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) − 3
When f(x) becomes −2 ⋅ f(x)

Answer 1

When f(x) becomes f(x) - 3, the y-intercept shifts down by 3 units. When f(x) becomes -2 * f(x), the curve is reflected about the x-axis. The changes to the y-intercept, increasing and decreasing regions, and end behavior depend on whether the function is even or odd.

Step-by-step explanation:

When a polynomial function f(x) is changed to f(x) - 3, the y-intercept shifts down by 3 units. The regions where the graph is increasing and decreasing remain the same, but the entire curve is shifted down vertically. The end behavior of the function remains the same as the original function.

When a polynomial function f(x) is changed to -2 * f(x), the y-intercept remains the same, but the curve is reflected about the x-axis. The regions where the graph is increasing and decreasing and the end behavior remain the same as the original function.

For even functions, the changes in y-intercept, increasing and decreasing regions, and end behavior are the same as described above. For odd functions, the changes in y-intercept and increasing and decreasing regions are the same as described above, but the end behavior is reflected about the x-axis.