Can someone help with a Algebra 2 question?

Question

This is a piecewise function below.

Determine non-zero values of a,b, and c so that the graph of h(x) is connected.
h(x)=2x+11, x<-2
ax^2+bx+c, -2<=x<=2
(x-1)(x-3), x>2

Answer 1

To make it connected, the values at the ends of each range have to match up.

x < -2:

You'll want this to connect up to the -2 <= x <= 2 graph, so you need to find what value 2x + 11 would have at x = -2:

2(-2) + 11 = -4 + 11 = 7

So the point (-2, 7) should fit the quadratic equation in -2 <= x <= 2.

x > 2:

Same deal here. When x = 2, (x - 1)(x - 3) = (2 - 1)(2 - 3) = 1(-1) = -1, so the point (2, -1) should also fit the quadratic in the middle.

-2 <= x <= 2:

We know that (2, -1) and (-2, 7) have to fit the equation:

ax² + bx + c

So I think we have 2 equations and 3 unknowns...

a(2)² + b(2) + c = -1
4a + 2b + c = -1

a(-2)² + b(-2) + c = 7
4a - 2b + c = 7

And now we have an issue.

This is an "underdetermined" system. This means that there are fewer equations than there are unknowns, so there are an infinite number of solutions.

This makes sense, because you have to have 3 points in order to uniquely find a parabola...and you only have 2.

The most general way to do this, would be to solve for two of the three variables in terms of the other one:

4a + 2b + c = -1
4a - 2b + c = 7

Add these and you get:

8a + 2c = 6
2c = 6 - 8a
c = 3 - 4a

Subtract the same 2 equations and you get:

4b = -8
b = -2

So any polynomial of the form:

ax² - 2x + (3 - 4a)

should work.

For instance, if a = 1, then

x² - 2x + (3 - 4) = x² - 2x - 1

would be the solution. If a = 2, then

2x² - 2x + (3 - 8) = 2x² - 2x - 5

would be the solution.

Hope this makes sense...sorry it's more complicated than I thought.

Answer 2

At -2, 2x+11 = 7, so you need a(-2)² + b(-2) + c = 7, that is, 4a - 2b + c = 7.

Also, at 2, (x-1)(x-3) = 1•-1 = -1, so you also need a(2)² + b(2) + c = -1, that is, 4a + 2b + c = -1.

So you solve the system:
4a - 2b + c = 7
4a +2b + c = -1 by subtraction to get
-4b = 8
b = -2

Plugging -2 in, we have 4a + 4 + c = 7, so 4a + c = 3. That means we have infinite possibilities to choose from. We have to hang a parabola from (-2,7) and (2,-1). So let a = 1, then c = -1. That gives us

x² - 2x -1, which = 7 when x = -2 and -1 when x = 2.

Answer 3

4a - 2b + c = -22
4a + 2b + c = - 1
4b = 21
b = 21/4
8a - 21 + 2c = -44
8a + 2c = -23
If the function is made continuous at x = -2,
2 = -4a + 21/4
4a = 29/4
a = 29/16
2c + 29/2 = -23
c = - 75/4