Respuesta :
Answer:
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)
Answer with Explanation
Let the polynomial function which is an odd function, be
[tex]f(x)=x^5+x^3+x[/tex]
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 [tex]\frac{-f(x)}{2}[/tex]
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 [tex]\frac{1}{2}[/tex].
Here also, Y intercept =0
Function is a decreasing function in it's domain [-∞, ∞]
Now, taking the even function
[tex]f(x)=x^6+x^4+x^2[/tex]
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 [tex]\frac{-f(x)}{2}[/tex]
The function will pass through third and fourth Quadrant,due to negative sign before it, and function expands by the value of [tex]\frac{1}{2}[/tex] on both sides of Y axis.
Here also, Y intercept =0
Function is increasing in [-∞,0) and decreasing in (0, ∞].
