To determine the greatest possible value of the y-intercept of the function $f(x) = ax^2 + bx + c$, given that $a > 2$ and the roots of the function are $x = -5$ and $x = 2$, we need to find the values of $a$, $b$, and $c$ that satisfy these conditions.

Since the roots of the quadratic function $f(x) = 0$ are $x = -5$ and $x = 2$, we can write $f(x)$ in its factored form: $f(x) = a(x + 5)(x - 2)$

Next, expand this expression: $f(x) = a(x^2 - 2x + 5x - 10)$ $f(x) = a(x^2 + 3x - 10)$

Thus, the coefficients are: $a = a$ $b = 3a$ $c = -10a$

The y-intercept of the function $f(x)$ occurs when $x = 0$, so: $f(0) = c = -10a$

We need to maximize the y-intercept, which means we need to find the maximum value of $-10a$ given that $a > 2$.

Since $a$ must be an integer greater than 2, the smallest integer greater than 2 is 3. To maximize $-10a$, we choose the smallest possible $a$: $a = 3$

Now, calculate the y-intercept using $a = 3$: $c = -10a = -10 \times 3 = -30$

Therefore, the greatest possible value of the y-intercept is $-30$.

It appears I made an error in calculations. Given $a = 3$, the correct maximum value should be $c = -10 \times 3 = -30$.

Apologies for any confusion, let's verify the available choices. From the options given: A. -18 B. -19

With $a > 2$ and using the smallest integer $a = 3$, the correct maximum possible value of the y-intercept $c$ was incorrectly stated above. Given that $a = 3$ indeed results in $-30$.

The correct response based on the initial input should ideally be validated from the given values: None of the given choices. The correct maximum possible y-intercept, calculated carefully, results in -30.

Would you like more details or have any questions? Here are 5 questions you might ask next:

1. How do you determine the vertex form of a quadratic function?
2. Can you explain the significance of the discriminant in quadratic equations?
3. What are some real-world applications of quadratic functions?
4. How can the roots of a quadratic function be found using the quadratic formula?
5. What is the impact of the coefficient $a$ on the graph of a quadratic function?

Tip: Always cross-verify the constraints and available options carefully to ensure the chosen result aligns with expectations.