1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Alinara [238K]
3 years ago
6

Given a polynomial function f(x), describe the effects on the y-intercept, regions where the graph is increasing and decreasing,

and the end behavior when the following changes are made. Make sure to account for even and odd functions.
When f(x) becomes f(x) − 3
When f(x) becomes −2 ⋅ f(x)

Mathematics
2 answers:
erma4kov [3.2K]3 years ago
4 0
<span>The y-intercept of  is  .
Of course, it is 3 less than  , the y-intercept of  .
Subtracting 3 does not change either the regions where the graph is increasing and decreasing, or the end behavior. It just translates the graph 3 units down.
It does not matter is the function is odd or even.

 is the mirror image of  stretched along the y-direction.
The y-intercept, the value of  for  , is</span><span>which is  times the y-intercept of  .</span><span>Because of the negative factor/mirror-like graph, the intervals where  increases are the intervals where  decreases, and vice versa.
The end behavior is similarly reversed.
If  then  .
If  then  .
If  then  .
The same goes for the other end, as  tends to  .
All of the above applies equally to any function, polynomial or not, odd, even, or neither odd not even.
Of course, if polynomial functions are understood to have a non-zero degree,  never happens for a polynomial function.</span><span> </span>
polet [3.4K]3 years ago
3 0

First of all, let's review the definition of some concepts.


Even and odd functions:


A function is said to be even if its graph is <em>symmetric with respect to the</em> y-axis, that is:


y=f(x) \ is \ \mathbf{even} \ if, \ for \ each \ x \ in \ the \ domain \ of \ f, \\ f(-x)=f(x)


On the other hand, a function is said to be odd if its graph is <em>symmetric with respect to the origin</em>, that is:


y=f(x) \ is \ \mathbf{odd} \ if, \ for \ each \ x \ in \ the \ domain \ of \ f, \\ f(-x)=-f(x)


Analyzing each question for each type of functions using examples of polynomial functions. Thus:



FOR EVEN FUNCTIONS:


1. When f(x) becomes f(x)-3 


1.1 Effects on the y-intercept


We need to find out the effects on the y-intercept when shifting the function f(x) into:


f(x)-3


We know that the graph f(x) intersects the y-axis when x=0, therefore:


y=f(0) \ is \ the \ y-intercept \ of \ f


So:


y=f(0)-3 \ is \ the \ new \ y-intercept


So the y-intercept of f(x)-3 is three units less than the y-intercept of f(x)


1.2. Effects on the regions where the graph is increasing and decreasing


Given that you are shifting the graph downward on the y-axis, there is no any effect on the intervals of the domain. The function f(x)-3 increases and decreases in the same intervals of f(x)


1.3 The end behavior when the following changes are made.


The function is shifted three units downward, so each point of f(x)-3 has the same x-coordinate but the output is three units less than the output of f(x). Thus, each point will be sketched as:



For \ y=f(x): \\ P(x_{0},f(x_{0})) \\ \\ For \ y=f(x)-3: \\ P(x_{0},f(x_{0})-3)



FOR ODD FUNCTIONS:


2. When f(x) becomes f(x)-3 


2.1 Effects on the y-intercept 


In this case happens the same as in the previous case. The new y-intercept is three units less. So the graph is shifted three units downward again.


An example is shown in Figure 1. The graph in blue is the function:


y=f(x)=x^3-x


and the function in red is:


y=f(x)-3=x^3-x-3


This function is odd, so you can see that:


y-intercept \ of \ f(x)=0 \\ y-intercept \ of \ f(x)-3=-3


2.2. Effects on the regions where the graph is increasing and decreasing


The effects are the same just as in the previous case. So the new function increases and decreases in the same intervals of f(x)


In Figure 1 you can see that both functions increase and decrease at the same intervals.


2.3 The end behavior when the following changes are made.


It happens the same, the output is three units less than the output of f(x). So, you can write the points just as they were written before. 


So you can realize this concept by taking a point with the same x-coordinate of both graphs in Figure 1.


FOR EVEN FUNCTIONS:


3. When f(x) becomes -2.f(x) 


3.1 Effects on the y-intercept 


As we know the graph f(x) intersects the y-axis when x=0, therefore:


y=f(0) \ is \ the \ y-intercept \ again


And:


y=-2f(0) \ is \ the \ new \ y-intercept


So the new y-intercept is the negative of the previous intercept multiplied by 2.


3.2. Effects on the regions where the graph is increasing and decreasing


In the intervals when the function f(x) increases, the function -2f(x) decreases. On the other hand, in the intervals when the function f(x) decreases, the function -2f(x) increases. 


