Increasing & Decreasing Intervals: Y=x²-4x, Y=-x²-2x+1

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Hey guys! Today, we're diving into how to find the intervals where functions are increasing or decreasing. This is a super useful skill in calculus and helps you understand the behavior of different functions. We'll be looking at two specific examples: y = x² - 4x and y = -x² - 2x + 1. Let's break it down step-by-step!

1. Understanding Increasing and Decreasing Intervals

Before we jump into the examples, let's quickly recap what increasing and decreasing intervals are. A function is said to be increasing on an interval if, as you move from left to right along the x-axis, the y-values are getting larger. Conversely, a function is decreasing on an interval if, as you move from left to right, the y-values are getting smaller. You can visualize this by imagining walking along the graph of the function. If you're walking uphill, the function is increasing; if you're walking downhill, it's decreasing.

The key to finding these intervals lies in the function's derivative. The derivative, denoted as f'(x) or dy/dx, tells us the slope of the tangent line at any point on the function. If the derivative is positive (f'(x) > 0), the function is increasing. If the derivative is negative (f'(x) < 0), the function is decreasing. If the derivative is zero (f'(x) = 0), we have a critical point, which could be a local maximum, a local minimum, or a point of inflection. These critical points are crucial because they often mark the boundaries between increasing and decreasing intervals.

To find the increasing and decreasing intervals, we typically follow these steps:

  1. Find the derivative of the function. This gives us an expression for the slope of the function at any point.
  2. Find the critical points by setting the derivative equal to zero and solving for x. These are the points where the function's slope is flat.
  3. Create a number line and mark the critical points on it. This divides the number line into intervals.
  4. Choose a test value within each interval and plug it into the derivative. If the derivative is positive, the function is increasing on that interval. If the derivative is negative, the function is decreasing on that interval.
  5. Write the increasing and decreasing intervals based on the signs of the derivative.

This method allows us to systematically determine where a function is going up or down, giving us a better understanding of its behavior. Now, let's apply this to our specific examples.

2. Example 1: y = x² - 4x

Let's start with the function y = x² - 4x. Our goal is to find the intervals where this function is increasing and decreasing. Follow along, and you'll see how straightforward it can be!

Step 1: Find the Derivative

The first step is to find the derivative of the function. Using the power rule, which states that the derivative of x^n is n*x^(n-1), we can find the derivative of y = x² - 4x:

dy/dx = 2x - 4

So, the derivative of our function is 2x - 4. This expression tells us the slope of the tangent line at any point x on the original function.

Step 2: Find the Critical Points

Next, we need to find the critical points. These are the points where the derivative is equal to zero. Set the derivative equal to zero and solve for x:

2x - 4 = 0 2x = 4 x = 2

We found one critical point: x = 2. This is where the function's slope is zero, indicating a potential turning point.

Step 3: Create a Number Line

Now, we'll create a number line and mark our critical point x = 2 on it. This divides the number line into two intervals: (-∞, 2) and (2, ∞). These intervals represent the regions where the function could be either increasing or decreasing.

Step 4: Choose Test Values

We need to determine the sign of the derivative in each interval. To do this, we'll choose a test value from each interval and plug it into the derivative 2x - 4.

  • Interval (-∞, 2): Choose x = 0 as a test value. 2(0) - 4 = -4. The derivative is negative in this interval.
  • Interval (2, ∞): Choose x = 3 as a test value. 2(3) - 4 = 2. The derivative is positive in this interval.

Step 5: Write the Intervals

Based on the signs of the derivative, we can now determine the increasing and decreasing intervals:

  • Decreasing Interval: (-∞, 2) because the derivative is negative.
  • Increasing Interval: (2, ∞) because the derivative is positive.

So, for the function y = x² - 4x, it is decreasing from negative infinity to 2 and increasing from 2 to infinity. The critical point x = 2 corresponds to a local minimum of the function.

3. Example 2: y = -x² - 2x + 1

Now, let's tackle the second function: y = -x² - 2x + 1. We'll follow the same steps as before to find its increasing and decreasing intervals.

Step 1: Find the Derivative

First, we find the derivative of y = -x² - 2x + 1. Again, using the power rule:

dy/dx = -2x - 2

So, the derivative of this function is -2x - 2.

Step 2: Find the Critical Points

Next, we find the critical points by setting the derivative equal to zero:

-2x - 2 = 0 -2x = 2 x = -1

We have one critical point: x = -1.

Step 3: Create a Number Line

We create a number line and mark the critical point x = -1 on it. This divides the number line into two intervals: (-∞, -1) and (-1, ∞).

Step 4: Choose Test Values

We choose a test value from each interval and plug it into the derivative -2x - 2.

  • Interval (-∞, -1): Choose x = -2 as a test value. -2(-2) - 2 = 4 - 2 = 2. The derivative is positive in this interval.
  • Interval (-1, ∞): Choose x = 0 as a test value. -2(0) - 2 = -2. The derivative is negative in this interval.

Step 5: Write the Intervals

Based on the signs of the derivative, we determine the increasing and decreasing intervals:

  • Increasing Interval: (-∞, -1) because the derivative is positive.
  • Decreasing Interval: (-1, ∞) because the derivative is negative.

Thus, for the function y = -x² - 2x + 1, it is increasing from negative infinity to -1 and decreasing from -1 to infinity. The critical point x = -1 corresponds to a local maximum of the function.

4. Conclusion

Alright, guys, we've successfully found the increasing and decreasing intervals for both functions! Remember, the key steps are finding the derivative, finding the critical points, creating a number line, and testing values in each interval. This method works for a wide range of functions and is a fundamental tool in calculus.

Understanding where a function is increasing or decreasing helps you sketch its graph and understand its behavior. Keep practicing, and you'll become a pro at this in no time!

If you have any questions, feel free to ask. Happy calculating!