Question

Which statements about the graph of the function f(x) = -x2 - 4x + 2 are true? Select three options.

The domain is {x\x 5-2}
The range is vy s 6)
The function is increasing over the interval (-0, -2).
The function is decreasing over the interval (-4, 0).
The function has a positive y-intercept

Answer 1

Domain is all real numbers.

The range is
`\y\le 6\`

The function is increasing over
`(-\infty,-2)`.

The function is decreasing over
`(-2,\infty)`

The function has a positive y-intercept.

-----------------This is a guess if I had interpreted your choices correctly:

Second option: The range is
`\y`

Third option: The function is increasing over
`(-\infty,-2)`.

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
`f(x)=-x^2-4x+2`.

Explanation:

`f(x)=-x^2-4x+2` 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
`-x^2-4x+2`.

Now since the is a quadratic then it is a parabola. We know it is a quadratic because it is comparable to
`ax^2+bx+c`,
`a \\eq 0`.

This means the graph sort of looks like a U or an upside down U.

It is U, when
`a>0`.

It is upside down U, when
`a<0`.

So here we have
`a=-1` so
`a<0` 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
`a<0`) or the lowest point (if
`a>0`).

The vertex here will be the highest point, again since
`a<0`.

The vertex's x-coordinate can be found by evaluating
`(-b)/(2a)`:

`(-(-4))/(2(-1))=(4)/(-2)=-2`.

So the y-coordinate can be found by evaluate
`-x^2-4x+2` for
`x=-2`:

`-(-2)^2-4(-2)+2`

`-(4)+8+2`

`4+2`

`6`

So the highest y-coordinate is 6. The range is therefore
`(-\infty,6]`.

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
`(-\infty,-2)` and decreasing on
`(-2,\infty)`.

So let's look at the intevals they have:

So on
`(-\infty,-2)` the function is increasing.

Looking on
`(-4,\infty)` 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
`x`:

`-(0)^2-4(0)+2`

`-0-0+2`

`0+2`

`2`

The y-intercept is positive since 2>0.