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6 days ago

The graph of the function f(x) = (x + 2)(x + 6) is shown below.

Mathematics

2 answers:

zzz [9K]6 days ago

4 0

Answer:

Detailed explanation:

f(x) = (x + 2)(x +6)

1) The function yields positive values for all real x where x > –4:

COUNTER-EXAMPLE: x = -3 → -3>-4 but (-3+2)(-3+6) = -1 ×3 =-3, not positive.

2) The function is positive for all real x where

x < –6 or x > –3.

COUNTER-EXAMPLE: x = -2.5 → -2.5>-3 but (-2.5+2)(-2.5+6) = -0.5 ×3.5 =-1.75, not positive.

the same analysis applies to the statement: "The function is negative for all real values of x where

x < –2."

conclusion: the assertion regarding the function is valid: "The function is negative for all real x where

–6 < x < –2."

Leona [9.2K]6 days ago

4 0

Answer:

The function takes negative values for all real x where

–6 < x < –2

Detailed explanation:

the function f(x) = (x + 2)(x + 6)

Let's find the x intercepts of the function

0 = (x+2)(x+6)

x+2=0, resulting in x=-2

x+6 = 0 giving x=-6

The x intercepts are -6 and -2

Now, we will draw a number line using the x intercepts, creating 3 intervals

- infinity to -6, -6 to -2 and -2 to infinity

Next, select a number from each interval to evaluate with f(x)

For x<-6, choose -7

f(-7) = (x + 2)(x + 6)

f(-7)=(-7 + 2)(-7 + 6) =5, which is positive

–6 < x < –2, choose -4

f(-4)=(-4 + 2)(-4 + 6) =-4, which is negative

For x>-2, choose 0

f(0)=(-0+ 2)(0 + 6) =12, which is positive

The function is negative for all real x where

–6 < x < –2

Explanation

This can be simplified as follows:

10 times a given quantity of _____ hundreds results in 60 hundreds
60 hundreds convert to _____ thousands

We will divide this into two segments:

$\boxed{ \ 10 \times (N \times 100) = 60 \times 100 \ }$ ... Equation-1

$\boxed{ \ 60 \times 100) = M \times 1,000 \ }$ ... Equation-2

This straightforward multiplication relates to the place value system. We need to perform calculations to find the values of N and M.

Initially, let's focus on Equation-1.

We will move the variable M to one side of the equation in order to isolate it and solve for its value.

$\boxed{ \ 10 \times (N \times 100) = 60 \times 100 \ }$

$\boxed{ \ 1,000 \times N = 6,000 \ }$

Both parts will be divided by 1,000, essentially being multiplied by $\frac{1}{1,000}$ .

$\boxed{ \ \frac{1}{1,000} \times 1,000 \times N = \frac{1}{1,000} \times 6,000 \ }$

Thus, we conclude with $\boxed{\boxed{ \ N = 6 \ }}$

Next, we process Equation-2 to derive M's value.

$\boxed{ \ 60 \times 100) = M \times 1,000 \ }$

$\boxed{ \ 6,000 = M \times 1,000 \ }$

Rearranging this equation to place M on the left side appears as follows:

$\boxed{ \ M \times 1,000 = 6,000 \ }$

Again, both sides undergo division by 1,000, which translates to multiplication by $\frac{1}{1,000}.$ .

$\boxed{ \ \frac{1}{1,000} \times M \times 1,000 = \frac{1}{1,000} \times 6,000 \ }$

This results in $\boxed{\boxed{ \ M = 6 \ }}$

- - - - - - -

Alternative approach for the second step:

60 hundreds equals to __ thousands

$\boxed{ \ 6 \times 10 \ hundreds = \ M \ thousands} \ }$

$\boxed{ \ 6 \times thousands = \ M \ thousands} \ }$

$\boxed{\boxed{ \ M = 6 \ }}$

Additional resources

A more detailed version of this topic
2 thousands 7 tens divided by 10 equals what?
100 is equivalent to 1/10 of which number?

Keywords: 10 times as many as, blank, hundreds, 60 hundreds, or, thousands, isolate, operations, multiply, divide, fraction, both sides, equal, the opposite, both sides are multiplied by, divided by, rearrange

4 0

1 month ago

Read 2 more answers

The fish population of Lake Collins is decreasing at a rate of 3% per year. In 2004 there were about 1,300 fish. Write an expone

Inessa [9006]

In this scenario, we have a function of the following form: A: initial amount, b: rate of decrease, x: time in years. Plugging in the values results in an exponential decay function suitable for this situation: y = 1300 * (0.97) ^ x. The estimated fish population in 2010 is then calculated to be approximately 1083.

6 0

4 days ago

Read 2 more answers

Angie baked 100 cookies and 20 brownies. She wants to split them into equal groups for the big sale. Each group must have the

zzz [9093]

Avoid repeating the question as it can be perplexing.

Given 100 cookies and 20 brownies,
What is the highest number you can divide them equally by?

To find the greatest common factor,
Factor them:
100 = 2*2*5*5
20 = 2*2*5
Thus, the greatest common factor, which is shared between them, is 2*2*5 or 20.

Calculating further, 100/20=5
20/20=1

This means there are 20 groups, each containing 5 cookies and 1 brownie.

20 groups

7 0

25 days ago

During batting practice, two pop flies are hit from the same location, 2 s apart. The paths are modeled by the equations h = -16

Inessa [9006]

Answer:

2.5 seconds after the initial ball was struck.

Step-by-step explanation:

The height equations are:

h = -16t² + 56t

h = -16t² + 156t - 248

At the point where the balls intersect, both heights are alike, therefore:

-16t² + 56t = -16t² + 156t - 248

-16t² + 56t + 16t² - 156t + 248 = 0

- 100t + 248 = 0

248 = 100t

t = 248/100

t = 2.48 ≈ 2.5

4 0

1 month ago

1-i need to mix 25% mineral spirits to the varnish i am using what ratio of spirit to varnish am i using? 2-if i use 240 ml of v

Svet_ta [9518]

Answer:

(1) The necessary proportion of spirit to varnish stands at 1:3.

(2) A total of 80 ml of mineral spirit is required.

(3) The ratio of the heights to the widths of the tapestries is 3:2.

Step-by-step explanation:

(1) To achieve a mixture containing 25% mineral spirit, the blend consists of 25% spirit and 75% varnish.

Therefore, $\frac{spirit}{varnish}=\frac{25}{75}$

$=\frac{1}{3}$

Consequently, the proportion of spirit to varnish is 1:3.

(2) Using 240ml of varnish indicates this is 75% of the entire solution, which means the ratio of varnish to the total solution is 3:4.

Let the total solution quantity be $x$ .

Thus, $240:x=3:4$ .

⇒ $\frac{240}{x}=\frac{3}{4}$

⇒ $3x=240*4$

⇒ $x=\frac{240*4}{3}$

⇒ $x=320$

This means the total solution amounts to 320 ml.

Now, calculating Spirit = Total solution - Varnish

⇒Spirit = 320ml-240ml

⇒Spirit = 80ml

Therefore, when using 240 ml of varnish, you will require 80 ml of mineral spirit.

(3) The dimensions of Robison's tapestries are uniform at 1.5m in length and 1m in width. Initially, to obtain whole numbers, we multiply these dimensions by 10.

Thus, the dimensions become 15m long and 10m wide.

Now, the proportion of Height to Width is 15:10

⇒ $\frac{Height}{Width} =\frac{15}{10}$

⇒ $\frac{Height}{Width} =\frac{3}{2}$

Thus, the proportion of the heights of the tapestries to their widths is 3:2.

8 0

1 day ago