Using the Factor Theorem, the quadratic functions are given as follows:
- Opens upward: f(x) = x² - 5x - 24.
- Opens downward: f(x) = -x² + 5x + 24.
What is the Factor Theorem?
The Factor Theorem states that a polynomial function with roots [tex]x_1, x_2, \codts, x_n[/tex] is given by:
[tex]f(x) = a(x - x_1)(x - x_2) \cdots (x - x_n)[/tex]
In which a is the leading coefficient.
For this problem, the roots are given as follows:
[tex]x_1 = -3, x_2 = 8[/tex]
Hence:
f(x) = a(x + 3)(x - 8)
f(x) = a(x² - 5x - 24).
It opens upward with a positive leading coefficient, that is, a > 0, hence one example is a = 1 as follows:
f(x) = x² - 5x - 24.
It opens downward with a negative leading coefficient, that is, a < 0, hence one example is a = -1 as follows:
f(x) = -x² + 5x + 24.
More can be learned about the Factor Theorem at https://brainly.com/question/24380382
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