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**Unformatted text preview: **Example 8.2.3. Find the quadratic function which passes through the points (−1, 3), (2, 4), (5, −2).
Solution. According to Deﬁnition 2.5, a quadratic function has the form f (x) = ax2 + bx + c
where a = 0. Our goal is to ﬁnd a, b and c so that the three given points are on the graph of
f . If (−1, 3) is on the graph of f , then f (−1) = 3, or a(−1)2 + b(−1) + c = 3 which reduces to
a − b + c = 3, an honest-to-goodness linear equation with the variables a, b and c. Since the point
(2, 4) is also on the graph of f , then f (2) = 4 which gives us the equation 4a + 2b + c = 4. Lastly,
the point (5, −2) is on the graph of f gives us 25a + 5b + c = −2. Putting these together, we obtain
a system of three linear equations. Encoding this into an augmented matrix produces a−b+c =
3
1 −1
1
3 Encode into the matrix
4a + 2 b + c =
4 −− − − − − −→ 4
2
1
4
−−−−−−− 25a + 5b + c = −2
25
5
1 −2
7
Using a calculator,4 we ﬁnd a = − 18 , b = 13 and c = 37 . Hence, the one and only quadratic which
18
9
7
ﬁts the bill is f (x) = − 18 x2 + 13 x + 37 . To verify this analytically, we see that f (−1) = 3, f (2) = 4,
18
9
and f (5) = −2. We can...

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