Question

the maximum value of the function $f(x) = -2x^2 + 4x + 5$ occurs at what y - coordinate?
a. 2
b. 7
c. 8
d. 1

Explanation:

Step1: Identify the vertex formula for a quadratic function

For a quadratic function in the form \( f(x) = ax^2 + bx + c \), the x - coordinate of the vertex (where the maximum or minimum occurs) is given by \( x = -\frac{b}{2a} \). In the function \( f(x)=-2x^{2}+4x + 5 \), we have \( a=-2 \) and \( b = 4 \).

Step2: Calculate the x - coordinate of the vertex

Using the formula \( x=-\frac{b}{2a} \), substitute \( a=-2 \) and \( b = 4 \):
\( x=-\frac{4}{2\times(-2)}=-\frac{4}{-4} = 1 \)

Step3: Find the y - coordinate by substituting x = 1 into the function

Substitute \( x = 1 \) into \( f(x)=-2x^{2}+4x + 5 \):
\( f(1)=-2\times(1)^{2}+4\times(1)+5=-2 + 4+5=7 \)

Answer:

b. 7