Let's analyze the given quadratic functions step-by-step to answer the questions.
### Function 1: [tex]\( f(x) = -3x^2 + 12x - 7 \)[/tex]
(a) What is the vertex of Function 1?
The vertex of a quadratic function in the form of [tex]\( ax^2 + bx + c \)[/tex] can be found using the vertex formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Here, [tex]\( a = -3 \)[/tex], [tex]\( b = 12 \)[/tex], and [tex]\( c = -7 \)[/tex].
1. Calculate the x-coordinate of the vertex:
[tex]\[ x = -\frac{12}{2 \cdot -3} = -\frac{12}{-6} = 2 \][/tex]
2. Substitute [tex]\( x = 2 \)[/tex] back into the function [tex]\( f(x) \)[/tex] to find the y-coordinate:
[tex]\[ f(2) = -3(2)^2 + 12(2) - 7 = -3(4) + 24 - 7 = -12 + 24 - 7 = 5 \][/tex]
Therefore, the vertex of Function 1 is:
[tex]\[ (2, 5) \][/tex]
### Function 2: Given points are symmetrical around [tex]\( x = -2 \)[/tex] and [tex]\( y = 8 \)[/tex]
(b) What is the vertex of Function 2?
The provided points for Function 2 show symmetry around [tex]\( x = -2 \)[/tex] and reach a maximum at [tex]\( y = 8 \)[/tex]. Thus, the vertex is:
[tex]\[ (-2, 8) \][/tex]
### Comparing Maximum Values
(c) Which function has the larger maximum value?
Compare the y-coordinates of the vertices obtained from both functions:
- Vertex of Function 1: [tex]\( (2, 5) \)[/tex]
- Vertex of Function 2: [tex]\( (-2, 8) \)[/tex]
The maximum value of Function 1 is [tex]\( 5 \)[/tex] and the maximum value of Function 2 is [tex]\( 8 \)[/tex].
Therefore, Function 2 has the larger maximum value.
The solutions summarize:
- (a) The vertex of Function 1 is [tex]\( (2, 5) \)[/tex].
- (b) The vertex of Function 2 is [tex]\( (-2, 8) \)[/tex].
- (c) Function 2 has the larger maximum value, which is [tex]\( 8 \)[/tex].
### Final Answers:
(a) The vertex of Function 1: [tex]\( (2, 5) \)[/tex].
(b) The vertex of Function 2: [tex]\( (-2, 8) \)[/tex].
(c) The function with the larger maximum value is Function 2, and the larger maximum value is [tex]\( 8 \)[/tex].
