Find the sum of a polynomials -4xsquare 9x-3 and 7xsquare-5x 4

So for every batch, you need 2 cups of flour, and you have 20 cups of flour. You cannot make a batch if you don't have the sufficient amount. So using our information, our inequality is $2x \leq 20$ . However, this can be further simplified.

For this inequality, divide both sides by 2, and <u>your inequality is $x\leq 10$ .</u> <u>In context, you can make up to 10 batches of cookies.</u>

8 0

3 years ago

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

Mashcka [7]

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.

6 0

3 years ago