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
It is known that 10% of the calculators shipped from a particular factory are defective. In a random sample of ten calculators,
umka2103 [35]

Answer:

0.7361

Step-by-step explanation:

In this question we have

number to be 10

Then we have a probability of 10% = 0.10

We have q = 1-p

= 1-0.10 = 0.90

Then the probability of not more than 1 being defective:

P(x=0) + p(x= 1)

(10C0 x 0.1⁰ x 0.9^10-0)+(10C1 x 0.1¹ x 0.9^10-1)

= 1 x1 x0.3487 + 10 x 0.1 x 0.3874

= 0.3487 + 0.3874

= 0.7361

This is the the required probability and this answers the question.

probability = 10 percent = 0.1

q= 1- 10percent = 90% = 0.9

n = 4

To get the required probabiltiy for this question is

P(not greater than one is defective )=P(x=0)+P(x=1)

= 4C0x(0.1)⁰x(0.9)⁴+4C1x(0.1)¹x(0.9)³

= 0.9477

The required probability is 0.9477

5 0
3 years ago
ΔXYZ ≅ΔFED where m∠X = 50°, m∠Y=30°, m∠D=2y+10, XY =9 and EF=3x−12. Solve for x and y.
Delicious77 [7]

Answer:

The value of x is 7 and value of y is 45

Step-by-step explanation:

In ΔXYZ

∠X = 50°, ∠Y=30°

To Find ∠Z we will use angle sum property of triangle (Sum of all angles of triangles is 180°)

So,∠X + ∠Y + ∠Z = 180°

So,50°+30°+ ∠Z = 180°

∠Z=180°-80°

∠Z=100°

Now we are given that ΔXYZ ≅ΔFED

So, the corresponding sides are equal and the corresponding angles are equal.

So, ∠Z= ∠D

So, 100=2y+10

100-10=2y

90=2y

45=y

Now XY = EF

So, 9 = 3x-12

21=3x

7=x

Hence The value of x is 7 and value of y is 45

5 0
3 years ago
Please help me with this,I will give you brainliest (only if you anwser first)
aleksklad [387]
79.56 would be the answer, (i’m pretty positive) because you take all three numbers and you multiply them together.
3 0
3 years ago
A 9-kg bag of mangoes for $12
Blababa [14]
A 9-kg bag of mangoes for $12
5 0
3 years ago
Liam bought 5/8 pound of cherries.Harrison bought more cherries than Liam .Which could be the amount of cherries that Harrison b
Yanka [14]

Answer: Harrison bought any answer higher than 5/8 pound or 62.5% of a pound. He has to have bought more than these two numbers. (they are equal one is a fraction one is a percent)

Step-by-step explanation:

5/8= 62.5%

5 0
3 years ago
Other questions:
  • Give another point on the curve where the slope of the line is approximately equal to the slope of your tangent line in
    12·1 answer
  • Tim worker went to his bank. He deposited $72.15 and the teller credited $5.79 to his account for interest. If Tim's initial bal
    13·2 answers
  • Can y’all help me on This num 20
    13·1 answer
  • A set of average city temperatures in July are normally distributed
    8·1 answer
  • What is log2x=22 in exponential form
    13·1 answer
  • Find the term that must be added to the equation x2−10x=1 to make it into a perfect square
    10·1 answer
  • An item is regularly priced at $85. it is on sale for 60% off the regular price. How much ( in dollars) is discounted from the r
    12·1 answer
  • What is the value of cos (L)?​
    6·1 answer
  • An iron ball has a density of 4 g/cm3.
    14·1 answer
  • A glacier in Alaska moves about 29.2 meters a day.
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!