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Damm [24]
3 years ago
9

HELP PLS Given a polynomial function f(x), describe the effects on the Y-intercept, regions where the graph is increasing and de

creasing, and the end behavoir 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 X f(x)
Mathematics
1 answer:
Mashcka [7]3 years ago
6 0
1. Remarks:

f(x) to f(x)-3 is the whole graph of f(x), shifted 3 units down. 

f(x) to -2f(x): 

The effect of "multiplication by -" is that the whole graph is reflected with respect to the x axis, so it is turned upside down.
 
The effect of "multiplication by 2" is that every point is "stretched vertically by a factor of 2" . So for example the point (-1, -4) in the original function, becomes (-1, -8) in the second one. Or (2, 5) would become (2, 10). 

The only points that do not change (are not streched vertically) are the roots. For example if (4,0) is an x-intercept (a root) in the original function, (4,0) is still a root in the second one because  2 times 0 is still 0.


2. Consider the polynomial function of degree n: 

f(x)= a_{n} x^{n} +a_{n-1} x^{n-1}+....+a_{2} x^{2}+a_{1} x^{1}+a_{0}

a. Y-intercept

The y - intercept is the value of the polynomial function at x=0. 
So it is f(0)=a_{0}, that is, the constant term of f(x)

in f(x)-3 the y intercept is shifted 3 units down as any other point, so it becomes  a_{0}-3

In -2f(x), the y-intercept a_{0} becomes -2a_{0}

b. Regions of f decreasing or increasing

f(x)-3 is f(x) just shifted down 3 units, so they are both increasing and decreasing in the same intervals of x

-2f(x) is f(x) turned upside down, so -2f(x) is increasing in all intervals f(x) is decreasing and it is decreasing in all intervals f(x) is increasing.

c. End behaviors

By now it is clear that end behaviors of f(x) and f(x)-3 are same, and f(x) with -2f(x) are opposite

d. Evenness, oddness

If f(x) is even, then f(x)=f(-x)

Let g(x)=f(x)-3

g(x)=f(x)-3=f(-x)-3=g(-3), so in this case f(x)-3 is even

If f(x) is odd, then f(-x)=-f(x)

g(x)=f(x)-3=-f(-x)-3,

so -g(x)=f(-x)+3

g(-x)=f(-x)-3,  

so g(-x) is not equal to -g(x). Which means f(x)-3 is not odd if f(x) is


Consider f(x)=-2f(x)

If f(x) is even, f(x)=f(-x)

g(x)=-2f(x)=-2f(-x)
g(-x)=-2f(-x)

So g(x)=g(-x), which means -2f(x) is even if f(x) is even

If f(x) is odd, f(x)=-f(-x)

let g(x)=-2f(x)=-2(-f(-x))=2f(-x)

g(-x)=-2f(-x)=-2(-f(x))=2f(x)

so g(-x) is not equal to -g(x), thus -2f(x) is not odd if f(x) is odd.

The conclusions about oddness and evenness can be also derived from the discussions about the graphs.
 

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Question 11
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Which of the following is NOT a variation of a Pythagorean identity?​
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The expression that is not a variation of the Pythagorean identity is the third option.

<h3>What is the Pythagorean identity?</h3>

The Pythagorean identity can be written as:

sin^2(x) + cos^2(x) = 1

For example, if we subtract cos^2(x) on both sides we get the second option:

sin^2(x) = 1 - cos^2(x)

Which is a variation.

Now, let's divide both sides by cos^2(x).

sin^2(x)/cos^2(x) + cos^2(x)/cos^2(x) = 1/cos^2(x)\\\\tan^2(x) + 1 = sec^2(x)\\\\tan^2(x) - sec^2(x) = -1

Notice that the third expression in the options looks like this one, but the one on the right side is positive. The above expression is in did a variation of the Pythagorean identity, then the one written in the options (with the 1 instead of the -1) is incorrect, meaning that it is not a variation of the Pythagorean identity.

Concluding, the correct option is the third one.

If you want to learn more about the Pythagorean identity, you can read:

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Which answer is a solution of the equation?
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2x + y = 5

Put the coordinates of the points to the equation and check it:

A. 5 - not make sense

B. (1, 2) - x = 1, y = 2

2(1) + 2 = 2 + 2 = 4 ≠ 5  NOT

C. (2, -1) - x = 2, y = -1

2(2) + (-1) = 4 - 1 = 2 ≠ 5    NOT

D. (3, -1) - x = 3, y = -1

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<h3>Answer: D. (3, -1).</h3>
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