SAT Math Problem (14)

Question

If the graph of the equation y = ax²+bx+c passes through the points (0,2), (-6,-70, AND (8,-14), what is the value of a+b+c?

Answer 1

Create a system of equations using the 3 given points.

 

a(0)² + b(0) + c = 2

c = 2                          eq1       ---->    also the y-intercept

 

a(-6)² + b(-6) + c = -7

36a - 6b + c = -7                   eq2

 

a(8)² + b(8) + c = -14

64a + 8b + c = -14                       eq3

 

 

Substituting eq1 (the value of c) into the other equations,

 

36a - 6b + 2 = -7

 

36a - 6b = -9                new eq2

 

 

64a + 8b + 2 = -14

 

64a + 8b = -16                 new eq3

 

 

Use the elimination method to solve for a and b from the new equations.  Once you found a and b, evaluate

 

a + b + c

 

Answer 2

From (0, 2), c = 2

 

-70 = 36a - 6b + 2

-14 = 64a - 8b + 2

 

Solve by method of your choice

Answer 3

If the graph passes through a point (X1,Y1) means that:   Y1 = aX₁^2 + b X₁ + c

 

Then we have for the points (0,2), (-6,-7) and (8,-14) the following equations are true:

 

Point (0,2):    2 = a (0)^2 + b(0) + c     or    c =2

 

Point (-6,-7):  -7 = a(-6)^2 + b(-6) + c   =   36a - 6b +2

 

Point (8,-14):   -14 = a(8)^2 + b(8) + c   =  64a + 8b + 2

 

Then we have two linear equations:

 

36a -6b +2 = -7      or       36a -6b = -9            (x8)  288a -  48b = -72   (we multiply both sides by 8)

 

64a +8b +2 = -14    or      64a + 8b = -16        (x6)  384a + 48b = -96   (we multiply both sides by 6)

                                                                            ----------------------

                                                                             672a          =  -168

 

                                                                                   a = -0.25 = -1/4

 

Substituting the value of variable a in one of the above equations:

 

36 (1/4) - 6b = -9        9 - 6b = -9     or   -6b = -18      b = 3