Question

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) + 2
-When f(x) becomes −(1 / 2) * f(x)

Answer 1

Answer with Explanation

Let the polynomial function which is an odd function, be

`f(x)=x^5+x^3+x`

f(-x)= - f(x)

So,it is an odd function.

The function will pass through first and fourth quadrant.

Y intercept =0

Function is increasing in it's domain [-∞, ∞]

1. When f(x) becomes f(x)+2

g(x)=f(x) +2

This function will shift 2 units up, and it will not pass through the origin and has Y intercept equal to 2.

This function will also pass through first and fourth quadrant.

Function is an increasing function in it's domain [-∞, ∞]

Now, when f(x) becomes
`(-f(x))/(2)`

The function will pass through second and fourth Quadrant,due to negative sign before it, and distance from y axis either in second Quadrant or in fourth Quadrant increases by a value of
`(1)/(2)`.

Here also, Y intercept =0

Function is a decreasing function in it's domain [-∞, ∞]

Now, taking the even function

`f(x)=x^6+x^4+x^2`

f(-x)= f(x)

So,it is an even function.

The function will pass through first and second quadrant equally spaced on both side of y axis.

Y intercept =0

Function is decreasing in [-∞,0) and increasing in (0, ∞]

1. When f(x) becomes f(x)+2

g(x)=f(x) +2

This function will shift 2 units up, and it will not pass through the origin and has Y intercept equal to 2.

This function will also pass through first and third quadrant.

Function is decreasing from [-∞,2) and increasing in (2, ∞]

Now, when f(x) becomes
`(-f(x))/(2)`

The function will pass through third and fourth Quadrant,due to negative sign before it, and function expands by the value of
`(1)/(2)` on both sides of Y axis.

Here also, Y intercept =0

Function is increasing in [-∞,0) and decreasing in (0, ∞].

Answer 2

f(x) + 2 is translated 2 units up and -(1/2)*f(x) is reflected across x-axis.

Step-by-step explanation:

We have f(x) becomes f(x) + 2.

The y-intercept of f(x) is f(0), implies that y-intercept of f(x) + 2 is f(0) + 2. This means that the graph of f(x) is translated 2 units upwards.

Moreover, the region where f(x) increases will be the same region region where f(x) + 2 increases and there will not any change in the size of the figure.

Now, we have f(x) becomes -(1/2)*f(x).

The y-intercept of -(1/2)*f(x) is -(1/2)*f(0). This means that the graph is dilated by 1/2 units and then reflected across x-axis.

Moreover, the region where f(x) increases will be the opposite region region where -(1/2)*f(x) increases and the size of the figure will change as dilation of 1/2 is applied to f(x)