Answer:
Step-by-step explanation:
For this problem, imagine the standard form, which is:
f(x) = ax² + bx + c = y
Using the table, and adding that y at the end, we can plug in and write out the following equations. (Writing it out is important):
f(-1) = a(-1)² + b(-1) + c = 14
f(0) = a(0)² + b(0) + c = 7
f(1) = a(1)² + b(1) + c = 4
f(2) = a(2)² + b(2) + c = 5
Now if this doesn't look familiar, it's actually a systems of equations using the a, b, and c elements as your three variables! If you simplify the equations:
f(-1) = a - b + c = 14
f(0) = c = 7
f(1) = a + b + c = 4
f(2) = 4a + 2b + c = 5
Something unique just happened. We have already defined what ' c ' is!
c = 7
Setting that aside, if you remove the f(x) portion of the equations, you're left with:
a - b + c = 14
a + b + c = 4
4a + 2b + c = 5
Using the two upper equations, if we add them together (you can do that as it doesn't change the values of the variables) you get:
2a + 2c = 18
Note: the ' b ' variables cancelled out in the addition [ b + (-b) ]
If you further simplify the equation:
a + c = 9
Awesome. Now we already know that c = 7, so if you plug that into the equation:
a + 7 = 9
Solve for a. So then a = 2
Now that we know the following:
a = 2
c = 7
We can then use the equation:
a + b + c = 4
And solve for b!
2 + b + 7 = 4
Simplify.
b + 9 = 4
Simplify.
b = -5
Now at this point, since you know what a, b and c are, you can write the equation!
f(x) = 2x² - 5x + 7
You can confirm your work by putting any of the x values in the table through!
