Question

Please help with algebra #16

Answer 1

Short answer: It is a quadratic

Preamble
It's not linear. A function of the form y = ax + b does not give the same y value for 2 different x values unless it is a line of the form y = a. That would mean that when x is 0, the y value should be (in this case) - 10. It is 8. So this  function is not linear.

Could it be a quadratic? Maybe. The y intercept is at y = -8 Since it is a point, the y intercept is (0,-8). 

So let's keep looking. There's a root between x = 1 and x = 2. How do you know? The sign changes. Is that root possible? again maybe.

If this is a quadratic, it's formula will be
y = ax^2 + bx + c
The y intercept is c. That means when x = 0 y = c
y = ax^2 + bx - 8 Now use any two points to solve for a and b

But before we do that, we should take a look at the points more closely. The only way that - 10 could be the same y value for 2 different x values is if there is a local minimum (in this case) between them. 

We know it has to be a minimum because -8 (the y part of the y intercept) is above - 10. and the value of the quadratic is above - 10

for (1,-4) we get
y = ax^2 + bx - 8
-4 = a(1)^2 + b(1) - 8
-4 = a + b - 8 Add 8 to both sides.
4 = a + b

For (2,2) we get
y = ax^2 + bx - 8
2 = a(2)^2 + b(2) - 8
2 = 4a + 2b - 8
10 = 4a + 2b Divide through by 2
5 = 2a + b
b = 5 - 2a Put this in the equation above.
a + b = 4
a + 5 - 2a = 4
-a = -1
a = 1

5 = 2a + b
5 = 2(1) + b
b = 3

So the formula according to this is 
y = x^2 +3x - 8
Is this possible? If it is, then (-2,-10) should be a point on the quadratic graph.
Is it?

y = x^2 + 3x - 8
-10 =?  (-2)^2 + 3(-2) - 8 Does the right side give - 10
-10 =? 4 - 6 - 8
-10 =? - 10

It works. There is one last thing we should check. What are the roots.
a = 1
b = 3
c = - 8

Using the quadratic formula (and I will leave this to you). 
one root is (1.70,0) and the other one is (-4.7,0)

The first root confirms what we said about a root being between x = 1 and x = 2.

Conclusion
The points are on a quadratic. One last thing, I've included a graph just to confirm everything I've written.