Use the quadratic formula to find the zeros

of f. (b) Find the maximum or minimum value of .

(c) Sketch the graph of f

f共x兲 		-3x2-6x-6

Let's tackle the problem step by step.

### Given function:
\[ f(x) = -3x^2 - 6x - 6 \]

### (a) Finding the Zeros of \( f(x) \) using the Quadratic Formula:
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the function \( f(x) = -3x^2 - 6x - 6 \), the coefficients are:
- \( a = -3 \)
- \( b = -6 \)
- \( c = -6 \)

Substituting these into the quadratic formula:
\[
x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(-3)(-6)}}{2(-3)}
\]

Simplify the equation step by step:
\[
x = \frac{6 \pm \sqrt{36 - 72}}{-6}
\]
\[
x = \frac{6 \pm \sqrt{-36}}{-6}
\]
\[
x = \frac{6 \pm 6i}{-6}
\]

Thus, the zeros of the function are:
\[
x = -1 \pm i
\]
Since the discriminant (\(b^2 - 4ac\)) is negative, the function has complex roots, meaning there are no real zeros.

### (b) Finding the Maximum or Minimum Value:
Since the coefficient of \(x^2\) (i.e., \(a = -3\)) is negative, the parabola opens downwards, meaning the function has a maximum value.

The vertex of the parabola, which gives the maximum or minimum value, is found using:
\[
x = -\frac{b}{2a}
\]
Substitute \(a = -3\) and \(b = -6\):
\[
x = -\frac{-6}{2(-3)} = 1
\]
The \(x\)-coordinate of the vertex is \(x = 1\).

Now, find \(f(1)\) to determine the maximum value:
\[
f(1) = -3(1)^2 - 6(1) - 6
\]
\[
f(1) = -3 - 6 - 6 = -15
\]
So, the maximum value of \(f(x)\) is \(-15\).

### (c) Sketching the Graph of \(f(x)\):
To sketch the graph:

1. The parabola opens downwards because \(a = -3\) (negative).
2. The vertex (maximum point) is at \((1, -15)\).
3. The function has no real zeros, indicating the parabola does not intersect the x-axis.

The graph will be a downward-opening parabola with a maximum point at \((1, -15)\).

---

Do you want further details on any step or have any questions? Here are 5 questions that could deepen your understanding:

1. How do you determine the direction in which a parabola opens?
2. Why does a negative discriminant result in complex roots?
3. How can the vertex formula be used to find the maximum or minimum value of a quadratic function?
4. What does the vertex of the parabola represent in the context of a quadratic function?
5. How do complex roots affect the graph of a quadratic function?

**Tip:** Always check the sign of the leading coefficient (\(a\)) to know if the parabola opens upwards or downwards.

Learn how to use the quadratic formula to find the zeros and maximum value of a quadratic function. Explore step-by-step solutions with detailed explanations of the discriminant and vertex form. Suitable for high school students.

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