We are given the quadratic function:

$f(x) = -3(x + 5)(x + 7)$

Step 1: Determining if the function has a minimum or maximum.

To determine whether a quadratic function has a minimum or maximum, we check the coefficient of the $x^2$ term. The general form of a quadratic equation is:

$f(x) = ax^2 + bx + c$

• If $a > 0$, the parabola opens upwards, and the function has a minimum.
• If $a < 0$, the parabola opens downwards, and the function has a maximum.

Since the function is factored, we first recognize that the leading coefficient $a$ comes from expanding the product of the two terms. The coefficient of the $x^2$ term will be the product of $-3$ (the leading coefficient) and the squared terms from expanding the binomials.

Here, $a = -3$, which is negative, meaning that the function has a maximum.

Step 2: Finding the maximum value.

The maximum or minimum of a quadratic function occurs at its vertex. Since the function is given in factored form, we can find the vertex by using the midpoint of the roots.

The roots of the function are found by solving $x + 5 = 0$ and $x + 7 = 0$:

$x = -5 \quad \text{and} \quad x = -7$

The midpoint (x-coordinate of the vertex) is:

$x_{\text{vertex}} = \frac{-5 + (-7)}{2} = \frac{-12}{2} = -6$

Step 3: Finding the maximum value (y-coordinate of the vertex).

Now, substitute $x = -6$ into the function to find the maximum value:

$f(-6) = -3(-6 + 5)(-6 + 7) = -3(-1)(1) = -3(-1) = 3$

Conclusion:

• The function has a maximum value.
• The maximum value of the function is 3, and it occurs at $x = -6$.

Would you like further details or have any questions?

Related Questions:

1. How do you expand the factored form of a quadratic function into standard form?
2. What is the general process for determining the vertex of a quadratic function in standard form?
3. Can you solve the same problem by completing the square instead of factoring?
4. How do you find the axis of symmetry of a quadratic function?
5. Can you graph this function and label its key points?

Tip:

When analyzing quadratic functions in factored form, the vertex can easily be found by taking the average of the roots (zeros).