepts (roots) are:

\[
(0.39, 0) \quad \text{and} \quad (-3.89, 0)
\]

---

### Summary of Results:
- **A) Line of symmetry:** \(x = -1.75\)
- **B) Vertex:** \((-1.75, -18.25)\)
- **C) Vertical intercept:** \((0, -6)\)
- **D) x-intercepts:** \((0.39, 0)\) and \((-3.89, 0)\)

Would you like further explanation on any part?

---

Here are five related questions:
1. How can we determine if a quadratic function opens upwards or downwards?
2. What is the discriminant, and how does it affect the number of roots?
3. How do you find the vertex form of a quadratic function?
4. Can every quadratic function have real x-intercepts?
5. How would the function change if the coefficient of \(x^2\) was negative?

**Tip:** The vertex represents the minimum or maximum point of a quadratic function, depending on whether it opens upward or downward.

Consider the quadratic function: f(x) = 4x^2 + 14x − 6. Find the following for this parabola.
A) The line of symmetry
B) The vertex
C) The vertical intercept is the point
D) Give the coordinates of the two x intercepts of the parabola as ordered pairs. Round your values to two decimal places for this part, if the roots are irrational.

Learn how to solve the quadratic function f(x) = 4x^2 + 14x - 6 by finding the line of symmetry, vertex, vertical intercept, and x-intercepts. This step-by-step guide covers key algebraic concepts including the quadratic formula and parabola properties. Suitable for students in Grades 9-11.

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Consider the quadratic function:
f ( x )

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