Sure, let’s analyze the function [tex]\( f(x) = -x^2 + 4x - 5 \)[/tex] in detail:
1. Recognizing the Parabola: The given function [tex]\( f(x) = -x^2 + 4x - 5 \)[/tex] is a quadratic function that represents a parabola. The coefficient of [tex]\( x^2 \)[/tex] is negative, indicating that the parabola opens downwards. This type of parabola will have a maximum point (vertex).
2. Finding the Vertex:
- For a quadratic function of the form [tex]\( ax^2 + bx + c \)[/tex], the x-coordinate of the vertex is given by [tex]\( x = -\frac{b}{2a} \)[/tex].
- Here, [tex]\( a = -1 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = -5 \)[/tex].
The x-coordinate of the vertex:
[tex]\[
x = -\frac{b}{2a} = -\frac{4}{2(-1)} = \frac{4}{2} = 2
\][/tex]
3. Evaluating the Function at the Vertex:
- To find the y-coordinate of the vertex, substitute [tex]\( x = 2 \)[/tex] back into the function [tex]\( f(x) \)[/tex].
[tex]\[
f(2) = -2^2 + 4(2) - 5 = -4 + 8 - 5 = -1
\][/tex]
Therefore, the vertex of the function [tex]\( f(x) \)[/tex] is at [tex]\( (2, -1) \)[/tex].
4. Conclusion:
- The maximum value of [tex]\( f(x) \)[/tex] is [tex]\( -1 \)[/tex].
- Given the function [tex]\( g(x) \)[/tex] whose graph is shown, we can now compare.
Without the graph of [tex]\( g(x) \)[/tex], it's not possible to numerically compare the maximum values.
In summary, the maximum value (absolute maximum) of the function [tex]\( f(x) \)[/tex] is [tex]\( -1 \)[/tex]. To compare [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], we would need the maximum value of [tex]\( g(x) \)[/tex]. If you can provide the graph or maximum value of [tex]\( g(x) \)[/tex], we can determine which function has the greater absolute maximum.
