Problem 89 - College Algebra with Modeling and Visualization

Question

If the graph of \(y=f(x)\) is increasing on \([1,4],\) then where is the graph of \(y=f(x+1)-2\) increasing? Where is the graph of \(y=-f(x-2)\) decreasing?

Step-by-Step Solution

Answer

The graph \(y = f(x+1) - 2\) is increasing on \([0, 3]\), and \(y = -f(x-2)\) is decreasing on \([3, 6]\).

Step 1: Understand the transformation for y=f(x+1)-2

Since the function is \(y = f(x+1) - 2\), it involves a horizontal shift left by 1 unit and a vertical shift down by 2 units. Horizontal shifts affect intervals of increase/decrease, but vertical shifts do not.

Step 2: Apply transformation to interval of increase

The interval \([1,4]\) shifts left by 1 unit: \([1-1, 4-1] = [0, 3]\). Thus, the interval of increase for \(y = f(x+1) - 2\) is \([0, 3]\).

Step 3: Understand the transformation for y=-f(x-2)

The transformation \(y = -f(x-2)\) involves a horizontal shift to the right by 2 units and a reflection across the x-axis. Horizontal shifts affect intervals of increase/decrease but reflection inverts them.

Step 4: Apply transformations to interval and reflect

Initially, apply the right shift by 2 units to \([1,4]\) giving \([3,6]\). The reflection across the x-axis turns increasing into decreasing on the same interval. Hence, \(y = -f(x-2)\) is decreasing on \([3, 6]\).

Key Concepts

Increasing Functions

When we talk about increasing functions, we are looking at where a function's outputs consistently grow as its inputs increase. Imagine you are climbing a hill and with each step up, you are higher than your previous step. This is similar to an increasing function. A function is said to be increasing on an interval if for any two numbers within that interval, the function's value at the larger number is greater than its value at the smaller number.
For example, if a function is increasing on the interval \[1, 4\], it means that as \(x\) moves from 1 to 4, the values of \(y = f(x)\) continue to rise. In function transformations, such as \(y = f(x+1) - 2\), horizontal shifts affect this increasing interval. In our exercise, shifting the original \([1, 4]\) interval to the left by 1 unit, changes it to \([0, 3]\). Therefore, the new interval \([0, 3]\) becomes where the function \(y = f(x+1) - 2\) is increasing.

Decreasing Functions

Decreasing functions are like walking down a hill; as you move forward, the value decreases. A function decreases on an interval if for every two numbers in the interval, the value at the higher number is less than at the lower number.
Consider the transformation \(y = -f(x-2)\). This involves two major changes: a right shift by 2 units, and a reflection. First, the shift means the initial increasing interval \([1, 4]\) moves to \([3, 6]\). Next, the reflection across the x-axis flips the direction from increasing to decreasing. Thus, where the function \(y = -f(x-2)\) decreases, it is over the interval \([3, 6]\). This reflection is like flipping an entire hill upside down; what was going up is now going down.

Interval Shifts

Interval shifts in functions occur when you adjust the input \(x\) by adding or subtracting a number, thereby shifting the graph horizontally. Such transformations can be visualized as moving the whole function left or right on a coordinate plane.
In our case, for the function \(y = f(x+1) - 2\), the graph moves to the left by 1 unit. This manipulation changes the increasing interval from \([1, 4]\) to \([0, 3]\) as explained earlier. Horizontal shifts are independent of vertical shifts; hence the vertical modification \(-2\) doesn’t affect the increasing or decreasing behavior.
Similarly, for \(y = -f(x-2)\), the factor \(x-2\) shifts the graph right by 2 units. Therefore, the interval \([1, 4]\) where the original function was increasing changes to \([3, 6]\), post-shift, before reflecting across the x-axis. This interval shifting is crucial because it tells us where the function's behavior changes in relation to its original state.