Big Ideas Math Algebra 2, 2014
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Big Ideas Math Algebra 2, 2014 View details
4. Modeling with Quadratic Functions
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Exercise 32 Page 82

Check the second difference sequence.

Function type: Quadratic, see solution.
Function rule: - 16x^2-x+320

Practice makes perfect

Since the data has equally-spaced inputs, we can check whether the data is linear by checking the first difference sequence.

The first difference sequence is not constant, so this data is not linear. Let's check the second difference sequence. This tells whether the data is quadratic or not.
Since the second difference sequence is constant, this data is quadratic. We do not have enough data to use either vertex form or the intercept form of a quadratic equation, so we will solve a system of three equations. We write the equations by substituting values from the table for x and y. (I): 303=a(1)^2+b(1)+c (II): 254=a(2)^2+b(2)+c (III): 320=a(0)^2+b(0)+c Because the third equation immediately simplifies to c=320, we can substitute it into remaining two equation.
303=a(1)^2+b(1)+c & (I) 254=a(2)^2+b(2)+c & (II)

(I), (II):c= 320

303=a(1)^2+b(1)+ 320 & (I) 254=a(2)^2+b(2)+ 320 & (II)

(I), (II):Calculate power and product

303=a+b+320 & (I) 254=4a+2b+320 & (II)
b=- 17-a & (I) 254=4a+2b+320 & (II)
b=-17-a & (I) 254=4a+2( - 17-a)+320 & (II)
b=-17-a & (I) 254=4a-34-2a+320 & (II)
b=-17-a & (I) 254=2a+286 & (II)
b=-17-a & (I) -32=2a & (II)
b=-17-a & (I) a=- 16 & (II)
Now the only unknown value is b, so let's find it by substituting a=- 16 into the second equation.
b=-17-a
b=-17-( - 16)
b=- 1
Since we have all of the coefficients we can complete the equation of the function. y=- 16x^2-x+320