Certainly! Let's go through the solution step by step for each part.
(a) Finding the vertex of Function 1
The function given is a quadratic function in the form of [tex]\( f(x) = -3x^2 + 12x - 5 \)[/tex].
To find the vertex of this quadratic function, we use the vertex formula for a parabola, given as [tex]\( f(x) = ax^2 + bx + c \)[/tex]:
- The x-coordinate of the vertex (h) can be found using the formula [tex]\( h = -\frac{b}{2a} \)[/tex].
- The y-coordinate of the vertex (k) is found by evaluating the function at [tex]\( x = h \)[/tex], i.e., [tex]\( k = f(h) \)[/tex].
For the given function:
- The coefficients are [tex]\( a = -3 \)[/tex], [tex]\( b = 12 \)[/tex], and [tex]\( c = -5 \)[/tex].
First, calculate [tex]\( h \)[/tex]:
[tex]\[ h = -\frac{b}{2a} = -\frac{12}{2 \times -3} = -\frac{12}{-6} = 2 \][/tex]
Next, calculate [tex]\( k \)[/tex]:
[tex]\[ k = f(2) = -3(2)^2 + 12(2) - 5 = -3(4) + 24 - 5 = -12 + 24 - 5 = 7 \][/tex]
So, the vertex of Function 1 is [tex]\( (2, 7) \)[/tex].
(b) Finding the vertex of Function 2
The vertex of Function 2 is given directly as [tex]\( (-2, 5) \)[/tex].
Therefore, the vertex of Function 2 is [tex]\( (-2, 5) \)[/tex].
(c) Comparison of Maximum Values
Quadratic functions have a maximum or minimum value at their vertices, depending on the direction in which the parabola opens. Since the leading coefficient [tex]\( a \)[/tex] in Function 1 is negative ([tex]\( a = -3 \)[/tex]), the parabola opens downwards, indicating that the vertex represents a maximum point.
For Function 2, since we are only given the vertex and not the full form of the function, we focus on the y-coordinate of the vertex which represents the function’s maximum value.
The y-coordinate of the vertices are:
- Function 1: Maximum value is [tex]\( 7 \)[/tex] (vertex [tex]\( (2, 7) \)[/tex]).
- Function 2: Maximum value is [tex]\( 5 \)[/tex] (vertex [tex]\( (-2, 5) \)[/tex]).
Comparing these maximum values:
- Function 1 has a maximum value of [tex]\( 7 \)[/tex].
- Function 2 has a maximum value of [tex]\( 5 \)[/tex].
Thus, Function 1 has the larger maximum value.
Summary of Findings:
(a) The vertex of Function 1 is [tex]\( (2, 7) \)[/tex].
(b) The vertex of Function 2 is [tex]\( (-2, 5) \)[/tex].
(c) Function 1 has the larger maximum value.
