Consider the quadratic function:
f ( x )

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:

$x = -1.75$

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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:

$(-1.75, -18.25)$

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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:

$(0, -6)$

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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:

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

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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?

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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.