Answer:
Domain is all real numbers.
The range is ![\{y|y\le 6\}](https://tex.z-dn.net/?f=%5C%7By%7Cy%5Cle%206%5C%7D)
The function is increasing over
.
The function is decreasing over ![(-2,\infty)](https://tex.z-dn.net/?f=%28-2%2C%5Cinfty%29)
The function has a positive y-intercept.
-----------------This is a guess if I had interpreted your choices correctly:
Second option: The range is ![\{y|y\le 6\}](https://tex.z-dn.net/?f=%5C%7By%7Cy%5Cle%206%5C%7D)
Third option: The function is increasing over
.
Last option: The function has a positive y-intercept.
I can't really read some of your choices. So you can read my above and determine which is false. If you have a question about any of what I said above please let me know.
Note: I guess those 0's are suppose to be infinities? I hopefully your function is
.
Step-by-step explanation:
is a polynomial function which mean it has domain of all real numbers. All this sentence is really saying is that there exists a number for any value you input into
.
Now since the is a quadratic then it is a parabola. We know it is a quadratic because it is comparable to
,
.
This means the graph sort of looks like a U or an upside down U.
It is U, when
.
It is upside down U, when
.
So here we have
so
which means the parabola is an upside down U.
Let's look at the range. We know the vertex is either the highest point (if
) or the lowest point (if
).
The vertex here will be the highest point, again since
.
The vertex's x-coordinate can be found by evaluating
:
.
So the y-coordinate can be found by evaluate
for
:
![-(-2)^2-4(-2)+2](https://tex.z-dn.net/?f=-%28-2%29%5E2-4%28-2%29%2B2)
![-(4)+8+2](https://tex.z-dn.net/?f=-%284%29%2B8%2B2)
![4+2](https://tex.z-dn.net/?f=4%2B2)
![6](https://tex.z-dn.net/?f=6)
So the highest y-coordinate is 6. The range is therefore
.
If you picture the upside down U in your mind and you know the graph is symmetrical about x=-2.
Then you know the parabola is increasing on
and decreasing on
.
So let's look at the intevals they have:
So on
the function is increasing.
Looking on
the function is increasing on (-4,-2) but decreasing on the rest of that given interval.
The function's y-intercept can be found by putting 0 in for
:
![-(0)^2-4(0)+2](https://tex.z-dn.net/?f=-%280%29%5E2-4%280%29%2B2)
![-0-0+2](https://tex.z-dn.net/?f=-0-0%2B2)
![0+2](https://tex.z-dn.net/?f=0%2B2)
![2](https://tex.z-dn.net/?f=2)
The y-intercept is positive since 2>0.