Finding Positive And Negative Intervals Of Quadratic Functions

Hey guys! Let's dive into the fascinating world of quadratic functions and learn how to figure out where they're positive or negative. This is super useful for understanding the behavior of these U-shaped (or upside-down U-shaped) curves. We'll go through some examples together, and by the end, you'll be able to tackle these problems like a pro. This guide will help you understand quadratic functions, positive and negative intervals, and how to find them. These concepts are fundamental in algebra and have applications in various fields, from physics to economics. So, let's get started!

Understanding Quadratic Functions

First things first, what exactly is a quadratic function? Simply put, it's a function that can be written in the form of y = ax^2 + bx + c, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The graph of a quadratic function is a parabola – a U-shaped curve that opens upwards if 'a' is positive and downwards if 'a' is negative. The key to understanding the positive and negative intervals lies in the parabola's shape and where it intersects the x-axis. The points where the parabola crosses the x-axis are called the x-intercepts or zeros of the function. These are the critical points that divide the x-axis into intervals where the function is either positive (above the x-axis) or negative (below the x-axis). When the parabola opens upwards, it is negative between the x-intercepts and positive everywhere else. Conversely, when the parabola opens downwards, it is positive between the x-intercepts and negative everywhere else. So, finding the x-intercepts is our first step in determining the positive and negative intervals.

Now, let's look at how to find these intervals for the problems you've given us. We'll use the principles of finding x-intercepts and analyzing the parabola's direction (up or down). Remember, the sign of the leading coefficient (the 'a' value) tells us whether the parabola opens upwards (positive 'a') or downwards (negative 'a'). This is crucial because it directly impacts where the function is positive or negative. Are you ready? Let's get started with the first problem, where we'll go through all of the steps together. This process will help you gain a better understanding of quadratic functions, which are fundamental in mathematics and have numerous real-world applications. By mastering these concepts, you'll be well-equipped to tackle more complex problems and appreciate the beauty of mathematical analysis.

Solving for Positive and Negative Intervals

Alright, let's get down to business and solve these problems step by step. We'll focus on how to determine where each quadratic function is positive or negative. Don't worry, it's not as scary as it sounds! We'll start by finding the x-intercepts, and then we'll analyze the behavior of the parabola to determine the intervals.

Let's start with problem 13: y = 2x^2 - 17x + 30. Here's the deal: The first step is to find the x-intercepts by setting y = 0. So, we have the equation: 2x^2 - 17x + 30 = 0. We can solve this by factoring, using the quadratic formula, or completing the square. Factoring is usually the easiest if it works. In this case, the equation can be factored as (2x - 5)(x - 6) = 0. Now, setting each factor to zero gives us the x-intercepts: 2x - 5 = 0 which leads to x = 5/2 or x = 2.5, and x - 6 = 0 which gives us x = 6. Now we know our x-intercepts. Next, we determine whether the parabola opens upwards or downwards. Since the coefficient of x^2 is positive (2), the parabola opens upwards. This means that the function will be positive for x-values less than 2.5 and greater than 6. The function is negative between 2.5 and 6. Therefore, the function y = 2x^2 - 17x + 30 is positive on the intervals (-∞, 2.5) and (6, ∞), and negative on the interval (2.5, 6). Great job, guys! Now we are going to dive into the next question.

Let's continue to the next problem, which is problem 14: y = -7x^2 + 35x - 28. To find the x-intercepts, we set y = 0: -7x^2 + 35x - 28 = 0. We can simplify this by dividing by -7, which gives us x^2 - 5x + 4 = 0. Factoring this, we get (x - 1)(x - 4) = 0. This gives us the x-intercepts: x = 1 and x = 4. Because the coefficient of the x^2 term is negative (-7), the parabola opens downwards. This means the function is positive between the x-intercepts (1 and 4) and negative everywhere else. So, the function is positive on the interval (1, 4) and negative on the intervals (-∞, 1) and (4, ∞). Keep up the amazing work.

Moving on to problem 15: y = -x^2 - 6x - 8. Setting y = 0, we get -x^2 - 6x - 8 = 0. Multiplying by -1, we have x^2 + 6x + 8 = 0. This factors to (x + 2)(x + 4) = 0. This yields the x-intercepts x = -2 and x = -4. Since the coefficient of x^2 is negative (-1), the parabola opens downwards. The function is positive between -4 and -2, and negative elsewhere. We're looking for where it's negative, so the intervals are (-∞, -4) and (-2, ∞). Well done! You are doing great.

Finally, let's tackle problem 16: y = 2x^2 - 4x - 16. Setting y = 0, we get 2x^2 - 4x - 16 = 0. Dividing by 2, we have x^2 - 2x - 8 = 0. Factoring, we get (x - 4)(x + 2) = 0. This gives us x-intercepts x = 4 and x = -2. Since the coefficient of x^2 is positive (2), the parabola opens upwards. We want to know where it's negative. This happens between the x-intercepts. So, the function is negative on the interval (-2, 4). You've made it to the end.

Conclusion and Practice

Congratulations, guys! You've successfully navigated the process of finding the positive and negative intervals of several quadratic functions. By understanding the relationship between the x-intercepts and the direction of the parabola, you can easily determine these intervals. Remember to always find the x-intercepts first, and then consider whether the parabola opens upwards or downwards. This will tell you whether the function is positive or negative between the intercepts and outside of them. Practicing with different examples is key to mastering these concepts. Try solving more problems on your own, and don’t hesitate to review the steps we've covered today. Keep up the amazing work, and you'll become a pro in no time!

This is a fundamental concept in algebra, and it forms the basis for understanding more advanced mathematical concepts. Being able to determine the positive and negative intervals of a function is crucial for graphing and analyzing the function's behavior. Keep practicing these problems. You're doing great! Keep up the good work!

Additional Example: Problem 17 (Conceptual)

Let's shift gears a bit and discuss the context provided in problem 17. It says, “A rock is thrown upward.” This isn't a typical quadratic function problem where we just find intervals. Instead, this describes a physical scenario modeled by a quadratic function, representing the rock's height over time. The key concept here is that the rock's path is a parabola. The function will be positive (the rock is above the ground) during the time the rock is in the air. The function will be zero at the start and end of the rock's trajectory (when it's at ground level). We'd need more information, like the initial velocity, to determine the exact equation and solve for the time intervals. However, the conceptual understanding is the crucial takeaway. Understanding that the rock's trajectory is a parabola, and its height can be described by a quadratic equation, is important. We can use the information given, and these core concepts, to describe the path and calculate various parameters such as the maximum height reached, or the total time the rock is in the air. The positive interval would be the time during which the rock is in the air (above the ground). The negative interval isn’t really applicable here, as the rock’s height cannot be negative (unless we consider the ground as the zero point). This example shows how quadratic functions and the concept of positive and negative intervals can be applied in real-world scenarios, bridging the gap between abstract mathematical concepts and tangible physical events. You are on the right track!