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Hi, could someone help me with this question?

 

Suppose f(x) is a quadratic function such that f(1) = -24, f(4) = 0, and f(7) = 60. Determine the value of f(-1).

 

Thank you so much!

 Jan 17, 2021
 #1
avatar+116126 
+2

We    have  the form

 

y = ax^2  + bx  +  c

 

Since   f(4)  = 0   then  4 is a root

And   f(1)  =  -24

And f(7)  = 60

 

So    we  have  this system of  equations

 

a(4)^2  + b(4)  + c   = 0

a(1)^2  + b(1) + c  = -24

a(7)^2 + b(7) + c  =   60         simplify

 

 

16a  + 4b +  c  =   0             (1)

a    +   b    +  c  =  -24          (2)

49a  + 7b + c  =  60             (3)

 

Subtract  (2) from (1)

15a + 3b  =  24  (divide through  by 3) ⇒   5a + b  = 8       ⇒  b = 8 - 5a    (4)

 

Subtract  (1)  from ( 3)

33a + 3b  =  60  (divide through by 3) ⇒ 11a + b  = 20         (5)

 

Sub (4) into (5)

 

11a  + (8 - 5a)  =  20

6a  =  12

a = 2

 

b = 8 - 5(2)  =  -2

 

a + b + c  = -24

 

2  - 2  + c =  -24

 

c = -24

 

So  our polynomial  is

 

f(x)   = 2x^2  - 2x  -  24

 

f(-1)  = 2(-1)^2 - 2(-1)  - 24   =   -20

 

Here's a grqph  : https://www.desmos.com/calculator/xzq6el9uj2

 

cool cool cool

 Jan 17, 2021
 #2
avatar+264 
+1

Thank you, CPhill!

Caffeine  Jan 17, 2021

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