Question

Use the drop-down menus to describe the key aspects of the function f(x) = –x2 – 2x – 1. The vertex is the . The function is increasing . The function is decreasing . The domain of the function is . The range of the function is

Answer 1

Let's dissect    f(x) = –x2 – 2x – 1.  

a) This is a quadratic equation whose graph is a parabola that opens down.  
b) The coefficients of x^2, x and 1 are {-1,-2, -1}.


                                                                             
c) Applying the quadratic formula results in

        -(-2) plus or minus sqrt ([-2]^2 - 4(-1)(-1) )
x = ----------------------------------------------------------
                            2(-1)

or x  = + 2 plus or minus sqrt(0)
           --------------------------------- = -1
                        -2       

Actually, we have a double root here:  x is {-1, -1}.

This means that the graph of this parabola touches the x-axis once, at (-1,0), but does not cross that axis.  (-1,0) is also the "root" or "solution" or "x-intercept"

The function is increasing from -infinity up to -1, and decreasing from -1 to positive infinity.

The domain is "the set of all real numbers."

The range is "the set of all real numbers from neg. infinity to zero."  y is never positive.

Please graph this function, so that you can verify this information.

Answer 2

Answer:

The vertex is (-1, 0),

The function is increasing and decreasing both for different intervals,

The domain of the function is set of all real numbers,

Range is (-∞, 0]

Step-by-step explanation:

Given function,

$f(x)=-x^2 - 2x - 1$

$f(x) = -(x^2 + 2x+1)$

$f(x) = -(x+1)^2+0$

Since, vertex form of a quadratic equation is,

$f(x) = a(x-h)^2 + k$

Where,

(h, k) is the vertex of the function,

By comparing,

Vertex of the given function = (-1, 0),

A quadratic function with negative leading coefficient is increasing from -∞ to its vertex and decreasing from its vertex to +∞,

Also, domain of a quadratic function is set of all real numbers,

While, quadratic function with negative leading coefficient is maximum at its vertex,

∴ Range = (-∞, 0]