Let's break down the process for the given quadratic function [tex]\( f(x) = -2x^2 + 16x - 9 \)[/tex]:
### Part (a)
Determine whether the function has a minimum value or a maximum value:
The general form of a quadratic function is [tex]\( ax^2 + bx + c \)[/tex]. In this case:
- [tex]\( a = -2 \)[/tex]
- [tex]\( b = 16 \)[/tex]
- [tex]\( c = -9 \)[/tex]
For quadratic functions:
- If [tex]\( a > 0 \)[/tex], the parabola opens upwards, and the function has a minimum value.
- If [tex]\( a < 0 \)[/tex], the parabola opens downwards, and the function has a maximum value.
Given that [tex]\( a = -2 \)[/tex] (which is less than 0), the parabola opens downwards, and hence, the function has a maximum value.
### Part (b)
Find the maximum value and determine where it occurs:
The vertex of the quadratic function [tex]\( ax^2 + bx + c \)[/tex] gives us the maximum or minimum value. The x-coordinate of the vertex can be determined using the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
For our function:
[tex]\[ a = -2 \][/tex]
[tex]\[ b = 16 \][/tex]
Thus:
[tex]\[ x = -\frac{16}{2 \times -2} = -\frac{16}{-4} = 4 \][/tex]
The x-coordinate of the vertex is [tex]\( x = 4 \)[/tex]. To find the maximum value, substitute [tex]\( x = 4 \)[/tex] back into the function:
[tex]\[ f(4) = -2(4)^2 + 16(4) - 9 \][/tex]
[tex]\[ f(4) = -2 \times 16 + 64 - 9 \][/tex]
[tex]\[ f(4) = -32 + 64 - 9 \][/tex]
[tex]\[ f(4) = 23 \][/tex]
Therefore, the maximum value is 23 and it occurs at [tex]\( x = 4 \)[/tex].
### Part (c)
Identify the function's domain and its range:
- The domain of any quadratic function is all real numbers because you can substitute any real number for [tex]\( x \)[/tex]. In interval notation, the domain is:
[tex]\[ (-\infty, \infty) \][/tex]
- The range of the function is all the possible values of [tex]\( y \)[/tex]. Since the quadratic function opens downwards and has a maximum value of 23, the range includes all real numbers less than or equal to 23. In interval notation, the range is:
[tex]\[ (-\infty, 23] \][/tex]
To summarize:
a. The function has a maximum value.
b. The maximum value is [tex]\(23\)[/tex] and it occurs at [tex]\( x = 4 \)[/tex].
c. The domain of [tex]\( f \)[/tex] is [tex]\( (-\infty, \infty) \)[/tex]. The range of [tex]\( f \)[/tex] is [tex]\( (-\infty, 23] \)[/tex].
