Intervals Where F(x) = -x^2 + 3x + 8 Is Increasing

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Intervals Where f(x) = -x^2 + 3x + 8 is Increasing

Hey guys! Today, we're diving into a classic calculus problem: finding the interval where the graph of the quadratic function f(x) = -x^2 + 3x + 8 is increasing. This is a fundamental concept in understanding the behavior of functions, and we'll break it down step-by-step to make sure you've got it. So, let's jump right in!

Understanding Increasing Functions

Before we tackle the specific function, let's make sure we're all on the same page about what it means for a function to be increasing. Simply put, a function is increasing over an interval if its y-values go up as its x-values go up. Think of it like climbing a hill – as you move to the right (increasing x), you're also going upwards (increasing y). Mathematically, we can say that a function f(x) is increasing on an interval (a, b) if for any two points x₁ and x₂ in the interval where x₁ < x₂, then f(x₁) < f(x₂). This means that if we pick any two x-values in our interval, the function's value will be greater at the larger x-value.

Key Concepts for Identifying Increasing Intervals

  • The Derivative: The derivative of a function, denoted as f'(x), gives us the slope of the tangent line at any point on the function's graph. This is our main tool for determining where a function is increasing or decreasing. If f'(x) > 0 on an interval, the function is increasing on that interval because the slope of the tangent line is positive, indicating an upward trend. Conversely, if f'(x) < 0, the function is decreasing.
  • Critical Points: Critical points are the points where the derivative is either zero or undefined. These points are crucial because they mark potential turning points where the function changes from increasing to decreasing or vice versa. To find critical points, we set f'(x) = 0 and solve for x, or we identify points where f'(x) is undefined.
  • Interval Testing: Once we find the critical points, we use them to divide the number line into intervals. We then pick a test value within each interval and evaluate the sign of f'(x) at that test value. This will tell us whether the function is increasing or decreasing in that entire interval. If f'(x) is positive, the function is increasing; if f'(x) is negative, the function is decreasing.

Analyzing the Function f(x) = -x² + 3x + 8

Now, let's apply these concepts to our specific function, f(x) = -x² + 3x + 8. This is a quadratic function, and its graph is a parabola. Because the coefficient of the x² term is negative (-1), the parabola opens downwards, meaning it has a maximum point. Our goal is to find the interval to the left of this maximum point where the function is increasing.

1. Find the Derivative

The first step is to find the derivative of f(x). We'll use the power rule, which states that if f(x) = axⁿ, then f'(x) = naxⁿ⁻¹. Applying this rule to each term in our function:

  • The derivative of -x² is -2x.
  • The derivative of 3x is 3.
  • The derivative of 8 (a constant) is 0.

So, the derivative of f(x) is: f'(x) = -2x + 3

2. Find the Critical Points

Next, we need to find the critical points. These are the points where the derivative is either zero or undefined. In this case, f'(x) = -2x + 3 is a linear function, so it's defined for all real numbers. Therefore, we only need to find where f'(x) = 0:

-2x + 3 = 0

Add 2x to both sides:

3 = 2x

Divide both sides by 2:

x = 1.5

So, our only critical point is x = 1.5. This point corresponds to the vertex of the parabola, which is the maximum point of the function.

3. Determine the Intervals of Increase

Now that we have our critical point, we can divide the number line into two intervals: (-∞, 1.5) and (1.5, ∞). We'll test a value in each interval to determine the sign of f'(x) and therefore whether the function is increasing or decreasing.

  • Interval (-∞, 1.5): Let's choose a test value of x = 0. Plug this into the derivative:
    • f'(0) = -2(0) + 3 = 3
    • Since f'(0) = 3 is positive, the function is increasing on the interval (-∞, 1.5).
  • Interval (1.5, ∞): Let's choose a test value of x = 2. Plug this into the derivative:
    • f'(2) = -2(2) + 3 = -1
    • Since f'(2) = -1 is negative, the function is decreasing on the interval (1.5, ∞).

4. Conclusion

From our analysis, we've found that the function f(x) = -x² + 3x + 8 is increasing on the interval (-∞, 1.5). This makes sense because, as a downward-opening parabola, the function increases until it reaches its vertex at x = 1.5, and then it decreases.

Why This Matters

Understanding where a function is increasing or decreasing is crucial in many areas of mathematics and its applications. Here are a few reasons why:

  • Optimization: In optimization problems, we often want to find the maximum or minimum value of a function. Knowing where a function is increasing or decreasing helps us identify potential maximum and minimum points.
  • Graphing: Determining intervals of increase and decrease is essential for accurately sketching the graph of a function. It gives us a clear picture of the function's behavior.
  • Real-World Applications: Many real-world phenomena can be modeled using functions, and understanding their increasing and decreasing behavior can provide valuable insights. For example, in economics, we might analyze the increasing or decreasing nature of a profit function.

Final Thoughts

So, there you have it! We've successfully determined the interval where the graph of f(x) = -x² + 3x + 8 is increasing. By finding the derivative, identifying critical points, and using interval testing, we were able to pinpoint the interval (-∞, 1.5). Remember, guys, this process can be applied to a wide variety of functions, so mastering these steps is a valuable skill in calculus and beyond. Keep practicing, and you'll be a pro in no time! If you have any questions, feel free to ask. Happy studying!