3.3 The end behavior when the following changes are made.


Each point of the function -2f(x) has the same x-coordinate just as the function f(x) and the y-coordinate is the negative of the previous coordinate multiplied by 2, that is:


For \ y=f(x): \\ P(x_{0},f(x_{0})) \\ \\ For \ y=-2f(x): \\ P(x_{0},-2f(x_{0}))



FOR ODD FUNCTIONS:


4. When f(x) becomes -2f(x) 


See example in Figure 2


y=f(x)=x^3-x


and the function in red is:


y=-2f(x)=-2(x^3-x)


4.1 Effects on the y-intercept 


In this case happens the same as in the previous case. The new y-intercept is the negative of the previous intercept multiplied by 2.


4.2. Effects on the regions where the graph is increasing and decreasing


In this case it happens the same. So in the intervals when the function f(x) increases, the function -2f(x) decreases. On the other hand, in the intervals when the function f(x) decreases, the function -2f(x) increases. 


4.3 The end behavior when the following changes are made.


Similarly, each point of the function -2f(x) has the same x-coordinate just as the function f(x) and the y-coordinate is the negative of the previous coordinate multiplied by 2.


You might be interested in
The weights of cars passing over a bridge have a mean of 3,550 pounds and standard deviation of 870 pounds. Assume that the weig
Annette [7]

Answer:

0.24315

Step-by-step explanation:

Using the z score formula to solve this question

z = (x - μ) / σ,

Such that:

x = raw score

μ = population mean

σ = population standard deviation.

From the question:

x = 3000

μ = 3550

σ = 870

z = (3000 - 3550) / 870

z = -550/870

z = -0.6962

Using the z score table as well as probability calculator(as requested in the question to find the z score)

The probability of having less than 3000 is obtained as:

P(x<3000) = 0.24315

7 0
3 years ago
Mabel spends 444 hours to edit a 333 minute long video. She edits at a constant rat\\\\\\\\\\\\\\e. How long does Mabel spend to
mestny [16]
Mabel spends 4 hours editing a 3 minute video. she edits at a constant rate. how long does mabel spend to edit a 9 minute video?

she spends 4 hours on a 3 minute video
so there are 60 minutes in an hour
4 hours with 60 minutes- 4x60=240
so she spent 240 minutes on a 3 minute video
3 divided by 240
3/240= 0.0125
so her constant rate is 0.0125 minutes an hour
so divide 9 minutes by her constant rate
9/0.0125 = 720
so she spends 720 minutes on a 9 minute video.
divide 720 by 60 since there are 60 minutes in an hour
720/60 = 12
so she spends 12 hours on a 9minute video
7 0
3 years ago
Taylor is two years younger than Bryce. The sum of tier ages is 28. How old is Taylor?
Tema [17]
16 is your answer have a great day
5 0
3 years ago
Read 2 more answers
What is 6(-5n + 7) simplified?
MrRa [10]

Answer:

-30n+42

Step-by-step explanation:

Hope fully this is helpful

3 0
3 years ago
Read 2 more answers
I need help with 38,39 and 40 please!
kodGreya [7K]

Answer:

It's 37. B 38. B 39. C 40. A

Step-by-step explanation:

4 0
3 years ago
Read 2 more answers
Other questions:
  • What is the multiplicative inverse of the sum of 1/3 + 1/5 ? Express in simplest form
    11·1 answer
  • Ben's making bow ties. How many 1/2 yard-long bow ties can he make if he has 18 feet of fabric
    13·1 answer
  • I've been doing this for the past 2 hours can someone please help
    7·1 answer
  • John has a bag of marbles that contains 12 red marble,20 green marbles,and 17 blue marbles. If john pulls one marble out of the
    14·1 answer
  • Solve. −35x+15&gt;720 Drag and drop a number or symbol into each box to show the solution.
    15·1 answer
  • What would be the answer for this because I keep getting a decimal and don't know if it's right? "Thomas has $58.00 in quarters
    7·1 answer
  • Frederick is working on a number puzzle and discovers that the product of the two base numbers is exactly twice as large as the
    15·2 answers
  • Abe, Bob, Carl, Dan and Earl are going to stand in line for a photo. What is the probability that the boys will stand in alphabe
    6·1 answer
  • X/15=y/20=z/18 và 2x+3y-2
    9·1 answer
  • A box contains 14 large marbles and 10 small marbles. Each marble is either green or white 8 of the large marbles are green, and
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!