Question

a quadratic function $f(x)$ is hidden from view. you must find the maximum value of the function $f(x)$. choose the form of the quadratic function $f(x)$ that you would like to see in order to answer the question most efficiently.

form: standard form $f(x) = -3x^2 + 24x - 45$

answer attempt 1 out of 2

Explanation:

Step1: Identify the coefficients

For the quadratic function \( f(x) = -3x^2 + 24x - 45 \), we have \( a = -3 \), \( b = 24 \), and \( c = -45 \). Since \( a < 0 \), the parabola opens downward, so the vertex is the maximum point.

Step2: Find the x-coordinate of the vertex

The formula for the x-coordinate of the vertex of a quadratic function \( f(x) = ax^2 + bx + c \) is \( x = -\frac{b}{2a} \). Substituting the values, we get \( x = -\frac{24}{2(-3)} = -\frac{24}{-6} = 4 \).

Step3: Find the maximum value (y-coordinate of the vertex)

Substitute \( x = 4 \) into the function \( f(x) \): \( f(4) = -3(4)^2 + 24(4) - 45 \). Calculate each term: \( -3(16) = -48 \), \( 24(4) = 96 \). Then \( f(4) = -48 + 96 - 45 = 3 \).

Answer:

3