The function f(x) = −(x + 5)(x + 1) is shown.What is the range of the function? all real numbers less than or equal to 4 all real numbers less than or equal to −3 all real number…

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asked May 19, 2017 233k views

2 Answers

Csnewb · Answer 1 · 2017-05-21T01:56:48+0000

The function f(x) = −(x + 5)(x + 1) is shown.What is the range of the function?

all real numbers less than or equal to 4

Shaydawg · Answer 2 · 2017-05-24T01:08:19+0000

we have

f(x)=-(x+5)(x+1)

we know that

The equation of a vertical parabola into vertex form is equal to

y=a(x-h)^(2)+k

where

(h,k) is the vertex of the parabola

if
a>0 ----> the parabola open upward (vertex is a minimum)

if
a<0 ----> the parabola open downward (vertex is a maximum)

In this problem convert the quadratic equation into vertex form

so

$f(x)=-(x+5)(x+1)\\f(x)=-(x^(2) +x+5x+5)\\f(x)=-( x^(2) +6x+5)$

f(x)=-x^(2)-6x-5

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

f(x)+5=-(x^(2)+6x)

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

f(x)+5-9=-(x^(2)+6x+9)

f(x)-4=-(x^(2)+6x+9)

Rewrite as perfect squares

f(x)-4=-(x+3)^(2)

f(x)=-(x+3)^(2)+4

This is a vertical parabola open down (vertex is a maximum)

the vertex is the point
(-3,4)

The range is the interval--------> (-∞,4]

$y\leq 4$

All real numbers less than or equal to

therefore

the answer is the option

All real numbers less than or equal to