To determine which function has the greater maximum value, we need to analyze the given function [tex]\( f(x) = -2x^2 + 4x + 3 \)[/tex] and compare it to the function [tex]\( g(x) \)[/tex] presented in the graph (though we do not have the actual graph provided).
Let's start by finding the maximum value of [tex]\( f(x) \)[/tex]:
1. Identify the quadratic function form:
The function [tex]\( f(x) = -2x^2 + 4x + 3 \)[/tex] is in the standard form [tex]\( ax^2 + bx + c \)[/tex], where [tex]\( a = -2 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = 3 \)[/tex].
2. Vertex formula for quadratic functions:
The vertex of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] is given by the x-coordinate at [tex]\( x = -\frac{b}{2a} \)[/tex].
3. Calculate the x-coordinate of the vertex:
[tex]\[
x = -\frac{b}{2a} = -\frac{4}{2 \times -2} = \frac{4}{4} = 1
\][/tex]
4. Find the maximum value by plugging the x-coordinate into [tex]\( f(x) \)[/tex]:
[tex]\[
f(1) = -2(1)^2 + 4(1) + 3
\][/tex]
Simplifying the equation:
[tex]\[
f(1) = -2(1) + 4 + 3 = -2 + 4 + 3 = 5
\][/tex]
We have now found that the maximum value of [tex]\( f(x) \)[/tex] is 5.
Next, without having the graph of [tex]\( g(x) \)[/tex] and its maximum value explicitly given, we need to make our conclusion based on the information about [tex]\( f(x) \)[/tex]:
- If the maximum value from the graph of [tex]\( g(x) \)[/tex] is higher than 5, then [tex]\( g(x) \)[/tex] has a greater maximum value.
- If the maximum value from the graph of [tex]\( g(x) \)[/tex] is lower than 5, then [tex]\( f(x) \)[/tex] has a greater maximum value.
- If the maximum value from the graph of [tex]\( g(x) \)[/tex] is exactly 5, then both functions have the same maximum value.
Given that we only know the maximum value of [tex]\( f(x) \)[/tex] is 5 from our calculation, and without further information on [tex]\( g(x) \)[/tex], we are unable to make a definitive comparison.
Thus, based on the information about the maximum value of [tex]\( f(x) \)[/tex], we must assume that to determine which function has the greater maximum value, we must refer to the graph of [tex]\( g(x) \)[/tex].
