The given quadratic function is:

$f(x) = -5x^2 - 5x - 15$

Step 1: Determine if there is a minimum or maximum value

A quadratic function in the form $ax^2 + bx + c$ has:

• A maximum value if $a < 0$ (the parabola opens downward).
• A minimum value if $a > 0$ (the parabola opens upward).

Here, $a = -5$, so the parabola opens downward, meaning the function has a maximum value.

Step 2: Finding the vertex (maximum value)

The x-coordinate of the vertex of a quadratic function is given by:

$x = \frac{-b}{2a}$

For the function $f(x) = -5x^2 - 5x - 15$, the coefficients are:

• $a = -5$
• $b = -5$

Substitute into the vertex formula:

$x = \frac{-(-5)}{2(-5)} = \frac{5}{-10} = -\frac{1}{2}$

So, the axis of symmetry is the vertical line $x = -\frac{1}{2}$.

Step 3: Finding the maximum value

To find the maximum value, substitute $x = -\frac{1}{2}$ into the original function:

$f\left( -\frac{1}{2} \right) = -5\left( -\frac{1}{2} \right)^2 - 5\left( -\frac{1}{2} \right) - 15$

First, calculate each term: $\left( -\frac{1}{2} \right)^2 = \frac{1}{4}$ $-5 \cdot \frac{1}{4} = -\frac{5}{4}$ $-5 \cdot \left( -\frac{1}{2} \right) = \frac{5}{2}$

Now, substitute into the function: $f\left( -\frac{1}{2} \right) = -\frac{5}{4} + \frac{5}{2} - 15$

Convert to a common denominator: $f\left( -\frac{1}{2} \right) = -\frac{5}{4} + \frac{10}{4} - \frac{60}{4}$ $f\left( -\frac{1}{2} \right) = \frac{5}{4} - \frac{60}{4} = \frac{-55}{4}$

Thus, the maximum value is:

$f(x) = -\frac{55}{4} \quad \text{or} \quad -13.75$

Final Answer:

• The function has a maximum value of $-\frac{55}{4}$ or $-13.75$.
• The axis of symmetry is $x = -\frac{1}{2}$.

Would you like more details or have any questions? Here are 5 related questions for deeper understanding:

1. How do you find the vertex of a quadratic function?
2. What is the significance of the coefficient $a$ in determining the direction of the parabola?
3. How can you rewrite quadratic functions in vertex form?
4. What is the relationship between the vertex and the maximum or minimum value of the function?
5. How can you graph a quadratic function based on its vertex and axis of symmetry?

Tip: The axis of symmetry for any quadratic function always passes through the vertex, and it is given by the formula $x = \frac{-b}{2a}$.