Some of quadratic functions whose graphs are passing through points (-1,18) and
(1,2) have only positive values. Which number can be the leading coefficient a of
that functions?
A
a=1
B.
a = 2
C
a=4
D
a = 9
Answers
Given : quadratic functions whose graphs are passing through points (-1,18) and (1,2) have only positive values.
To Find : which number can be the leading coefficient a of that functions?
a = 1
a = 2
a = 4
a = 9
Solution:
f(x) = ax² + bx + c
passing through points (-1,18)
=> a - b + c = 18
passing through points ( 1,2)
=> a + b + c = 2
-2b = 16 => b = - 8
a + c = 10
f(x) = ax² + bx + c
f(x) = ax² - 8x + 10 - a
=> f'(x) = 2ax - 8
2ax - 8 = 0
=> x = 8/2a = 4/a
minimum value at x = 4/a
f(x) = ax² - 8x + 10 - a > 0
x = 4/a
a(4/a)² - 8(4/a) + 10 - a
=> 16/a - 32/a + 10 - a > 0
=> 10 > a + 16/a
a = 1 => 1 + 16/1 = 17 > 10
a = 2 => 2 + 16/2 = 10 = 10
a = 4 => 4 + 16/4 = 8 < 10
a = 9 => 9 + 16/9 > 10
Hence a = 4 satisfies this
=> c = 6
f(x) = 4x² - 8x + 6
a = 4 can be the leading coefficient a of that functions
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