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 −one half ⋅ f(x)

Answer 1

Answer:

We are have the function f(x).

Part 1: It is given that the function f(x) is transformed to f(x)+2.

This gives us that the function is translated 2 units upwards.

As, the y-intercept of f(x) is f(0). So, the y-intercept of f(x)+2 is f(0)+2, which is also translated 2 units up.

Translation does not change the behavior of the function.

So, if f(x) is increasing (or deceasing), then f(x)+2 will be increasing(or decreasing) respectively.

Moreover, f(x)+2 will be even or odd if f(x) is even or odd respectively.

Part 2: It is given that the function f(x) is transformed to $\frac{-1}{2}f(x)$

This gives us that the function is shrinked vertically by a factor of $\frac{1}{2}$ followed by a reflection across x-axis.

Also, the y-intercept of is $\frac{-1}{2}f(0)$.

The graph of f(x) after changing to $\frac{-1}{2}f(x)$ will flip over the x-axis.

So, the function $\frac{-1}{2}f(x)$ will increase or decrease when f(x) decrease or increase respectively.

Moreover, $\frac{-1}{2}f(x)$ will be even if f(x) is even or odd respectively.