Given a quadratic equation y ( x ) and three points through which it passes, determine the

Alessandra Clarke

Alessandra Clarke

Answered question

2022-05-20

Given a quadratic equation y ( x ) and three points through which it passes, determine the value of a constant so that y ( x ) has a minimum
y = a x 2 + b x + c
passes through ( 1 , 1 ), ( 2 , m ) and ( m , 2 ) where m 2. Determine an interval for m such that y ( x ) has a minimum.
Progress:
If it has a minimum, then a > 0.
Also, by the points given:
a + b + c = 1
m = 4 a + 2 b + c
a m 2 + b m + c = 2
Using the fact that a is positive, I got the expressions:
a + b + c = 1 b + c < 1
a m 2 + b m + c = 2 b m + c < 2
m = 4 a + 2 b + c 2 b + c < m
But I don't know how to solve it.

Answer & Explanation

losing2mema

losing2mema

Beginner2022-05-21Added 7 answers

solving your system of equations we get
a = m 1 m , b = m 2 + m + 1 m 1 , c = m 2 + m 2 m 1
a > 0 means that
m 1 m > 0
case work yiels
0 < m < 1

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