Question

Which statements about the graph of the function f(x) = –x2 – 4x + 2 are true? Check all that apply.

The domain is x ≤ –2.
The range is y ≤ 6.
The function is increasing over the interval (–∞ , –2).
The function is decreasing over the interval (−4, ∞).
The function has a positive y-intercept.

Answer 1

we have

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

using a graph tool

see the attached figure

Statements

case 1) The domain is x ≤ –2

Is false. the domain is all real numbers---------> interval (-∞,∞)

case 2) The range is y ≤ 6.

Is true

Is a parabola open down, the vertex is the point
`(-2,6)`

the range is the interval ------> (-∞,-6]

case 3) The function is increasing over the interval (–∞ , –2).

Is true (see the graph)

case 4) The function is decreasing over the interval (−4, ∞)

Is false

In the interval (-4,-2) the function is increasing and in the interval (-2, ∞) the function is decreasing

case 5) The function has a positive y-intercept

Is true

The y-intercept is the point
`(0,2)`

therefore

the answer is

The range is y ≤ 6

The function is increasing over the interval (–∞ , –2)

The function has a positive y-intercept

Answer 2

For the function
f(x) = -x2 -4x + 2
The standard form is:
f(x) =-(x+2)2 + 6

Since this a quadratic function, the domain is all real numbers.
The vertex is at (-2, 6)
Since there is a negative 1 before the square term, the graph is curved downwards.
Equating the function to 0, the y-intercepts can be determined which are 0.45 and -0.45.

Therefore, from the given statements, these apply:
The range is y.
The function is increasing over the interval (–∞ , –2).
The function is decreasing over the interval (−4, ∞).
The function has a positive y-intercept.