The function f is defined by f(x) = ax^2 + bx + c, where a, b, and c are integer constants. The graph of y = f(x) in the xy-plane intersects the x-axis at the points (-5, 0) and (2, 0). If a is greater than 2, what is the greatest possible value of the y-intercept of function f?

A. -18
B. -19

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.

Explore how to maximize the y-intercept of a quadratic function defined by f(x) = ax^2 + bx + c, given specific roots and conditions. Understand the impact of coefficient a on the function and verify calculations. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation