Question

the function f(x) = –x2 − 2x 15 is shown on the graph. what are the domain and range of the function? a. the domain is all real numbers. the range is {y|y < 16}. b. the domain is all real numbers. the range is {y|y ≤ 16}. c. the domain is {x|–5 < x < 3}. the range is {y|y < 16}. d. the domain is {x|–5 ≤ x ≤ 3}. the range is {y|y ≤ 16}.

Answer 1

we have

$f(x)=-x^{2} -2x+15$

Convert to vertex form

Group terms that contain the same variable, and move the constant to the opposite side of the equation

$f(x)-15=-x^{2} -2x$

$f(x)-15=-(x^{2} +2x)$

Complete the square. Remember to balance the equation by adding the same constants to each side

$f(x)-15-1=-(x^{2} +2x+1)$

$f(x)-16=-(x^{2} +2x+1)$

Rewrite as perfect squares

$f(x)-16=-(x+1)^{2}$

$f(x)=-(x+1)^{2}+16$

This is a vertical parabola open downward

The vertex is the point $(-1,16)$ is a maximun

The domain is the interval--------> (-∞,∞)

All real numbers

The range is the interval-----> (-∞,16]

$y\leq 16$

All real numbers less than or equal to $16$

The graph in the attached figure

therefore

the answer is the option B

the domain is all real numbers. the range is {y|y ≤ 16}

Answer 2

-x^2 - 2x + 15 = -(x^2 + 2x - 15) = -(x^2 + 2x + 1 - 15 - 1) = -(x + 1)^2 - (-15 '- 1) = -(x + 1)^2 + 16
Vertex = (-1, 16)
Range = {y|y ≤ 16}
The domain is all real numbers. the range is {y|y ≤ 16}