Let's analyze the functions and the options provided.
### Function [tex]\( f \)[/tex]
Function [tex]\( f \)[/tex] is given by:
[tex]\[ f(x) = x^2 - 5x + 6 \][/tex]
#### Calculating the [tex]\( y \)[/tex]-intercept of [tex]\( f \)[/tex]
The [tex]\( y \)[/tex]-intercept is found by evaluating the function at [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 0^2 - 5 \cdot 0 + 6 = 6 \][/tex]
So, the [tex]\( y \)[/tex]-intercept of function [tex]\( f \)[/tex] is [tex]\( 6 \)[/tex].
### Function [tex]\( g \)[/tex]
Function [tex]\( g \)[/tex] is a downward-opening parabola with vertex at [tex]\( (2, 4) \)[/tex].
#### Understanding the vertex
Since the parabola opens downward, the vertex [tex]\( (2, 4) \)[/tex] represents the maximum value of the function [tex]\( g \)[/tex].
#### Calculating the [tex]\( y \)[/tex]-intercept of [tex]\( g \)[/tex]
Although the exact equation for [tex]\( g \)[/tex] is not given, we know the vertex form of a downward opening parabola:
[tex]\[ g(x) = a(x - 2)^2 + 4 \][/tex]
where [tex]\( a \)[/tex] is a negative value since the parabola opens downward.
To find the [tex]\( y \)[/tex]-intercept, we evaluate [tex]\( g \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = a(0 - 2)^2 + 4 = 4a + 4 \][/tex]
Since [tex]\( a \)[/tex] is negative, [tex]\( 4a \)[/tex] is also negative, making [tex]\( 4 + 4a \)[/tex] less than [tex]\( 4 \)[/tex]. This means the [tex]\( y \)[/tex]-intercept of [tex]\( g \)[/tex] is less than [tex]\( 4 \)[/tex].
### Comparing [tex]\( y \)[/tex]-intercepts of [tex]\( f \)[/tex] and [tex]\( g \)[/tex]
- The [tex]\( y \)[/tex]-intercept of [tex]\( f \)[/tex] is [tex]\( 6 \)[/tex].
- The [tex]\( y \)[/tex]-intercept of [tex]\( g \)[/tex] is less than [tex]\( 4 \)[/tex].
Given that [tex]\( 6 \)[/tex] (the [tex]\( y \)[/tex]-intercept of [tex]\( f \)[/tex]) is greater than [tex]\( 4 \)[/tex] (and thus greater than the [tex]\( y \)[/tex]-intercept of [tex]\( g \)[/tex]).
### Evaluating the provided options:
A. The minimum of function [tex]\( g \)[/tex] is at [tex]\( (2, 4) \)[/tex].
- Incorrect. Since function [tex]\( g \)[/tex] has a vertex at [tex]\( (2, 4) \)[/tex] and opens downward, this is the maximum point, not the minimum.
B. The [tex]\( y \)[/tex]-intercept of function [tex]\( f \)[/tex] is greater than the [tex]\( y \)[/tex]-intercept of function [tex]\( g \)[/tex].
- Correct. [tex]\( 6 \)[/tex] is greater than the [tex]\( y \)[/tex]-intercept of [tex]\( g \)[/tex], which is less than [tex]\( 4 \)[/tex].
C. The [tex]\( y \)[/tex]-intercept of function [tex]\( f \)[/tex] is less than the [tex]\( y \)[/tex]-intercept of function [tex]\( g \)[/tex].
- Incorrect. The [tex]\( y \)[/tex]-intercept of [tex]\( f \)[/tex] is [tex]\( 6 \)[/tex], which is greater than the [tex]\( y \)[/tex]-intercept of [tex]\( g \)[/tex].
D. The minimum of function [tex]\( f \)[/tex] is at [tex]\( -5, 6) \)[/tex].
- Incorrect. The minimum of [tex]\( f(x) = x^2 - 5x + 6 \)[/tex] can be found using the vertex form for a parabola [tex]\( x = -\frac{b}{2a} \)[/tex]. Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -5 \)[/tex], so [tex]\( x = \frac{5}{2} = 2.5 \)[/tex]. Thus, the minimum is not at [tex]\((-5, 6)\)[/tex].
### Conclusion
The correct answer is:
B. The [tex]\( y \)[/tex]-intercept of function [tex]\( f \)[/tex] is greater than the [tex]\( y \)[/tex]-intercept of function [tex]\( g \)[/tex].
